Comments on Rocket Scaling
Dr. Adam London, Ventions, LCC, President and CEO
Dr. Daniel Kirk, Florida Institute of Technology, Associate Professor
Overview:
On traditional liquid-fueled launch vehicles, the engines themselves tend to weight about twice
as much as the payload being delivered to orbit. At launch, they are required to produce a thrust
slightly larger than the total weight of the vehicle. If they could produce this same thrust while
weighing much less, this weight savings could be used to increase the size of the payload. There
are two ways that the thrust-to-weight ratio can be increased for a given propellant combination.
1. Higher combustion chamber pressure will lead to a smaller engine for a given thrust
level.
2. By making the engine smaller at constant chamber pressure, the thrust-to-weight ratio
will increase, everything else being equal. This is because the thrust produced is
proportional to the throat area, while the weight of the engine is proportional to its
volume. For perfect scaling, the ratio of the throat area to the overall volume will increase
as the rocket is made smaller.
Example:
If one takes a traditional ‘baseline’ engine and makes four copies of it, each exactly half the size
(which implies ¼ the exit area and 1/8th the volume) of the original, the four engines operating
together would produce the same thrust as the larger baseline engine, but only weight half as
much. One could then do the same with the half size engine, and make a total of 16 quarter-sized
engines, which when ganged together would still produce the same thrust as the original, but
together weigh a quarter of the original baseline engine. In theory, this process could be
continued indefinitely, leading to a massively-parallel thrust system with a very high thrust to
weight ratio.
The Scaling Laws:
Consider the case of a ‘baseline’ engine with a characteristic length dimension, L.
Quantity
Thrust, T
Weight, W
Comment
T scales with the throat area, At
W scales with the volume, V
T/W
Thrust-to-Weight Ratio
Scaling Law
T ~ At ~ L2
W ~ V ~ L3
A 1
T
~ t ~
W V
L
Equation
1
2
3
Now take the baseline engine and reduce the characteristic length dimension by 1/2, thus making
an exact copy, but ½ the size (again note that areas would be reduced by ¼, and volumes by
1/8th). We will call this length L2 and that of the baseline engine L1, so L2=L1/2. We can now use
the above scaling to show that:
1
1
T 
  
 W 1 L1
1
2
T 

  
 W  2 L2 L1
T 
T 
   2 
 W 2
 W 1
4a, 4b
5
This means that by reducing the size of the rocket by ½ we have doubled the thrust-to-weight
ratio between the two overall engine systems. We are interested in keeping the overall thrust
level of the sum of the ½ size copies the same as that of the baseline engine. We need to know
how much thrust each of these copies produces relative to the baseline engine. Examining
equation 1 for both cases, we find that:
T2 
6
T1
4
This shows that each of the ½ sized copies will produce ¼ the thrust of the baseline engine, and
hence we would need to use four of them to give the same overall thrust level as the single
baseline engine. Now, we need to check and see if there is a weight savings. One way to do this
is by comparing the overall thrust-to-weight ratios of the two engine systems, and equating the
thrust of each in equation 5.
W2 
7
W1
2
This indicates that we have reduced the weight of the overall new have sized system by a factor
of 2 and have maintained the overall thrust level of the original, single-engine baseline system.
The above provides how an example of how the scaling laws, but they are most powerful in
examining trends. By manipulating equations 1 and 2 slightly, we can come up with a more
useful representation, which shows the behavior of T/W very rapidly. Equations 1-3 can be
manipulated by to yield:
1
T
~T 2
W
1
T
~W 3
W
T/W scaling for variations in T
8
T/W scaling for variation in W
9
If I want to know how the thrust-to-weight ratio of my baseline rocket will change if I double,
halve, or increase by a factor of 10 the thrust level, I can use equation 8. For example, how will
T/W vary if I double the T of an engine? Using equation 8, T/W~2-1/2, gives a reduction in T/W
by a factor of 0.707. In general, we can see using equations 8 and 9 that increase the thrust or
weight of an engine will decrease the T/W ratio. Conversely, decreasing either of these values
will lead to an increase in T/W. Equations 8 and 9 can also be rearranged to look at variation in T
or W while varying the other quantity. For example, how does W change if I double T. These
manipulations are shown in equations 10 and 11.
2
W ~T
T ~W
3
2
2
W scaling for variations in T
10
3
T scaling for variation in W
11
The figure below shows the results of these equations plotted on a log scale where either T or W
is varied by a factor from 0.1 to 10. The black lines show the T/W scaling as either T (solid) or
W (dashed) is varied, and this quantity is read on the left-side scale. The red lines show how W
varies with T (solid) and how T varies with W (dashed) and is read on the right-side scale.
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T/W (vary T)
T/W (vary W)
Weight (vary T)
Thrust (vary W)
6
30
5
25
4
20
3
15
2
10
1
5
0
Resultant Thrust or Weight
Thrust / Weight Ratio Based on Scaling Law
7
0
0.1
1
10
Thrust or Weight Scale Factor x Baseline (Basline = 1)
Being able to understand and use the basic scaling laws is a very powerful tool.
Practicality of the Scaling Laws:
As is often the case, reality and practicality get in the way of theory. This approach was used to
varying degrees in both the US and Soviet moon programs. The first stage of the Saturn V was
powered by five F-1 engines, saving about half the weight of an equivalent single engine
according to the above argument. The Soviet launcher was to be powered by about 25 engines.
By the arguments above, one would have expected that together they weighed about a fifth of an
equivalent single engine. The rocket never had a successful flight as there were a number of
single engine failures that led to failure of the launch system. A system with a large number of
engines has the capability to provide redundancy in that the loss of thrust from one could have
only a small effect on total thrust level. However, if the failure of one engine cannot be
contained, additional engines will multiply the number of single point failure modes for the
launch system, leading to significantly reduced overall system reliability.
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Other practical issues arise as well. One must justify the additional complexity required in the
pluming and control of many versus few engines. Additionally, the traditional view is that there
is a minimum chamber residence time required for complete combustion in rocket engines,
which does not scale with size. This means that one cannot perform an exact scaling of the
engines without sacrificing efficiency, something launch vehicle designers are quite loathe to do.
Cost is another concern. Using traditional manufacturing methods, the cost of producing one
half-size engines is probably not much less than the cost of producing a full-sized engine, as in a
perfect scaling each of the pieces would have to be reproduced at half-scale. The cost of a
smaller engine might even exceed that of a larger one as it becomes harder to reproduce the
detail of the original pieces at smaller and smaller scales. Eventually, limits in fabrication
technology would prevent one form successfully making the smaller engine. In any case, the cost
per unit thrust of an engine would certainly increase.
It is quite clear that the reduction in scale of rocket engines is not a so-called ‘silver-bullet’
automatically leading to better system performance. As is usually the case, a high-level system
trade-off is required in choosing the appropriate number and size of engines for a given
propulsion system. The new concept of rockets which are micro-fabricated from silicon-carbide
appear provide substantial benefits to the scaling laws.
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