I see proponents of aerospike (altitude-compensating)
nozzles claiming that the improved efficiencies seen vs altitude down in the
atmosphere, extend out into vacuum, making such nozzles “perfect” for
single-stage to orbit vehicle designs. But,
such aerospike nozzles CANNOT perform well in vacuum! Compressible flow physics says so! Read on.
Aerospike is one type of a “free-expansion” nozzle. All these expand the flow partly against a
suitably-shaped surface, and partly as
an unbounded free-surface, both along
the expanding stream. As indicated in Figure
1 below (all figures are located at the end of this article), at the design point (at some altitude), the free surface of the stream pretty much
moves straight aft, getting its
expanding stream cross-section area from the shape of the spike or ramp. That design point operation is the top right
image in Figure 1.
All the momentum vectors at this design condition point
more-or-less axially aft, at the last
point of contact with the spike or ramp,
which is also where the expanded stream pressure matches the surrounding
atmospheric pressure. Only the cosine
components of these momentum vectors apply to axial thrust, but because the angles are nearly axial, those cosine factors are high, very nearly 1! Their averaged value is the “nozzle kinetic
energy efficiency” ηKE.
Thrust is F = m Vexp ηKE with no pressure term (Pexp
– Patm)*Aexp,
since Pexp = Pamb by design and the term is zero, where m is the mass flow rate through the
nozzle.
Below the design altitude,
less area expansion is required to reach the higher ambient atmospheric
pressure, and so the final size of the
stream at the last point of contact is smaller, and the expanded velocity Vexp is
less, although still high. The momentum vectors are all still rather
closely aligned with the axial direction,
so ηKE is still high.
One is still getting very close to the theoretically-possible
thrust. This condition is the top left
image in Figure 1.
Above design altitude,
more area expansion is required to reach Pexp = Pamb
by about the last point of contact with the spike or ramp. The expanded velocity Vexp will be
higher than design there. But
since the spike or ramp geometry is fixed,
the only way to get the extra expanded stream area is for the free
surface to angle outward, off the axis! This is the bottom left image in Figure 1.
Note that many of the momentum vectors now have significant
angles off of axial, reducing their
cosine-factored axial components. The
average of all those cosine factors is now lower, so that ηKE is lower, meaning thrust is reduced significantly below
what could be theoretically had. The
lesson here is that free expansion designs have high efficiencies at and below
their design altitudes, but those
efficiencies drop above design altitude!
The extremized case is taking the free-expansion nozzle out
into vacuum, which is the lower right
image in Figure 1. The stream
needs to expand to an “infinite” area in order to achieve an expanded pressure
Pexp that is zero, but it
cannot! It can deflect only to some
angle off the axis, if the free
expansion starts at the Mach 1 throat area of the nozzle. This is the limiting angle value as
determined by a part of compressible flow analysis called “Prandtl-Meyer
flow”. Vexp will be as high
as it can be, but probably somewhere on
a curved control volume surface intersecting the axis somewhat aft of the last point
of contact.
Look again at the bottom right image in Figure 1. Note that most of the momentum vectors are
significantly angled away from axial, out
past the last point of contact with the spike or ramp! That geometry will not change, even if the pressure reaches its minimum
somewhere aft of that last point of contact.
Some of those vectors are right out to the side, where they contribute nothing to axial thrust, or even past that, subtracting from thrust!
The average of all those cosine factors ηKE is so
very clearly sharply reduced below 1,
and so the deliverable thrust is far below what is theoretically
possible! The lesson here is
painfully obvious: you do NOT want to
use an aerospike (or by extension any free-expansion nozzle design) out in
vacuum!
There is a way to limit the vacuum fan-out angles to
something less. This is shown in Figure
2 below, where the design has a
partial fixed bell to a point where the Mach number is greater than 1, followed by a free expansion, as before.
The simplest form of this notion is the “scarf-cut bell”, but there can be aerospikes fed by
supersonics.
The outer free boundary angle of the stream in vacuum
is driven by the Prandtl-Meyer flow limitation starting from a supersonic Mach
number, which is a lower final angle
than when starting from Mach 1. But the
expanded velocity is less because the expanded stream area is less at the last
point of contact, reducing the theoretically-obtainable
thrust. The spread of momentum angles
is less, leading to a higher (but still
reduced) ηKE.
As Figure 2 indicates, the further supersonic you start the free
boundary expansion, the more you can
limit the fan-out angle. But you are
losing expanded speed doing this! It’s a
trade-off! And it does not change
the fundamental lesson of not using a free expansion nozzle in vacuum! Only the detailed numbers are different.
Conventional fixed-bell nozzles as illustrated in Figure
3 below, do not face this
free-surface fan-out problem, as their
stream is confined by the bell surface right up to the last point of contact
at the exit plane! THAT is where you
figure thrust, and thus you DO NOT CARE
that the plume fans out suddenly just downstream of the exit plane, by the same mechanisms as described
above. This free-surface fan-out
expansion, just aft of the exit plane, has been observed on every launch vehicle leaving
the sensible atmosphere since the space age began. It is quite real!
The momentum vector angles at the exit plane of a fixed bell
are constrained by the physical bell walls,
right up to the exit plane where the area is Aexp. The shapes and angles of those walls
determine the exiting angle into the free surface edge of the plume. The vector on the axis of the bell is
axially-directed at the exit. Thus all
the cosine factors are close to 1, and ηKE
is a high value, fixed by the bell geometry, and independent of the ambient atmospheric
pressure! The differences from all the free expansion
designs are (1) that there is now a
backpressure correction term on thrust:
Fth = m Vexp ηKE + (Pexp – Pamb)*Aexp, and (2) that ηKE is unaffected by
altitude ambient atmospheric pressure.
Figure 3 also shows that a curved bell and a conical
bell can have exactly the same high ηKE, it is just that the curved bell can be a
little shorter, but is more difficult to
design.
Because of the backpressure term on thrust, and how backpressure-induced separation
limits design options, for a true “sea
level” nozzle design, you set Pexp
= Pamb = PSL, and
accept the lower area expansion ratio and Vexp. That will be all the thrust you can get at
sea level, and while it increases as you
ascend, it does not increase by very
much, because the largest effect on
thrust is area expansion ratio to reach Vexp, not the backpressure term.
The so-called “vacuum-optimized” nozzle bell is nothing
of the sort! It is entirely
constraint-driven: you simply go to
the biggest nozzle exit area that will actually fit behind the stage or
vehicle! It’s the same constraint, whether one engine or several. Being able to thrust-vector-control the
vehicle with engine gimbal angles usually reduces that max nozzle exit size a
little. This vacuum design approach is also
shown in Figure 3.
The thrust equation above for a fixed bell can be converted
with compressible flow to a very convenient thrust coefficient form, as shown in Figure 3. This allows one to determine the expansion
ratio-related data without knowing the actual dimensions first! That is also shown in Figure 3. For the thrust coefficient equations shown in
the figure, there are 3 items related to
the expansion ratio selected. They are
the area expansion ratio Ae/A* itself,
the exit plane Mach number Me,
and the ratio Pe/Pc of the exit plane expanded pressure and the chamber
stagnation pressure Pc, that feeds the
nozzle. If you know any one of them, basic compressible flow quickly finds the
other two for you.
Thrust coefficient has two parts, as indicated in Figure 3. Those are the vacuum thrust coefficient CFvac, and a backpressure correction term that
varies with altitude ambient pressure Pa and the chamber pressure Pc. The actual thrust coefficient down in the
atmosphere CF = CFvac – (Pa/Pc)*(Ae/A*). And thrust Fth = CF Pc
A*, by definition. All of this presupposes no separated flow in
the bell (dealing with that is beyond scope here).
Now, Figure 4
below shows two altitude-range sketches for launch and ascent toward low
circular Earth orbit (LEO). The left one
is marked up to show what happens when using a free-expansion nozzle like an
aerospike for a single-stage to orbit (SSTO) vehicle, as aerospike proponents want. Most of the time (and propellant) is spent
down low, because the vehicle has yet to
accelerate to high speeds. As it leaves
the sensible atmosphere, the trajectory has
bent over. Most of the so-called
“delta-vee to orbit” is obtained essentially out in vacuum! For an aerospike, this is where ηKE is seriously degraded! It has good ηKE only early-on, down in the sensible atmosphere, as discussed thoroughly above.
The plot on the right in Figure 4 is marked for a
two-stage to orbit (TSTO) vehicle, using
sea level fixed-bell engines in the first stage, and vacuum fixed-bell engines in the second. Again,
most of the “delta-vee to orbit” is obtained essentially in vacuum by
the second stage, with engines that are
operating at high efficiency. In
contrast, the aerospike SSTO is getting
most of its “delta-vee to orbit” out in vacuum at low efficiency!
The only other point to make about Figure 4 is also
in the plot on the right. Down low in
the atmosphere during the TSTO first stage burn, both the fixed-bell sea level nozzle
and the aerospike nozzle would be operating at high efficiency! The aerospike might even outperform the
fixed bell in that role, if the staging
altitude is not too high!
Figure 5 below shows 2 sketches. The left sketch is thrust vs altitude for a
fixed-bell sea level nozzle, plus an
alternate ascent design option for that same role. The plot on the right is an altitude-range
plot marked for a TSTO vehicle, showing
where staging occurs as you leave the atmosphere at only-supersonic
speeds. (The entry interface altitude is
the edge of the sensible atmosphere only when moving at truly orbital-class
speeds.)
The sea level nozzle on the left shows an increase in thrust
at higher altitudes, because the Pamb
in the backpressure term is going down.
You can design for expansion to ambient at a higher altitude, say in the 5 – 10 km range. That thrust trend will have a larger
expansion ratio and a higher thrust going into vacuum, but at the cost of lower sea level thrust
when the vehicle is heaviest! It can
actually average a slightly higher thrust over the ascent to staging, than a true sea level nozzle. It is an engineering design trade available
to you!
Bear in mind that an aerospike (or other free-expansion
design) designed at high altitude will have an increasing Vexp and a
still-high ηKE, almost all
the way to staging, which is essentially
just outside the sensible atmosphere. That
makes such a nozzle a real competitor in the first stage role of a TSTO!
The second stage of the TSTO just needs to use a fixed-bell
vacuum design. That will be efficient, because all of that burn is outside the
sensible atmosphere, essentially in vacuum. And as thoroughly discussed above, a free-expansion design will be quite
inefficient at such altitudes, due to the
excessive stream-spreading angles of its free edge.
Conclusions
(#1) I have no doubt at all, that free expansion nozzle designs perform
quite well throughout the sensible atmosphere,
and probably better than sea level fixed bells, for ascent to staging altitudes.
(#2) Conclusion (#1)
extends to even the compromise fixed-bells designed at altitudes modestly above
sea level.
(#3) A fixed vacuum
bell will far out-perform any free-expansion design, when operating out in vacuum! That is because free-expansion efficiency
starts falling somewhere above design altitude,
and it has fallen drastically upon going into vacuum!
(#4) The best role
for free-expansion nozzle designs is in the first stage of a TSTO. They will do better in vehicles that have
lower staging altitudes, because the
vacuum free-surface spreading effects upon overall impulse delivery get
reduced.
Figure 1 – Free-Expansion Plume Behaviors (Aerospike Shown)
Figure 2 – Limiting the Fanning-Out In Vacuum Is a Tradeoff
Figure 3 – Fixed-Bell Nozzles Can Be Either Atmospheric or
Vacuum But Not Both
Figure 4 – SSTO to LEO With Aerospike vs TSTO to LEO With SL
and Vacuum Bells
Figure 5 – The Best Application of Aerospike Is First Stage
of a TSTO
Followup: Actual
Numbers Estimated
I went ahead and ran a study to see how bad the fan-out and
fan-in angles might be, and what effect
they might really have, on the ideal
expansion numbers. I ran it for an
axisymmetric aerospike designed for “perfect” expansion at 60,000 feet altitude
(18.29 km), with an arbitrary spike
radius at the throat station of 1.000 feet,
and a spike cone half-angle of 13 degrees. I presumed liquid oxygen (LOX) and liquid
hydrogen (LH2) as propellants, in a
full-flow cycle with no dumped bleed,
and a Pc of 4400 psia.
The basic calculations are depicted in Figure 6. At design, the streamtube boundary goes straight
back: all the area expansion ratio is
provided by the spike and lip geometry. The
last point of contact with the spike tip is where the control volume about the
engine must be located, in order to
exclude from thrust calculations the effects of the conical oblique shock that
turns the spike-angled surface flow back to straight axial.
Below design altitude,
the flow must “fan-in” from the throat lip by some angle C in order to
“hit” the right area ratio at lower altitudes.
Above design altitude, the flow
must fan-out by some angle B in order to “hit” the right area ratio at the
higher altitudes. The angle B may not
exceed the turning angle “v” from Prandtl-Meyer flow (P-M)! If it does,
the control volume must move aft of the tip, and you must account for the oblique shock
losses, too.
Figure 6 – How the Numbers Were Estimated
At design, a decent
estimate of the averaged cosine factor for correcting all streamline momentum
vectors back to axial, would be the
average of cosine(0) = 1 for the plume edge,
and cosine(A) for the axis. That
averaged cosine factor is the value of the nozzle kinetic energy efficiency ηKE
that applies to both thrust coefficient CF and specific impulse
Isp, as well as to thrust, which I did not explicitly figure for this
study. (F = CF Pc At.)
Skipping thrust, one
can go directly from CF to Isp as Isp = CF c* (1 – BF) / (gc
CD), where BF is the dumped
bleed fraction, and CD is the
effective-area factor applied to the geometric throat area, for computing massflow. In this study, BF = 0,
and I used a nice high CD = 0.995.
The design sizing numbers are shown in Figure 7. Sizing at 60,000 feet is why Re is only
slightly larger than Rt. This has to be
set iteratively, to hit the “right” Ae.
Figure 7 – Design Values for This Study
The ideal expansion numbers are very easy to
compute. One needs the ambient pressure
Pa at that altitude, and sets the
expanded pressure Pe equal to it. Thus, at any altitude, Pe/Pc is known, and can solved for the expanded Mach number
Me, at the last point of contact with
the spike tip. From that, the area ratio Ae/At is easily found. The thrust would be massflow x velocity x
averaged cosine factor m Ve ηKE, or in compressible flow variables F = γ ηKE
Pe Ae Me2, since the
(Pe-Pa)Ae term is zero. Dividing by Pc
and At, we have the ideal free-expansion
thrust coefficient:
CF = (Pe/Pc) (Ae/At) γ ηKE
Me2 for which ideally
ηKE = 1
The Isp is then Isp = Fth/w = Pc CF At / (Pc CD
At gc /c*(1 – BF)) = CF c* (1-BF)/(gc CD). For this study BF = 0.0 and CD =
0.995.
I used a power-function c* correlation to estimate c* at Pc
from some old data for LOX-LH2 at 1000 psia:
c* = 7950 ft/s (Pc/1000 psia)0.006051
In these units, one
uses gc = 32.174.
All of these ideal numbers figured at ηKE =
1 are totally uncorrected for the fan-in and fan-out angles that
inherently occur! Those angles affect
the average of the cosine factors that must be applied to the streamline
momentum at every point on the exit surface of the control volume. Not looking at the angle effects on ηKE is the mistake that proponents of aerospike
(and other free-expansion nozzle designs) make,
unless they are adept at modeling real-world effects in compressible
flow. Such are NOT easy to calculate!
