Black Arrow: Britains Satellite Launcher

The Black Arrow was Great Britain's first indigenously-developed rocket launcher. In four flights, two were successful. It successfully put one satellite, the Prospero, into orbit, making Great Britain the sixth country to orbit a satellite (after the Soviet Union, the United States, France, Japan and China)

SOURCES
Finding accurate, authoritative information on the Black Arrow was difficult (at least in the U.S.). Therefore, one should expect a wider error margin on the weights and performances listed here. However, the values make sense with regard to various sources so they probably aren't too far off.

Because of the conflicting and non-authoritative information published widely on the Internet, I'm going to approach this discussion a bit differently. Since the Internet references conflict or give obviously wrong information, I'm going to extract some of the design information from highly authoritative sources and first principles, confirming or denying information published from less authoritative sources. Otherwise, I'll use whatever other technical information from other sources
that appears correct with the models derived from more accurate sources.

There are two primary sources that I'll base much of the analysis on. The first is the table of weights [1] published by the Royal Airforce Establishment (RAE).


The second source is a trajectory diagram [2] also published by the RAE. This diagram illustrates the time of launch events and other significant parameters of the orbital ascent.



The problem with the trajectory data diagram that warrants caution is that there is no specific identification of this trajectory with a specific vehicle (there were five of them built) as well as whether or not this diagram represents desired or actual trajectory figures. However, the results obtained by using this diagram match well with the specifications from the weight table. Therefore, I'll use at as a basis for establishing authoritative derived parameters.

GENERAL
The Black Arrow was a three stage satellite launcher able to place a 300 pound (136 kg) satellite into a 135 mile altitude Low Earth Orbit. The first and second stages used liquid propellants and the third stage used a solid motor.

The following table shows the performances and weights of the stages and its constituent components. Because the second stage carried and then jettisoned the payload aerodynamic fairing during flight, it has been broken into two stages, BA2A and BA2B (for before and after fairing separation, respectively):



DETERMINING ACTUAL STAGE SPECIFIC IMPULSES
Most sources, even from the 1960's say that the Specific Impulse (Isp) is 265 seconds for both the first stage and the second stage. This is obviously wrong for several reasons because the first stage flies through a varying atmospheric pressure regime throughout its flight and therefore the Isp will vary. We need to know what the range of Isp variance is to properly model the vehicle. Additionally, although the first and second stage motors were very similar, there were significant differences that affected their performances. Specifically, although the second stage may have used the same motor chambers as the first stage, it has been stated in numerous sources that the second stage had a nozzle extension. This would make the Isp of the second stage different from that of the first. Therefore, I need to look at what realistic Specific Impulses might be for each of the stages.

There is authoritative information that might give clues to the likely performances of the first and second stage motors. Using the two primary sources, I can derive some bounds on the specific impulses of the engines in each stage.

For review, the formula for Isp (Specific Impulse) is:

Isp = Thrust * time / weight

Where the Isp (in seconds) is equal to the thrust (in force) times the number of seconds of that thrust divided by the weight of the propellant (also in force units) required to produce the thrust. Since you have force units over force units in the equation, they cancel out, leaving only seconds as the remaining unit.

The weight table says that the propellant weight of the first stage is 28735 lbs with a thrust of 50000 lb-f at sea level and 57640 lb-f in vacuum. The trajectory diagram says that the first stage burn duration was 131.2 seconds. The propellant weight of 28735 lbs divided by 131.2 seconds means that (on average), the propellant usage rate is 219.017 lbs/sec. Now, I have no idea whether or not the engines were throttled during the flight so it is possible that this only does represent an average usage, and not an actual per-second usage, but it's the best information I have. Based on these numbers and the Isp equation, we can establish that the effective Sea Level Isp is:

 Eff. Isp(SL) = 50,000 lb-f * 1 s / 219.017 lb-m
 Eff. Isp(SL) = 228.293 seconds

 Eff. Isp(VAC) = 57640 lb-f * 1 s / 219.017 lb-m
 Eff. Isp(VAC) = 263.176 seconds

The Weight table lists the second stage as having 6522 lbs of propellant and a vacuum thrust of 15340 lb-f. The trajectory diagram shows the burn time of the second stage as 116 seconds. With 6522 lbs of propellant burnt in 116 seconds, the average propellant usage is 56.224 lb-m/s. Using the Isp equation, we can specify the effective Isp as:

 Eff. Isp(VAC) = 15340 lb-f * 1 s / 56.224 lb-m
 Eff. Isp(VAC) = 272.84 seconds

These answers make a basic kind of sense because the first stage motor's Isp will vary throughout flight up to its maximum Isp performance in a vacuum. The second stage, having a nozzle extension, will have somewhat greater Isp in vacuum.

For the third stage, I could find no authoritative source for the motor efficiency; I used an Isp value which is widely cited on the Internet. The propellant weight values I used are derived from [5].

Whether these estimations are accurate representations of the actual usage or not, I can't say. But, a rocket which has these parameters would closely meet the basic performance characteristics specified in the reference documents.

