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]

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]

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

Isentropic Flow Calculator

Input

Output

Mach

Mach Angle

P-M Ang

p/pt

T/Tt

rho/rhot

q/p

A/A*

Wcor/A

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!