The conservation of mass is a fundamental
concept of physics. Within some problem domain, the amount of mass
remains constant; mass is neither created or destroyed. The
mass of any object is simply the
volume
that the object
occupies times the density of the object.
For a fluid (a liquid or a gas) the
density, volume, and shape of the object can all change within the
domain with time and mass can move through the domain.

mdot = (A * pt/sqrt[Tt]) * sqrt(gam/R) * M * [1 + M^2 * (gam-1)/2]^-[(gam+1)/(gam-1)/2]

During a flow process, like pushing a gas through a tube, the
state
of the thermodynamic variables of the gas can change. If the
specific heat capacity
is a constant value, the gas is said
to be calorically perfect and if the specific heat capacity
changes, the gas is said to be calorically imperfect.
At subsonic and low supersonic
Mach numbers, air is calorically perfect.
But under low hypersonic flow conditions
or high total temperature conditions, air is
calorically imperfect.
Derived flow variables, like the
speed of sound and the
isentropic flow relations
are slightly different for a calorically imperfect gas
than the conditions predicted for a calorically perfect gas
because some of the energy of the flow excites the vibrational
modes of the diatomic molecules of nitrogen and oxygen in the air.

The
continuity equation
can be easily integrated to give the mass flow rate equation for a
calorically perfect gas, as shown above. The procedure for
a calorically imperfect gas is more difficult, as described below.

The compressibility effects on mass flow rate have some
unexpected results.
We can increase the mass flow through a tube by
increasing the area, increasing the total
pressure, or decreasing the total temperature.
But the effect of increasing velocity (Mach number) is a little harder to figure out.
If we were to fix the area, total pressure and temperature, and graph
the variation of mass flow rate with Mach number, we would find that a
limiting maximum value occurs at Mach number equal to one.
Using calculus, we can
mathematically determine
the same result: there is a maximum airflow limit that occurs when the Mach
number is equal to one.
The limiting of the mass flow rate is called choking of the
flow.
The value of the mass flow rate at choked conditions is given by:

Mach number equal to one is called a sonic condition
because the velocity is equal to the speed of sound and
we denote the area for the sonic condition by "A*".

If we have a tube with changing area, like the
nozzle
shown on the slide, the maximum mass flow rate through the system
occurs when the flow is choked at the smallest area. This
location is called the throat of the nozzle.
The conservation of mass specifies that the mass flow rate through a nozzle is a constant.
If no heat is added, and there are no pressure losses in the nozzle,
the total pressure and temperature are also constant. By substituting the sonic
conditions into the mass flow rate equation, and doing some algebra,
we can relate the Mach number M at any location in the nozzle to the ratio between
the area A at that location and the area of the throat A*.
The resulting equation is shown in the box at the bottom of the slide.

A / A* = {[(gam+1)/2]^-[(gam+1)/(gam-1)/2]} / M * [1 + M^2 * (gam-1)/2]^[(gam+1)/(gam-1)/2]

For the calorically imperfect gas, we can not derive a simple equation for
area ratio. The procedure to determine the area ratio is shown on this slide:

From the continuity equation:

A / A* = [ a* (rho*) ] / [ a * rho * M]

where a* is the speed of sound at the sonic condition, rho* is the density
at the sonic condition, a is the speed of sound at the exit, and rho is
the density at the exit.
Mathematical models
based on a simple harmonic vibrator have been developed for the calorically
imperfect gas.
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA Report 1135.
To a first order approximation, the equation for the speed of sound for
a calorically imperfect gas is given by:

where gamma is the ratio of specific heats for a perfect gas and theta is
a thermal constant equal to 5500 degrees Rankine.
The relation for the total temperature is
given as:

The equation relating Mach number and total temperature must be solved iteratively
to obtain a value for the total temperature. Having the total temperature ratio,the relations between density and total temperature is:

where rho/rhot is the ratio of the static density to the total density.
This set of equations must be solved iteratively to determine the value for
the area ratio.

Here is a Java program that solves the area ratio equation.

By default, the program Input Variable is the
Mach number
of the flow. Since the area ratio depends only on the Mach number and
ratio of specific heats, the program can calculate the value of the
area ratio and display the results on the right side of the output
variables. You can also have the program solve for the Mach number
that produces a desired value of area ratio.
Using the choice button labeled Input Variable,
select "Area Ratio - A/A*".
Next to the selection, you then type in a value for A/A*.
When you hit the red COMPUTE button,
the output values change. The area ratio is double valued;
for the same area ratio, there is a subsonic
and a supersonic solution. The choice button at the right top selects
the solution that is presented.

If you are an experienced user of this calculator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

Considering a rocket nozzle, we can set the mass flow rate by setting the
area of the throat. And we can set the exit Mach number by setting
the area ratio of the exit to the throat. Using the isentropic
relations, we can determine the pressure and temperature at the
exit of the nozzle. And from the Mach number and temperature we can
determine the exit velocity. If we consider the rocket thrust
equation, we have now determined all the values necessary to
determine the thrust of the rocket. You can explore the operation of
a nozzle with our interactive thrust
simulator and design your own rockets!