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 and the velocity of the flow increases.
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.
The generation of
sound waves
is an isentropic process. 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The isentropic relations are no longer
valid and the flow is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.

During an isentropic process, the
state
of the thermodynamic variables of a 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 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.

On this page we have collected many of the important equations
which describe an isentropic flow. We present two figures and
two sets of equations; one for the calorically perfect gas and the
other for the calorically imperfect gas.
We begin with the calorically perfect gas and 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.

We now turn to the calorically imperfect gas equations.

Mathematical models
based on a simple harmonic vibrator have been developed for the calorically
imperfect gas.
The details of the analysis were given by Eggers 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. There is a similar relation
for the ratio of the static and total pressure p/pt:

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

You select an input variable by using the choice button labeled Input
Variable. Next to the selection, you then type in the value
of the selected variable. When you hit the red COMPUTE button,
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 high speed
inlets, nozzles and ducts.

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: