**It is intended to give readers a means to compute accurate and realistic thrusts. This plus a knowledge of chamber characteristic velocity c* is sufficient to do very elementary rocket ballistics.**

*This article applies to anything that thrusts from a subsonic chamber through (at least) a sonic throat.***see stuff added at the very end, below, past the original figures.**

__Update 11-16-18__:**see stuff added at the very end below, past the first update.**

__Update 1-26-19__:**see stuff added at the very, very end below, past the second update.**

__Update 1-31-19__:
Most (but not all) nozzles that have a sonic throat also have
a supersonic expansion bell. Scramjet is
excluded as being without a sonic throat:
the feed to the nozzle inlet is already supersonic, and there is no contraction in flow area to a
throat.

Rockets of any type are typically high pressure ratio PR chamber-to-exit, and high area ratio AR exit-to-throat. These can be ablatively cooled, or actively liquid-cooled.

Gas turbine engine nozzles are typically low pressure ratio PR
chamber-to-exit, and low area ratio AR
exit-to-throat. These are usually air-cooled, and variable geometry: anything from convergent-only to a mild
supersonic expansion bell. Lower turbine
inlet temperatures require lean mixtures and cooler flames, making air cooling possible, as long as the air itself isn’t too hot. That high speed air heat effect limits the flight
speeds achievable with gas turbine engines.

Ramjet engine nozzles are typically low pressure ratio PR chamber-to-exit, and low area ratio AR exit-to-throat. Modern missile designs are usually ablative. Some of the oldest designs were
air-cooled, similar to gas
turbines, but this approach is severely
limiting in a modern ramjet design,
which can run far richer, and at far
higher flight speeds, where the air
itself is far hotter.

**Fundamentals**

Conservation of mass: the same massflow exists throughout the nozzle
(any air cooling bleed effects or other injections or leaks are ignored, if any exist at all).

Conservation of momentum:
a

**control volume**drawn about the rocket engine is**pierced by the exit stream exactly at its exit area**, and the momentum of the propellant feeds are either inconsequential, or perpendicular to the thrust axis, or they come from tanks inside the control volume. This could be any combination of those situations, or even all three. Balancing stream momentum and the pressure forces against a restraining force, leads to evaluating the thrust.
Conservation of energy:
the drop (from chamber to exit) in enthalpy, as measured by the drop in static temperature,
equals the increase in kinetic energy of
the stream, with essentially zero
kinetic energy inside the chamber. There
is an accompanying drop in static pressure,
in an amount defined by the ideal gas assumptions and the corresponding equation
of state. See Figure 1.

__All figures are at the end.__

We use enthalpy “h” instead of internal energy “u”, because it includes the effects of pressure
change upon energy content, and internal
energy does not. Enthalpy difference Δh is
essentially the temperature difference ΔT,
multiplied by the specific heat at constant pressure c

_{p}. (Internal energy change uses the specific heat at constant volume c_{v}.)
Book-keeping: this is
done the easiest way in Mach number-pressure-temperature variables, instead of the primitive variables,

**. That last means we may use as our equation of state P = ρ R T, and we may use as the change in enthalpy Δh = c***as long as the ideal gas assumption applies*_{p}ΔT.
In this book-keeping scheme,
we make good use of total (or stagnation) pressures P

_{t}and temperatures T_{t}. Assuming no appreciable friction losses, flow is “isentropic”,__meaning total pressure and total temperature are constant through the nozzle__, a very good assumption in almost every conceivable case.
The ratio of specific heats γ = c

_{p}/c_{v}becomes a very useful value to relate totals to statics. At a location where the Mach number is M, the total/static temperature ratio TR = 1 + 0.5*(γ – 1) M^{2}, and the total/static pressure ratio is PR = TR^{exp}, where exp = γ / (γ – 1).
The streamtube area model is more complicated than the
simple mass conservation-derived relation in incompressible flow, and is based off of sonic conditions at the
throat area A

_{t}. If you know the Mach number M at another station where the area is A, you can find that area ratio AR = A/A_{t}as easily as the total/static ratios TR and PR. If you know instead the area ratio A/A_{t}, finding the Mach number M is inherently a transcendental (iterative) solution:
A/A

_{t}= (1 / M) [TR / const1]^{const2}
where TR is defined as above, const1 = 0.5 (γ + 1), and const2 = 0.5 (γ + 1) / (γ –
1)

Heat transfer:

**We***this is driven not by static temperature but by recovery temperature!*__must__do this because the supersonic flow in the nozzle is both highly compressible, and highly dissipative. At any given Mach number M, recovery temperature T_{r}is very nearly the same as total temperature T_{t}. How it varies depends upon laminar versus turbulent flow, and the gas property Prandtl number P_{r}:
T

_{r}= T + r (T_{t}– T) where the recovery factor r = P_{r}^{0.5}laminar, P_{r}^{0.33}turbulent
Only the heat transfer film coefficient h responds significantly
to the varying Mach number,
pressure, and temperature down
the nozzle profile. It does this in a
very empirical way. Multiple models
exist for this, not covered here. The local heat flux at any station is of the
form:

Q/A = h
(T

_{r}– T_{s}) where T_{s}is the material surface temperature
For heat transfer purposes,
failing real data, you can
estimate Prandtl number P

_{r}= 4 γ /(9γ – 5).**Conventional Nozzle Thrust Coefficient C**

_{F}

_{}
Your

__ideal gas model__of the gas flowing through the nozzle comprises its constant specific heat ratio γ, and its constant molecular weight MW. These can come from thermochemistry calculations, and need to reflect the high temperatures involved.
The thrust F of an idealized nozzle evaluated at its exit
plane is the momentum of the exiting gas,
plus the exit area A

_{e}times the difference in pressure between the exiting stream static pressure P_{e}and the ambient backpressure of the surroundings P_{b}. Ideally, all the streamlines are parallel to the thrust axis.
In the real
world, they are not. This
streamlines-off-angle effect is modeled with the nozzle kinetic energy
efficiency factor η

_{ke}. It applies to the momentum term in thrust,__but not the pressure term__, as long as the exit plane is perpendicular to the axis.
F = η

_{ke}m V_{e}+ (P_{e}– P_{b}) A_{e}where m = mass flow rate and V_{e}= exit velocity
To convert this to compressible flow variables, we make use of the m = ρ

_{e}A_{e}V_{e}massflow definition, the ideal gas equation of state P_{e}= ρ_{e}R T_{e}with R = R_{univ}/MW, and the exit plane speed of sound for an ideal gas c_{e}= (γ g_{c}R T_{e})^{0.5}. The variable g_{c}is the “gravity constant” to make the equation consistent with inconsistent mass and force units. If those units are consistent, g_{c}will be 1.
F = η

_{ke}(P_{e}/ R T_{e}) Ve^{2}Ae + (P_{e}– P_{b})A_{e}using massflow, then equation of state
F = γ η

_{ke}P_{e}A_{e}M_{e}^{2}+ (P_{e}– P_{b})A_{e}using speed of sound**Distribute the A**

*Note that the first term in the equation just above is the momentum term, and the second term is the static pressure difference term.*_{e}so that there are 3 separate terms, and regroup.

F = P

_{e}A_{e}[1 + γ η_{ke}M_{e}^{2}] – P_{b}A_{e}recombining terms such that P_{e}A_{e}factors out
Here, inside the
bracket, the 1 now corresponds to the exit static pressure term with P

_{e}A_{e}factored out, and the γ η_{ke}M_{e}^{2}corresponds to the momentum term with P_{e}A_{e}factored out. The backpressure effect is still a separate force term,*with the recombined bracket-containing force term really just being thrust into vacuum.*

Now we introduce the definition of thrust coefficient C

_{F}= F / P_{c}A_{t}with the understanding that the P_{c}is the total (stagnation) pressure feeding the nozzle. If the contraction from chamber to throat area is large enough, there is no measurable difference between total and static pressure at the nozzle entrance.
C

_{F}= F / P_{c}A_{t}= (P_{e}A_{e}/ P_{c}A_{t})[1 + γ η_{ke}M_{e}^{2}] – (P_{b}A_{e}/ P_{c}A_{t})
CF = (P

_{e}/P_{c})(A_{e}/A_{t})[1 + γ η_{ke}M_{e}^{2}] – (P_{b}/ P_{c})(A_{e}/ A_{t}) regrouping P’s and A’s together**CF = (AR / PR) [1 +**

**γ η**

_{ke}M_{e}^{2}] – AR / PR_{op}**(**

__the__thrust coefficient equation)
with PR

_{op }= P_{c}/P_{b}using the actual design P_{c}and P_{b}
with PR
= P

_{c}/P_{e}= (1 + 0.5 (γ – 1) M_{e}^{2})^{exp}= TR^{exp}where exp = γ / (γ – 1)
and AR
= A

_{e}/A_{t}= (1/M_{e})[ TR/const1]^{const2}
with const1 = 0.5 (γ +
1) and const2 = 0.5 (γ + 1) / (γ – 1)

This last formulation is particularly convenient when one
wants a certain exit Mach number M

_{e}, because AR = A_{e}/A_{t}and PR = P_{c}/P_{e}are easily calculated from M_{e}using the ideal gas γ. Otherwise, if conditions at a certain AR are desired, one iteratively tries M_{e}values until the desired AR obtains, then computes PR. Essentially, M_{e}and P_{c}/P_{e}are “locked in” by the AR value regardless of the value of P_{c}, although they are not most conveniently figured in that order.
The “operating pressure ratio” PR

_{op}= P_{c}/P_{b}depends directly upon your design choices for P_{c }and P_{b}. One had to choose a P_{c}to do the thermochemistry, and P_{b}is set by the altitude, or else 0 if vacuum.
Once γ, M

_{e}, PR, AR, and PR_{op}are all known, evaluating C_{F}is easy, per the above equation. If you have used a value of c* to size a throat A_{t}elsewhere in your fundamental ballistics, then the nozzle thrust is easily obtained as F = C_{F}P_{c}A_{t}. From ballistics, choked nozzle massflow w = P_{c}C_{D}A_{t}g_{c}/ c*, see Figure 2 below. C_{D}is the nozzle throat’s discharge coefficient (or efficiency).
If all the hot gas generated in the engine workings upstream
of the nozzle entrance goes through the nozzle,
then I

_{sp}= C_{F}c* / g_{c}.*If not, you must ratio down your calculated I*_{sp}, F, and A_{t}by 1 + f, where f is the fraction of generated hot gas massflow that does__not__go through the nozzle.