For ideal flow at ηKE = 1, the performance looks like that depicted in Figure
8. This “looks great” all the way
from the surface into vacuum! And that
is what non-adept proponents of aerospikes use to make their claims, including that these nozzles are “perfect”
for single-stage to orbit usage. But, ηKE is NOT 1 all the way to vacuum; it CANNOT BE!
Figure 8 – Ideal Expansion Performance Looks Deceptively Great
You have to account as best you can for the exit
area-averaged cosine factors of all the streamline momenta. In other words, ηKE is a function of the fan-in
angle C below design, the fan-out angle
B above design, and the spike half angle
A. I showed some
well-inside-the-ballpark approximations for those dependencies in Figure 6
above.
This angle-dependent ηKE has little effect below
design altitude, but the effect
increases rapidly above design altitude,
becoming quite severe, as
indicated in Figure 9. The fan-in
angle C is shown vs altitude top left,
the fan-out angle B is shown vs altitude top right, and the values of ηKE and CF
vs altitude that result, are shown
bottom left. The resulting Isp for this
study that results from this properly-corrected CF trend, is shown bottom right.
Figure 9 – Actual Angle-Corrected Performance Falls Off at
High Altitudes
Two things should “jump off the page at you” immediately, looking at Figure 9 just above:
(#1) These aerospike- (and by extension any free-expansion-)
design nozzles are NOT good vacuum engines! They start losing performance in a serious
way somewhere above design altitude,
which for this example is in the vicinity of 150,000 feet altitude
(45.72 km). Such is actually below
typical staging altitudes for two-stage-to-orbit (TSTO) vehicles, which usually fall in the 50-70 km
range, and it is far below any proper
orbital altitude (150+ km).
(#2) Given (#1), the
“best application” for the aerospike- (or any other free-expansion-) nozzle
design, is in the first stage of a
TSTO vehicle, NOT in an SSTO
vehicle!
And (#2) confirms what I said, in the non-numerical article above this “Followup”
section.
Final Remarks
Otherwise, I may not
have run the “right” aerospike design.
My throat area is very small, to
get from a very high Pc down to a very low Pa at the high altitude of 60,000
feet. Both a lower Pc and a lower design
altitude would act to increase throat area,
showing up as a larger difference between Rt and Re in Figure 7 above.
However, that
does not matter to the basic trends outcome here!
The
trends identified with fan-in and fan-out angles would remain, with only the numbers shifting a little. The trend here that is important, is the free-surface fan-out angle that
increases with increasing altitude above design, no matter what else is done! That WILL ALWAYS happen! And it will always dominate performance
somewhere above design!
Which in turn means aerospike nozzle designs (and by
extension any free-expansion designs) WILL ALWAYS show performance degradation,
far below what is theoretically
possible, while leaving the sensible
atmosphere at only-supersonic speed.
That will inherently happen somewhere near 45 km altitude, FAR BELOW orbital altitudes, just because the air is so thin up there!
It is very easy to estimate the theoretical performance of
free-expansion-type nozzles. It is not
an easy task to estimate that plume fan-out angle, unless you are very well-versed in such
aspects of compressible flow.
Further, you must understand
where and how (and why “there”) to draw the control volumes that let you
properly calculate thrust.
What I
have repeatedly seen is that most proponents of free-expansion designs as
vacuum-capable, do not have those
compressible flow skills, beyond the
very elementary theoretical expansion calculation! Further, they simply do not understand that
the fanning-out of the free surface of their plume strongly lowers thrust
efficiency, by the cosine component
effects upon the momentum vectors of every streamline. Or that this effect is “real” precisely
because it happens INSIDE the control volume that one MUST draw!
And, usually at least
some of them will really dislike being shown the error of their claims! To them I can only say “sorry, but the truth will prevail”.
Meanwhile, the aerospike nozzle has great application
down in the atmosphere, both in rockets
and in jet propulsion!
In
point of fact, it has been flying in
some turbojet engines for decades. The
ones with the spike sticking out of the nozzle “turkey feathers”, those are axisymmetric aerospikes, when the “turkey feather” outlet is
converging and choked! It works great, down in the atmosphere!
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