FIRST STAGE
The first stage utilized a Gamma-8 type rocket motor which burned 90% Hydrogen Peroxide and Kerosene. The Gamma 8 consisted of 8 separate chambers in a cruciform shape which gimballed on one axis each. These were fed through a peroxide-driven turbopump which raised the chamber pressure to about 640 PSI (reported as 44 bar/638 PSI in several sources but as high as 47.4 bar/687 PSI in other sources). Each chamber had an expansion ratio of about 10 giving them good high altitude performance: a Sea Level Isp of about 228 seconds and a vacuum Isp of about 265 seconds.

Regarding the Gamma-8 motors, various sources say that they had expansion ratios anywhere from 60 to 80. However, no motor running at sea level will have stable exhaust if the exit exhaust pressure is below about 40% of the atmospheric pressure. With a chamber pressure of 640 PSI, the exhaust pressure of a 60:1 nozzle will be about 1 PSI versus the 14.7 PSI atmospheric pressure. Sixty-to-one or greater appears to be too high of an expansion ratio for the first stage motors. Historical sources tell us that the Gamma-8 motor chambers were derived from the Gamma 301. A cross sectional diagram of the Gamma 301 is available in [4]:



Measuring the Gamma-301 ratio of the diameters (squared) of the nozzle exit versus the throat shown in this picture gives an area ratio of 10.27. All pictures of the first stage motors appear to have this ratio in all pictures that I've seen. Therefore, I reject the various claims that the first stage motors had an expansion ratio of 60, 80 or more. The expansion ratio appears to be about 10 which is consistent with exhaust stability theory, the computed Isp's and various images of the motors.



If I model the motors using a combustion analysis program with the published values, then I get the following results:

These values very closely match the values derived by using the flow rates and thrusts. So, if the motors ran closer to 100% efficiency and the turbopumps used about 6% of the calculated efficiency to operate, then we would see the kinds of Isp values that I've calculated.

SECOND STAGE
The second stage used two of the same chambers that made up the 8 motors of the Gamma-8. A nozzle extension was added to the motors to give a reported expansion ratio of 350:1 (according to some sources). These motors are reported as having a vacuum Isp of 265 seconds but my analysis shows it's likely closer to 273 seconds.

The second stage carried the payload fairing. The trajectory table shows the payload fairing being ejected after 180 seconds. The weight of the fairing can be found in the weight table[1].

For the second stage motor, there does not appear to be any authoritative source referencing the expansion ratio of the nozzle. But using the 94% combustion efficiency derived for the first stage, and the derived value of the vacuum Isp as being 272.84 seconds, then the motor maximum theoretical Isp would have to be something like 272.84 s / 0.94 or 290.26 seconds. Again using a combustion analysis software, and the pressure and mixture ratios used in the first stage motors, we can estimate that the second stage expansion ratio had to be at least 16:1 as the minimum necessary expansion ratio. The various pictures of the second stage nozzles seem to suggest that the nozzle expansion ratio may have been closer to 60 or 80. Maybe this was the value that was wrongly attributed to the first stage expansion ratio in the various sources. As it stands, I have no idea what the accurate expansion ratio was.

THIRD STAGE
The third stage used a solid propellant rocket motor (called Waxwing) without guidance but which was spun to be gyroscopically stabilized. Again, sparse authoritative information exists for its specifications.

GUIDANCE AND CONTROL
Like the few rockets we have looked at so far, the guidance system was kept in the second stage, and the third stage was unguided. Using the familiar "point and fire" technique, the second stage would orient itself to the desired angle for the orbit, small rockets were used to spin up the third stage to ensure orientation stability and then the third stage would fire its motor.

The first stage control system utilized motor gimbaling to provide full yaw-pitch-roll control. The second stage also utilized gimbaling of two motors to provide full control during motor firing. After the second stage propellant was burned out, the second stage used a pressurized nitrogen cold-gas reaction control system to orient the third stage for firing.

AERODYNAMIC AND GRAVITATIONAL LOSSES
Simulations using the specifications above result in an observed first stage aerodynamic loss of 587 feet per second (fps) and gravitational losses of about 4218 fps. The reason for these kinds of losses are due to the Black Arrow's relatively large diameter relative to its length (it had a fineness ratio of 7) and its relatively low acceleration (takeoff thrust was only 25% more than GLOW).

SUMMARY
I can't say how accurate these results are, but they fit and describe the vehicle fairly well. A vehicle with the specifications described above would have similar performance to those described in the various sources. Nonetheless, the derived specifications are still useful metrics for comparison against other vehicles.

REFERENCES

[1] Weight Table, derived from reference [3] page 28.
[2] Trajectory Table, derived from reference [3] page 50.
[3] The Black Arrow rocket: a history of a satellite launch vehicle and its engines 
    Millard Douglas, 2001, ISBN 1900747413
[4] History of Liquid Propellant Rocket Engines
    Sutton George Paul, 2006, ISBN 1563476495
[5] A Vertical Empire
    Hill C.N., 2001, ISBN 1-86094-267-9