**Example Problem: Conventional Nozzle, Sea Level and 20 kft Designs**

I automated these calculations into a spreadsheet, and verified the numbers with hand
calculations. An image of the spreadsheet for the sea level
design is given in Figure 3. In
the spreadsheet, items highlighted
yellow are the user inputs, and items
highlighted blue are the principal outputs from the sizing calculations. These are used to generate the performance
table versus altitude, which is not
highlighted.

For this example, I
assumed Pc = 1800 psia, and a conical
nozzle of 15 degree half angle. I used
specific heat ratio γ = 1.20, and a c* =
5900 ft/sec so that specific impulse would be near 300 sec, similar to LOX-RP1. I used A

_{t}= 1.0 square inch, with a nozzle C_{D}= 0.99 to size flow rate. The resulting design is a nominal 3000 lb thrust design, completely immune to backpressure-induced separation, since it is never over-expanded. How the nozzle kinetic energy efficiency is calculated from half angle is discussed below.
Keeping all the data the same except for the design
backpressure, I ran the spreadsheet again
for perfect expansion at 20 kft instead of sea level. The effect is to increase the expansion ratio
for a higher momentum term, and then accept
the negative pressure difference term reducing thrust below 20 kft
altitude. The gas generating chamber and
throat are exactly the same. An image of the 20 kft design
spreadsheet is given as Figure 4.
The spreadsheet includes a separation backpressure estimate (see that
discussion just below), which shows the
risk starts at pressures about 9 psi larger than sea level air pressure. So,
this design is also very likely immune to backpressure-induced
separation risks.

**Flow Separation Risks**

These can only be estimated empirically. There are many correlations. My preferred one uses the inverse of PR = P

_{c}/P_{e}. P_{sep}is the estimated backpressure, at and above which nozzle flow separation is to be expected. It is empirical, and it is a rough estimate.__The designer should allow significant margin__.
P

_{sep}/ P_{c}= (1.5 P_{e}/P_{c})^{0.8333}
For the 20 kft design example just above, P

_{c}/P_{e}= 266.3, so that P_{e}/P_{c}= 0.003755. Thus P_{sep}/P_{c}= 0.013355, and for P_{c}= 1800 psia, the expected P_{sep}= 24.04 psia, quite a margin above sea level pressure. We can conclude that there is no risk of separation in the example nozzle, all the way down to sea level, where P_{b}is only 14.7 psia. The risky backpressure is even higher at about 45 psia for the sea level design.**KE-Efficiency Correlations**

Most conventional nozzles are axisymmetric. Those streamlines near the axis are aligned
along that axis, so that the cosine
factor for off-angle alignment is cos(0

^{o}) = 1.00. Those near the nozzle wall are aligned__at the angle of that wall off the axis__. For a conical nozzle, this is the half angle of the cone. The cosine factor for off-angle alignment is cos(a) where “a” is the half angle of the cone. See Figure 1 again.
Thus, there is a
distribution of local off-axis alignments for the streamlines across the exit
plane. While the “correct” way to
determine the

__effective cosine factor for the distribution__would be to integrate them for an average,**. Simply compute the arithmetic average of the centerline cosine factor value (1.00) and the wall cosine factor value cos(a), and call that the nozzle kinetic efficiency factor:***there is an easier model that is just as good***η**

_{ke}= 0.5 [1 + cos(a)] where “a” is the effective average half-angle of the nozzle wall

For a conical nozzle,
“a” is the cone’s geometric half-angle.
At 15 degrees, η

_{ke}= 0.983. For a curved bell, there is a local “a” near the throat, and a smaller local “a” at the exit lip. One simply averages the two local a’s, and uses that average as “a” in the kinetic energy efficiency formula. For most practical curved bell designs, that average “a” won’t be far from 15 degrees. See again Figure 1.**Free-Expansion Designs By “Last Point of Contact = Perpendicular Exit Plane Model”**

There are multiple techniques and geometries by which a
nozzle can be made self-compensating for perfect expansion at any altitude
backpressure. They all share two
features: (1) a free streamtube surface
unconfined by a physical shell before the “exit plane”, and (2) a point of last contact with physical
structure that is wetted by the propulsion stream that locates the exit plane.

*We want the components of the actual distribution of exhaust velocities, that are aligned with the engine axis.*
Most, if not
all, these free-expansion designs can be
analyzed for expected performance using the very same ideal gas compressible
flow techniques used for conventional nozzles.

