As a gas is forced through a tube, the gas molecules are deflected
by the walls of the tube. If the speed of the gas is much less than
the speed of sound of the gas, the
density
of the gas remains
constant. However, as the speed of the flow approaches the
speed of sound
we must consider
compressibility effects
on the gas. The density of the gas varies from
one location to the next.
Considering flow through a tube, as shown in
the figure, if the flow is very gradually compressed (area decreases) and then
gradually expanded (area increases), the flow conditions return to their
original values. We say that such a process is reversible.
From a consideration of the
second law
of thermodynamics,
a reversible flow maintains a constant value of
entropy.
Engineers call this type of flow an isentropic flow;
a combination of the Greek word "iso" (same) and entropy.

Isentropic flows occur when the change in flow variables is small
and gradual, such as the ideal flow through the
nozzle
shown above. A
supersonic flow that
is turned while the flow area increases is also isentropic.
We call this an isentropic
expansion
because of the area increase.
If a supersonic flow is turned
abruptly and the flow area decreases,
shock waves
are generated and the flow is irreversible.
The isentropic relations are no longer
valid and the flow is
governed by the oblique or normal
shock relations.

On this slide we have collected many of the important equations
which describe an isentropic flow. We begin with the definition
of the Mach number since this
parameter appears in many of the isentropic flow equations.
The Mach number M is
the ratio of the speed of the flow v to the speed of sound a.

where R is the gas constant from the
equations of state. If we begin with the
entropy equations for a gas, it can be
shown
that the pressure and density of an isentropic flow are related as follows:

Eq #3:

p / r^gam = constant

We can determine the
value of the constant by defining total conditions to be the
pressure and density when the flow is brought to rest isentropically.
The "t" subscript used in many of these equations stands for "total
conditions". (You probably already have some idea of total conditions
from experience with Bernoulli's equation).

Eq #3:

p / r^gam = constant = pt / rt^gam

Using the equation of state, we can easily
derive
the following relations from equation (3):

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

The starred conditions occur
when the flow is choked and the Mach number is equal to one.
Notice the important role that the Mach number plays in all the
equations on the right side of this slide. If the Mach number of the
flow is determined, all of the other flow relations can be
determined. Similarly, determining any flow relation (pressure ratio
for example) will fix the Mach number and set all the other flow
conditions.

Here is a Java program that solves the equations given on this slide.

You select an input variable by using the choice button labeled Input
Variable. Directly below the selection, you then type in the value
of the selected variable. When you hit the Enter key on your keyboard,
the output values change. Some of the variables (like the area ratio) are double
valued. This means that 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.
The variable "Wcor/A" is the
corrected airflow per unit area
function which can be derived from the
compressible mass flow.
This variable is only a function of the Mach number of the flow. The
Mach angle and
Prandtl-Meyer angle
are also functions of the Mach number.
These additional variables are used in the design of rocket nozzles.

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: