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

RESIDUAL PROPELLENTS AND THEIR EFFECTS ON LAUNCH VEHICLE DESIGN

INTRODUCTION

Recently Ben Brockert, a professional rocket engineer, suggested that I had ignored an important issue in the design of small launch vehicles: residual propellants. He was right and it is worth considering the effects of residual propellants on small launch vehicle performance.

Residual propellants are unutilized propellants. It is not possible (or sometimes desirable) to burn 100% of the propellants. This restriction results in unused propellants which do not contribute to propulsion, yet whose mass remains with the rocket stage.

With solid rocket engines, nearing the end of their burn duration, it is possible for the chamber pressure to drop below a point where neither reasonable combustion nor propulsion occurs. This can result in slivers of unburned propellant which do not contribute to the thrust of the vehicle in any meaningful way.

With liquid propellant rocket engines, problems can also occur during the end of the propellant feeding period. Conditions can arise where gases instead of liquids are fed into the rocket engine or its pumping systems. When the propellant in a tank gets low, effects like sloshing can result in gases being ingested into the propellant feed pipes instead of the propellants. Uneven mixture ratios can result in a surplus of one propellant or the other. These situations can result in catastrophic conditions which often must be avoided by the designer.

To avoid the effects of these conditions, rocket designers will often purposely design for there to be residual propellants as part of the normal usage of the rocket. In any case, the existence of residual propellants is almost guaranteed. Their effect on the performance of launch vehicles must be considered. They also effect the lower mass boundary for a useful small launch vehicle.

RESIDUAL PROPELLANTS IN THE ROCKET EQUATION

The rocket equation can take the following form:

eqn(1)

Where:

= delta V or change in velocity

= gravitational acceleration

= specific impulse

= + +

= +

= propellant mass

= structural mass

= payload mass

To account for residual propellants, we need to break up the propellant mass into two components:

= +

= mass of utilized propellant

= mass of residual propellant

We can express this relationship by specifying a percentage term:

= Propellant Residual Percentage

and then the representation of the propellant becomes:

= +

=

=

Putting these terms back into the rocket equation, we get:

Eqn(2)

We can see that the residual propellant adds to the inert mass of the rocket and effects its mass ratio. Additionally, only the utilized propellant effects the actual delta V of the vehicle.

DESIGNING FOR RESIDUAL PROPELLANTS

We often know a delta V we wish the rocket stage to attain and we want to calculate the masses of the various components which will produce that performance given known, reasonable mass ratios that can be attained. Therefore, we're interested in calculating the budget for the inert mass and the amount of propellant for a desired performance and payload.

Knowing that residual propellants have an effect on the performance, we would like to include that in the calculations. Therefore, we must rearrange the rocket equation to consider these realities.

We can identify the inert mass of the vehicle as a function of the propellant mass using a parameter known as lambda:

=

or

=

For a nice explanation of lambda, see: [http://selenianboondocks.com/2010/02/rocket-equation-mod-1/].

Knowing the "technology level" or structural sophistication of the rocket design, we can pick characteristic values of lambda and apply it to a newer design. Historical values of lambda for various launch vehicles can be found at [ http://home.earthlink.net/~apendragn/atg/coef/ ].

We can represent the entire rocket equation as a function of these new component ratios:

eqn(3)

Where:

= delta V (velocity change)

= gravitational acceleration

= specific impulse

= total propellant mass

= Residual Propellant Percentage

= payload mass

=

From this, we would like to solve the equation to calculate the total propellant mass as a function of the other components:

Then, given our Pr and lambda ratios, we can calculate all other mass component values from that results:

=

=

=

If we let MR be the mass ratio:

then we can simplify our representation of the rocket equation

eqn(4)

The derivation of the equation for calculating the total propellant mass, knowing the other mathematical components is:

eqn(5)

We now have an equation which calculates the total propellant mass as a function of Ml, dV, g, Isp, Pr, and lambda.

REALISTIC RESIDUAL PROPELLANT FRACTIONS

Exact figures on realistic expectations for residual propellants are hard to obtain. Ultimately it depends on your feed system design. However, Sutton [1] suggests 0.5% to 2%. For the Saturn V S1C stage, residual values of ~1% LOX and ~1.7% RP1 were seen for an overall Pr of 1.3% [3].

But, a pump-fed design with regenerative cooling in the combustion chamber will reasonably have more residual propellants than a pressure-fed ablative design. It all depends on the design and other factors. However, it is not unrealistic to expect upwards of 2% residual propellant in the initial design phase.

UNDERSTANDING THE EFFECTS OF RESIDUAL PROPELLANTS ON ROCKET DESIGN

One thing we can do to understand the effects of residual propellants on a rocket design is to see the effects on one particular design.

Suppose we have a stage which has a payload of 700 lbs which we want to have a delta V of 7,500 feet per second (FPS) and which has an average Isp over flight of 250 seconds. Suppose also that we have a lambda of 48% (this is approximately the value of lambda for the Redstone missile).

We can look at the results of designing the rocket with and without residual propellant considerations.

Without consideration of residual propellants (Pr = 0), using equations 4 and 5 we get:

Therefore, the vehicle has the following specifications:

(GLOW = Gross Lift Off Weight)

Now, let's presume that we have 2% residual propellant left at the end of flight, but we want a rocket that has the same performance capabilities:

Therefore, the same performance vehicle with 2% Pr has the following specifications:

Two percent in residual propellants required a vehicle which is about 22% heavier to get the same performance as a vehicle which doesn't have any residual propellants (e.g. utilizes 100% of its propellants).

Using the above vehicle values as a baseline, we can graph the effects of residual propellant versus the vehicle glow:

As can be seen, in this example, the GLOW just about doubles between 0% and 5% residual propellants for the same payload, lambda, delta V and Isp.

For a multistage vehicle, these residual propellant effects compound, resulting in a significantly larger overall vehicle.

If we consider a 3 stage vehicle, where each stage has the following specifications:

Stage 3	Payload	2.2 lbs
	delta V	13750 fps
	Isp	300 s
	lambda	0.20
Stage 2	Payload	Stage 3
	delta V	13750 fps
	Isp	300 s
	lambda	0.20
Stage 1	Payload	Stage 2
	delta V	7500 fps
	Isp	250
	lambda	0.48

then we can compare results for the GLOW. The results I get are:

		Residual Propellants Across the Stages (units in lbs)
STAGES		0%	1%	2%	5%
Stage 3	Ml	2.2	2.2	2.2	2.2
	Mp	18.825	21.216	24.302	43.120
	Ms	3.765	4.243	4.860	8.624
	Mf	24.791	27.659	31.363	53.945
Stage 2	Ml	24.791	27.659	31.363	53.945
	Mp	212.133	266.740	346.454	1075.325
	Ms	42.427	53.348	69.291	211.465
	Mf	279.351	347.747	447.107	1322.735
Stage 1	Ml	279.351	347.747	447.107	1322.735
	Mp	1652.266	2279.107	3285.383	15270.945
	Ms	793.088	1093.971	1576.984	7330.054
	Mf	2724.704	3720.825	5309.475	23923.734
GLOW		2724.704	3720.825	5309.475	23923.734
Multiplier from 0% Pr		1X	1.37X	1.95X	8.78X

The design in the first column is unrealistic whereas the designs in the other columns are more realistic. It is unrealistic to expect 100% propellant utilization and the effects of residual propellants can be significant. Using a realistic value of 2% results in an almost doubling of the vehicle GLOW over the design without residual propellant considerations.

SUMMARY

This quick analysis of the effects of residual propellants on the rocket equation and on particular rocket designs suggests that residual propellants cannot be ignored for practical rocket designs. The effects also illustrate that realistic multistage rocket designs are likely double the take-off mass to those of designs which do not consider residual propellants.

REFERENCES

1. Sutton & Biblarz, Rocket Propulsion Elements, Ninth Edition.

2. Selenian Boondocks, A Simple Modification of the Rocket Equation [ http://selenianboondocks.com/2010/02/rocket-equation-mod-1/ ].

3. NASA, Saturn V Flight Manual - SA 503, MSFC-MAN-503 [http://hdl.handle.net/2060/19750063889 ].