**Note that the very same gas-generating chamber and throat area serves as the feed to the free-expansion “nozzle” at all values of P***It is just that the order in which things need to be done is revised.*_{b}.
Conceptually, we are
interested in an effective planar exit area located at the “point of last
contact” (just as the exit lip is the “last point of contact” with conventional
bell nozzles), and oriented perpendicular to the engine axis. This is shown in Figure 5.

Unlike conventional nozzles,
these are always perfectly expanded,
so that P

_{e}= P_{b},__as long as P__! Once a P_{b}is not exactly zero_{b}is known, then PR = P_{c}/P_{e}= P_{c}/P_{b}is known. One solves the PR equation for M_{e}at this value of PR, which is__not__a transcendental iteration, just a simple direct solution:
M

_{e}= { 2/(γ – 1) [PR^{(}^{γ-1)/}^{γ}– 1]}^{0.5}
With M

_{e}now known, find the area ratio from the streamtube relation, and use it with the throat area to find the effective value of the exit area A_{e}:
TR = 1+
0.5 (γ
– 1) M

_{e}^{2}
const1
= 0.5 (γ
+ 1)

const2
= 0.5 (γ
+ 1) / (γ
– 1)

AR = (1
/ M

_{e}) [TR / const1]^{const2}
A

_{e}= AR A_{t}
Referring
again to Figure 5, there is
obviously a distribution of streamline directions at the exit plane,

**.***which is different for each backpressure*__Each geometry is different__, but the idea is to find the largest half-angle off of axial and use it as “a”. This goes into the correlation for kinetic energy efficiency. That correlation is generally for “a” < 30 degrees,__so we are misusing this here!__But, it is the best I have at this time to offer.*Any such “a”-dependent model, even if flawed, is better than no model at all!*

Instead of a thrust coefficient, we estimate thrust directly from the
calculated exit plane conditions,
remembering that P

_{e}= P_{b}, and from that thrust, the thrust coefficient (to use with c* for I_{sp}):
F = η

_{ke}γ P_{e}A_{e}M_{e}^{2}
C

_{F}= F / P_{c}A_{t}
One should note that neither A

_{e}nor η_{ke}are constants here, as P_{b}changes. At high backpressures (low altitudes), “a” is small, η_{ke}is high, and A_{e}and M_{e}are lower. At low backpressures (high altitudes), “a” is quite large, η_{ke}is lower, and A_{e}and M_{e}are high.*Exactly how “a” varies is quite geometry-dependent.*
If P

_{b}= 0 (vacuum of space), PR = infinite, leading to infinite M_{e}and A_{e}. There can be no planar exit plane, and Prandtl-Meyer expansion says “a” > 90 degrees by a small amount.**, quite unlike a conventional nozzle! (Which means this free-expansion design approach is inappropriate in vacuum!)***There is no point trying to use this compressible flow analysis technique on a free-expansion nozzle in vacuum*
However, for an
axisymmetric center-spike design (aerospike nozzle), one could estimate a = tan

^{-1}[(R_{e}-R_{t})/L_{spike}]. For this, R_{e}= (A_{e}/pi)^{0.5}, and R_{t}= approximately (A_{t}/pi)^{0.5}. L_{spike}is the distance from throat plane to exit plane. Longer is lower effective “a”, but higher weight, and a tougher cooling design.
I made another worksheet in the spreadsheet for axisymmetric
aerospike nozzles, embodying the above
calculation techniques, and I verified
it with hand calculations. It lays out
differently, since the sequence is
different, and more items vary with
altitude. The same grouping of design point
data vs altitude performance is maintained,
and the same color-coding for highlighted items. However,
the volume of data is larger,
requiring two figures (vs one) to display herein.

**Example Axisymmetric Aerospike Problem**

The fairest way to compare this type of nozzle design with any
conventional nozzle design is to size both with the same P

_{c}, A_{t}, and γ. If thrust is the issue, and it usually is for launch vehicles, then the preferred performance variable to examine is thrust.
For the example problem,
we use P

_{c}= 1800 psia, A_{t}= 1.0 in^{2}, and γ = 1.20, same as the conventional nozzle examples earlier. The same c* and nozzle throat C_{D}are used.__In effect, this engine shares the very same gas generator as the two conventional examples__. The same altitude backpressures are also used, so that this design can be compared directly to the earlier examples,**.***except that vacuum performance cannot be included*
The spreadsheet results are given in Figures 6 and 7 below. The two figures together provide the image of
the spreadsheet. I have repeated the
altitude data in Figure 7 for convenience.

**Comparisons Among the Example Nozzle Designs**

How these designs compare,
especially as regards altitude performance, does not “jump off the page” from tabular
data. That takes plots, something this spreadsheet software offers. I used the same altitudes and air pressure
data for all 3 examples. Copying selected
data from each worksheet into yet another worksheet provides a way to directly
plot performance from all 3 nozzles on the same page. I did this for thrust, specific impulse, thrust coefficient, and nozzle kinetic energy efficiency.

