An important property of any gas
is its
pressure.
Because understanding what pressure is and how it works is so
fundamental to the understanding of
rocketry,
we are including
several slides on pressure in the Beginner's Guide.
There are two ways to look at pressure: (1) the small scale action
of individual air molecules or (2) the large scale action of a large
number of molecules.
On the the small scale, from the
kinetic theory of gases, a gas is composed
of a large number of molecules that are very small relative to the
distance between molecules. The molecules of a
gas
are in constant, random
motion and frequently collide with each other and with the walls of
any container.
During collisions with the walls, there is a change in velocity and
therefore a change in momentum of the molecules. The change in momentum
produces a force on the walls which is related to the gas pressure.
The pressure of a gas is
a measure of the average linear momentum
of the moving molecules of a gas.
On the large scale, the pressure of a gas is a
state variable,
like the
temperature and the
density.
The change in pressure during any process is
governed by the laws of
thermodynamics.
Although pressure itself is a
scalar quantity,
we can define a
pressure force
to be equal to the pressure (force/area) times the surface
area
in a direction perpendicular to the surface.
If a gas is static and not flowing, the measured pressure is the same in
all directions. But if the gas is moving, the measured pressure depends on
the direction of motion. This leads to the definition of the dynamic pressure.
To understand dynamic pressure, we begin with a one dimensional version of the
conservation of linear momentum
for a fluid.
r * u * du/dx =  dp/dx
where r is the density of the gas,
p is the pressure,
x is the direction of the flow,
and u is the velocity in the x direction.
Performing a little algebra:
dp/dx + r * u * du/dx = 0
For a constant density (incompressible flow) we can take the "r * u" term inside the
differential:
dp/dx + d(.5 * r * u^2)/dx = 0
and then gather all of the terms:
d(p + .5 * r * u^2)/dx = 0
Integrating this differential equation:
ps + .5 * r * u^2 = constant = pt
This equation looks exactly like the incompressible form of
Bernoulli's equation.
Each term in this equation has the dimensions of a pressure (force/area);
ps is the static pressure,
the constant pt is called the total pressure, and
.5 * r * u^2
is called the dynamic pressure because it is a pressure term
associated with the velocity u of the flow.
Dynamic pressure is often assigned the letter q in aerodynamics:
q = .5 * r * u^2
The dynamic pressure is a defined property of a moving flow of gas.
We have performed this simple derivation to determine the form of the
dynamic pressure, but we can use and apply the idea of dynamic pressure in
much more complex flows, like
compressible flows
or viscous flows. In particular, the
aerodynamic forces
acting on an object as it moves through the air are
directly proportional to the dynamic pressure.
The dynamic pressure is therefore used in the definition of the
lift coefficient
and the
drag coefficient.
As we have seen, dynamic pressure appears in
Bernoulli's equation
even though that relationship was originally derived using
energy conservation.
The dynamic pressure depends on both the local value of the density
and the velocity of the flow, or rocket .
The density of the air decreases with altitude in a
complex manner.
The velocity of a rocket during launch is constantly increasing with
altitude. Therefore, the dynamic pressure on a rocket during launch
is initially zero because the velocity is zero. The dynamic pressure
increases because of the increasing velocity to some maximum value,
called the maximum dynamic pressure, or Max Q. Then the dynamic
pressure decreases because of the decreasing density.
The Max Q condition is a design constraint on full scale rockets. You
can investigate the variation of dynamic pressure with altitude and
velocity by using our
atmosphere simulator.
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