Max Q and Dynamic Pressure

How fast should a rocket accelerate to get to orbit or high altitudes? The answer is always "it depends," but there are obvious trends that are worth considering. This article reviews various acceleration rates and their associated dynamic pressures to identify answers to the question of how fast a rocket should accelerate.
The math behind accelerating to orbit is obvious. You need to accelerate from zero miles per hour on the ground to ~17500 miles per hour (~25500 feet per second) in orbit within about 5 minutes (300 seconds) or more. Therefore:
The average acceleration needs to be at about 2.6 g's over the ascent. But acceleration in the atmosphere leads to increasing velocity. Ultimately, the increased velocity leads to air drag.
The equation representing the magnitude of Air Drag is:
It is possible to rearrange the drag equation to get:
Let then
Q is known as the dynamic pressure. As can be seen it is the result of the air density () and the velocity squared. Because of the velocity being squared, the dynamic pressure can increase significantly for small increases in velocity.
Dynamic Pressure arises as a result of the air being pressurized in front of the rocket as it moves. When Q reaches its maximum value, it is known as "Max Q."

Rocket Ascent

As the rocket accelerates in the atmosphere, its velocity increases and so does the dynamic pressure, but the pressure starts to decrease as altitude is gained and the atmospheric density decreases. However, the dynamic pressure effects can become quite large. Since the force exerted by a pressure is:

Force = Area * Pressure
The larger the area, the larger the force.
Taking the above mentioned 2.6 g's as a acceleration, we can see the following trend in the dynamic pressure:
Around 26 seconds, the dynamic pressure reaches a maximum of about 14 PSI (2016 PSF or 96.5 kPa). This is on top of the normal atmospheric pressure at the given altitude. When that pressure is applied over the frontal area of the vehicle, it can be a substantial force.

In modern practice, rockets usually decrease acceleration through the density & velocity region where the dynamic pressure becomes too great. For a vehicle like the space shuttle, the acceleration profile can be rather complicated. Table 1 [ref 1] and Figure 1 [ref 2], below, show the acceleration profile of Space Shuttle STS 121 during its ascent phase.

Table 1	Figure 1

Using the data from table 1, we can calculate the dynamic pressure as Space Shuttle STS 121 ascended:

In general, the attempt was made to keep the shuttle's dynamic pressure below about 700 lbs/cuft. If you think about it, the shuttle's tank had a diameter of about 8.4 meters ( 27.6 feet ) so the total frontal area of just the tanks was on the order of 86153 square inches. At a dynamic pressure of 700 lb/ft^2, there is a pressure of 4.86 PSI and therefore a force of 86153 sq in * 4.86 PSI = 418703 pounds. This is a very large force. Without slowing down to avoid high dynamic pressures, it is possible for the aerodynamic forces to become so large that structural failure occurs. As Figure 1 above shows, the solution is to throttle back the engines during this period to keep the dynamic pressure below the prescribed amounts.

In order to maintain dynamic pressure within acceptable levels, the engine must be throttled back a significant amount at the proper time. Generally, before about 30 seconds into flight, the thrust needs to be quickly dropped down to about 60% or less of its maximum value. Once the Max Q is passed, the throttle can be slowly increased towards its maximum value, but it still must be done with consideration of the dynamic pressure.

SUMMARY

Dynamic pressure is a force that can overwhelm the structural integrity of rockets. It is necessary to control the thrust and acceleration of the vehicle to maintain some designed-for structural integrity limit.

REFERENCES

1. SPACE SHUTTLE ASCENT - Math & Science @work, NASA & Texas Instruments
2. SPACE SHUTTLE LAUNCH MOTION ANALYSIS - Math & Science @work, NASA & Texas Instruments