Bear in mind that all three share the same gas generator at
P

_{c}= 1800 psia, A_{t}= 1 square inch, γ = 1.20, chamber c* = 5900 ft/sec, and nozzle throat discharge coefficient C_{D}= 0.99. All three are roughly the same 3000 lb thrust__at their design points__, within a percentage point or three.
The thrust
comparison is given in Figure 8 below.
The conventional sea level design has slightly better thrust at sea
level ( by about 82 lb out of a nominal 3000 lb) than the conventional 20 kft
design. This reflects the effects of the
negative pressure difference term at sea level, for the slightly-overexpanded 20 kft
design.

The 20 kft design has about a 107 lb thrust advantage, above 100 kft, over the sea level design. This reflects the larger expansion ratio of the
20 kft design, and the fact that the
exit momentum term dominates

__by far__over the pressure difference term, in thrust.
The axisymmetric aerospike design is “right in there” with
the other two, up to about 50 kft or 60
kft altitude. Then its performance drops
dramatically with increasing altitude,
something the free expansion is supposed to compensate! It is a little better than the conventional
sea level design at sea level, and it
remains superior all the way up to about 55 kft. It is equivalent or very slightly better to
the conventional 20 kft design at sea level,
and remains essentially equivalent to about 20 kft. Its downturn in thrust performance is quite
dramatic, and starts at about 40 kft or
50 kft.

It should not surprise anyone that the specific impulse trends in
Figure 9 tell the same tale as the thrust in Figure 8, since all three share the same gas generator
with the same propellant massflow. Nor
should it surprise anyone that the thrust coefficient trends in Figure 10 also tell exactly the
same tale, since all 3 designs share the
same gas generator operating at the same chamber pressure.

The reason for the dropoff in aerospike performance, versus the conventional designs,

**, something that in turn depends upon the effective average half-angle of the propulsion stream bondary. This is really nothing but the cosine factors of streamlines that are aligned off-axis. Kinetic energy efficiency trends are given in Figure 11.***traces directly to the trends of nozzle kinetic energy efficiency*
Remember, for the
conventional designs, half-angle is

__locked-in by the physical bell__, right up to the exit plane. Downstream of the exit lip, gas expands laterally into the vacuum, but this happens__downstream__of the “last point of contact”, where thrust is actually calculated. This is implied by how we draw the control volume about the engine and nozzle, something shown in the lower right corner of Figure 1,__touching at that last point of contact__.
For the axisymmetric aerospike free-expansion design, the last point of contact is the tip of the
spike. The free expansion surface of the
plume is

__inside__the control volume, as is the bell of the conventional nozzle. At high altitudes where the air pressure is low, the plume boundary must expand quite far laterally, between the throat, and the “exit plane” at the last point of contact. This is precisely how large AR and M_{e}are achieved, in order to match P_{e}= P_{b}. Since the length of the free-expansion zone is fixed, the boundary half-angle must be quite large at high AR. That reduces kinetic energy efficiency.
The two conventional designs share a constant kinetic energy
efficiency of 98.3%, as shown. The aerospike starts out slightly better at
99.1% (due to the choice of L

_{spike}used), but drops below conventional at about 20 kft, and falls ever more rapidly to only about 77.7% at 100 kft.__This traces directly to the effective half-angle__of the plume boundary between the throat, and the exit plane at last point of contact.
That is why I included a plot of the axisymmetric aerospike half-angle vs altitude as
Figure 12. Looking at this, please remember that half-angle is constant-with-altitude
at 15 degrees for the two conventional designs.
At 100 kft, cos(56.335

^{o}) = 0.5543. Averaging that with 1 inherently produces η_{ke}= 77.7%.**Conclusion**

I don’t
see any significant advantage to the free-expansion nozzle approach. The small performance improvement is
restricted to the lower atmosphere, and
this design approach is

__entirely inappropriate for use in vacuum__! The complications with cooling the spike outweigh any tangible performance benefits, which are low (unless you cheat by not accounting for the streamline divergence effects).**This conclusion is correct as far as it goes, but it is also incomplete. See the update just below the original 12 figures. I explored the free-expansion nozzle approach a lot further, in two different forms. It can be made to work better at high altitudes, but NOT in vacuum, and not really significantly better than a conventional nozzle sized at the highest altitude consistent with not flow-separating at sea level.**__Update 11-16-18__:Figure 2 – Modeling Nozzles with Compressible Flow

Figure 3 – Spreadsheet Image for 15 Degree Conical Nozzle As Sea Level Design

Figure 4 – Spreadsheet Image for 15 Degree Conical Nozzle As 20 Kft Design

Figure 5 – Analogous Procedure for Free-Expansion Designs

Figure 6 – Example Axisymmetric Aerospike Nozzle Results, Part A

Figure
8 – Thrust Comparison Among the 3 Designs vs Altitude

Figure 10 – Thrust Coefficient Comparison Among the 3
Designs vs Altitude

Figure 12 – Trend of Effective Half-Angle “a” for
Axisymmetric Aerospike Design

__Update 11-16-18__:

When I did the original article, I made the axisymmetric aerospike design at
the same spike length as the nozzle bell length for the 20 kft conventional
nozzle design. This made its performance
comparable to the 20 kft conventional design up to around 40 kft altitude, then above that, the ever-increasing streamline divergence
angles “killed” its nozzle kinetic energy efficiency, reducing its performance below conventional, at the higher altitudes.

This design approach essentially determined 20 kft as the
altitude at which the effective average boundary half-angle of the circular
cross-section free-expansion streamtube was 15 degrees, same as the 15 degree conical conventional
nozzles. The fixed input here is the
aerospike expansion surface’s length.
The area ratio (and expanded diameter) are determined by expanding to
local pressure. The difference between
exit and throat diameters (ignoring the spike),
divided by the spike length, is
the tangent of the streamtube half-angle,
presuming a conical shape.

I did not originally check separation pressures for the
conventional nozzles, but added that
later. As it turns out, for this study’s 1800 psia chamber
pressure, you do not want to attempt
conventional designs above about 30 kft.
This is because the margin between expected separation backpressure and
sea level pressure gets to be too small to trust.

To the three designs in the original article, I added a 30 kft conventional design, and a revised axisymmetric aerospike with a
much longer spike that has a 15 degree half angle at 100 kft altitude. These five designs (are depicted in Figure 13 (

**).***all figures at the end of this update*
There is a different idea about the free-expansion aerospike
geometry

**: I call it the “twin aerospike”. Instead of being axisymmetric with the spike immersed along the plume centerline, in the twin aerospike, the nozzle bell is conceptually cut away, from throat to exit, top and bottom, leaving two symmetrically-placed spikes along each side, at the physical bell half angle. This is shown in Figure 14.***that limits expanded half-angle better*

*I have never before seen a proposal like this; therefore,*__it is my idea__. Please give me credit for it, if you pursue it.
This alternate approach leads to an elliptical plume cross
section, with one diameter fixed by the
position of the twin spikes. The other
diameter is smaller at low altitudes,
and larger at high altitudes. The
effective average boundary half angle is the arithmetic average of the fixed
spike angle, and the variable angle
produced by plume diameter, throat
diameter, and spike length. The plume cross section is circular at the
design point, where the free expansion
half angle matches the physical spike half-angle.

I ran this twin spike design approach at a spike half-angle
same as the conventional bell (15 degrees),
and two spike lengths, one set by
a design altitude the same as the 30 kft conventional nozzle, the other 100 kft, same as the added axisymmetric aerospike
design. That makes a total of 7 designs
to compare, as given in the following
table:

**, in order to understand what happens to nozzle kinetic energy efficiency. That in turn governs the thrust and impulse performances that can be achieved.**

*It is very important to understand what happens to effective average boundary half-angle for these various designs*
Accordingly, the
first comparison plot (Figure
15) is effective average boundary half-angle vs altitude for the 7
designs.

**That item is plotted vs altitude in the second comparison plot (Figure 16) for all 7 designs. The other performance measures are thrust, specific impulse, and thrust coefficient, as given in Figures 17, 18, and 19.***Its cosine averaged with 1 is the nozzle kinetic energy efficiency.*
Looking at Figure 15, note that all
three conventional designs share exactly the same 15 degree half-angle. They fall right on top of each other, so only the 30k design is visible in the plot, being the last group plotted.

Both of the lower-altitude free-expansion designs show
similar upward trends of half-angle to about 70-80 kft, with only the twin aerospike 30k design trend
“bending over” with increasing altitude.
This is the beneficial effect of the elliptical cross-section shape, with one longitudinal section fixed at 15
degrees, and averaged with the other
that has a variable boundary half angle,
to meet exit area.

The two high-altitude free-expansion designs have the same
design altitudes of 100 kft, and very
similar trends of effective average boundary half-angle with increasing altitude. The differences attribute to the averaging of
effective boundary half-angle that takes place in the twin aerospike, but not in the axisymmetric aerospike.

I did not have available atmosphere data higher than 200
kft; indeed such is questionable, as we are heading into something more like
free-molecule flow, than continuum flow,
at such altitudes. Had there been atmospheric pressure data available
between 200 kft and 300 kft, the twin 100k
design would have “flattened out” the way the twin 30k design did. Neither axisymmetric design can do that.

What those half-angle data represent is the effective
average boundary values whose cosines average with 1 in the nozzle kinetic
energy efficiency correlation equation.
Those data for the 7 designs are given in Figure 16. Note
that all 3 conventional designs share a constant half angle of 15 degrees, and thus a constant nozzle efficiency of 98.3%. There is not much room above that for the
free-expansion designs to do “better” at low altitudes, although they certainly do.

However, above about
30-40 kft, the two low-altitude
free-expansion designs (“axi a 20k” and “twin 30k”) have efficiencies that drop
below conventional. These drop to around
80% efficiency, although at two
different altitudes. That difference is
an artifact of the two design altitudes,
and (more importantly) how the plume cross section forces the averaging
(or not) of the effective half angle. The
“twin 30k” design sort-of “bottoms out” around 80% efficient, while the “axi a 20k” design does not. We could “eyeball-extrapolate” the “twin 30k”
design to an efficiency near 80% at 300 kft (effectively out in space); the “axi a 20k” design’s efficiency evidently
continues to drop.

**The nozzle efficiencies in Figure 16 show both those designs with similar trends to 200 kft altitude. An “eyeball guess” says the twin 100k design might “bottom out” somewhere near 70% efficiency at 300 kft, while the “axi a 100k” design’s efficiency will probably just continue to fall precipitously.**

*Designing the free expansion nozzles at higher altitude (getting far longer spikes) is quite evidently a better deal*__from a fluid mechanics standpoint__.
Free expansion nozzle efficiencies are equal to, or very slightly better than, conventional,
all the way up to the design altitude;
and backpressure-induced separation is by definition no risk at
all, all the way down to sea level. Then, at
altitudes above the design point, the
nozzle efficiencies must inherently fall off,
according to one or the other behavior,
depending upon whether an axisymmetric aerospike or a twin aerospike. There is an altitude above which
free-expansion efficiency is less than conventional.

These half-angle and nozzle efficiency behaviors combine
with the fluid mechanics of expansion, to produce the thrust vs altitude curves of Figure 17.

*These show quite clearly that we do*__not__want the lower-altitude free-expansion designs, since by about 40 kft,**. The 30k twin aerospike design does better than the 20k axisymmetric aerospike design, mainly because of the half-angle averaging, but neither maintains equal or better performance than conventional, past 50 kft “for sure”.***the conventional nozzles outperform them, just as I said in the original article*
The two high-altitude free-expansion designs do indeed show
equal or slightly superior performance to the conventional designs, but only up to about 170-180 kft
altitudes. They “peak” at their design
altitudes of 100 kft. The “axi a 100k”
design’s performance will fall precipitously past 200 kft, while the “twin 100k” design’s performance
may (or may not) “bottom out” well below conventional performance levels, at 300 kft.

This outcome also shows in the delivered specific impulse
and thrust coefficient trends of Figures 18 and 19. That
is because “all else really is equal”,
in particular chamber pressure,
throat area, c* velocity, and thus nozzle mass flow rate.

Could we “push” the free-expansion advantage to higher
altitudes still? Probably, by making the effective average boundary half
angle = 15 degrees at higher altitudes than 100 kft. The cost,
as indicated in Figures
13 and 14, is very long spikes
indeed!

*There is no fluid mechanical optimum here!*
But,

**from both the constructional, and the thermo-structural, viewpoints. It is***increasingly-long spikes are increasingly infeasible*__that__trade-off,__not fluid mechanics__, which determines whether either of the free-expansion design approaches is “better” than a conventional nozzle design. And, as shown in Figures 17-19, the performance advantage of the free-expansion design over a 30 kft conventional design is never very large at all!__Just as the original article conclusion indicated__.**Updated Conclusions**

**Free-expansion designs can NEVER outperform conventional designs in vacuum, just as originally concluded. The trends are just wrong to support such a conclusion.**

*The conclusions in the original article are correct, but a bit incomplete.*
Designed at sufficiently high altitude, the performance of free-expansion designs can
be made to equal, or to slightly exceed,
conventional designs up to some fairly-useful
altitude in the 200-300 kft range. But, there will always be a performance penalty to
pay in full vacuum!

This performance gain at some altitudes with free-expansion
designs comes at a cost: an ever-longer
expansion spike (or spikes). These can
easily be infeasible for constructional,
or thermo-structural reasons.
Fluid mechanics

__does not__limit this!
Of the two free-expansion designs considered here, the twin aerospike approach offers somewhat better
performance potential than the axisymmetric aerospike. This is because of the way the
fundamentally-elliptic cross-section of the twin aerospike forces the average
boundary half-angle to behave, relative
to the fundamentally-circular cross-section of the axisymmetric aerospike.

Two-dimensional linear aerospike geometries were not
included in this study. However, their performance characteristics should fall
within the bounds of the axisymmetric aerospike and the twin aerospike. There will be no “breakthroughs” with the 2-D
linear form, relative to the other two.

Conventional nozzle technology is well-established and has
been flying for over a century. Free-expansion
nozzle designs (of any kind) have never been flown

__up to this time__. They thus cannot be considered a well-established technology, the failed X-33 program notwithstanding.
So, for the time
being, my personal recommendation is
just continue with conventional nozzles,
designed to (at most) about 30 kft perfect expansion, as long as the somewhat lower sea level
thrust is tolerable, relative to a sea
level-expanded design. If not, reduce the design altitude, and accept the penalty at high
altitudes. Simple. Effective.
Well-proven. “KISS”, which means “Keep It Simple, Stupid”.

Figure
13 – Pertinent Dimensions and Conditions for Conventional and Axisymmetric
Aerospike Designs

Figure 14 – Pertinent Dimensions and Conditions for Twin Aerospike Designs

Figure 15 -- Comparison of Effective Half-Angle vs Altitude for the 7 Designs

Figure 16 -- Comparison of Nozzle Kinetic Energy Efficiency vs Altitude for the 7 Designs

Figure
17 – Comparison of Thrust vs Altitude for the 7 Designs

Figure 18 -- Comparison of Specific Impulse vs Altitude for the 7 Designs

Figure 14 – Pertinent Dimensions and Conditions for Twin Aerospike Designs

Figure 15 -- Comparison of Effective Half-Angle vs Altitude for the 7 Designs

Figure 16 -- Comparison of Nozzle Kinetic Energy Efficiency vs Altitude for the 7 Designs

Figure 18 -- Comparison of Specific Impulse vs Altitude for the 7 Designs

Figure 19 -- Comparison of Thrust Coefficient vs Altitude
for the 7 Designs

Figure A – Geometry and Ballistic Analysis Assumptions

Figure
B – Effective Average Half-Angle Trends vs Altitude

Figure C – Effective Nozzle Kinetic Energy Efficiency Trends vs Altitude

Figure D – Thrust vs Altitude Trends

Figure E – Specific Impulse vs Altitude Trends

__Update 1-26-19__:
In response to questions from a friend, I revisited the nozzle performance
comparisons with one more free-expansion design: an axisymmetric aerospike designed at 150 kft
altitude. I already had an axisymmetric
100 kft design, and a twin aerospike 100
kft design, both of which performed very
much better at very high altitudes, than
the corresponding free-expansions designed at lower altitudes.

These numbers in this article and its updates were run for a
US 1962 standard day atmosphere table that I happened to have, that only extends to 200 kft. It’s not quite a hard vacuum at 300 kft, but it is getting fairly close. The difference is essentially irrelevant for
a conventional bell nozzle, but not for
the free expansion designs. So, I added the axisymmetric 150 kft design to
see what happened to it at 200 kft. This
verifies the trends. A 200 kft design is
possible, but will have an even longer
spike (see below).

I eliminated all the conventional designs except the 30 kft
bell, for better clarity in the
plots. That design expands perfectly at
30 kft on a standard day, while still
being unseparated at sea level. It has
the best average performance, and with a
very modest-length bell of normal expansion ratio. Because of the negative backpressure
term, its sea level thrust is lower than
for a bell optimized there, but is quite
comparable to the sea level performance for all the free expansion
designs.

The way all of these rocket engines were analyzed is at
constant chamber pressure Pc = 1800 psia,
with a constant chamber c* = 5900 ft/sec, a fixed throat area A

_{t}= 1 square inch, at a fixed throat discharge coefficient C_{D}= 0.99. Whether conventional bell or free-expansion, the streamtube area ratio at design is such that expanded pressure equals ambient atmospheric pressure at the design altitude.
The conventional bell is conical at fixed 15 degree half
angle. That geometry sets its
length, which is quite short in
comparison to the spike lengths of the high-altitude free-expansion designs. The free-expansion designs had their spike
lengths set such that the effective average half-angle of the free-expansion
plume boundary was 15 degrees at design.
The details of exactly how that geometry was estimated is in the
original article above. For these
high-altitude free-expansion designs,
those spikes are very long indeed.
Whether such long designs are practically achievable thermally and
structurally is NOT addressed here. Only
the fluid mechanics is considered.

The axisymmetric aerospikes feature a circular cross section
geometry, while the twin aerospike has
an elliptical cross section geometry;
thus they are different in geometry analysis and altitude behavior. Because the twin is such a novel
geometry, I didn’t do a 150 kft version
of it. Most folks’ perceptions are more
accustomed to the axisymmetric version,
or its very close cousin, the 2-D
linear aerospike. However, they are all
more-or-less comparable in overall performance.

Figure A shows a direct comparison of the conventional and
free-expansion engines. All share EXACTLY the SAME chamber and throat
areas. Note the 10 inch bell versus the
45 to 108-inch long spike geometries. For the record, the original lower-altitude
axisymmetric aerospike design featured an 8-inch spike at about 20 kft. The original lower-altitude twin spike design
featured a 10.44 inch spike at 30 kft.
These expansions compare quite closely with 20 kft and 30 kft design
conventional bells, actually.

Figure B shows the results of the expanded plume geometries
in terms of effective average half-angle “a” that goes into the kinetic energy
efficiency as nke = 0.5*(1 + cos(“a”)).
The conventional bell is, of
course, the constant one, at 15 degrees. The two 100 kft free expansion designs show
slightly-different but very comparable trends that rise rather rapidly to near
50 degrees at 200 kft. The
higher-altitude / longer spike 150 kft axisymmetric design lowers that to
nearer 30 degrees at 200 kft, but it is
still obviously rising rapidly.

Figure C shows the resulting trends of nozzle kinetic energy
efficiency nke, which is the multiplier
upon the exit momentum term in thrust.
It does NOT multiply the pressure-difference term in the conventional bell, while the free-expansion designs do not have
a pressure-difference term in thrust.

Figure D shows thrust versus altitude for all these four
designs (conventional 30 kft design,
axisymmetric aerospike 100 kft design,
twin aerospike 100 kft design,
and axisymmetric aerospike 150 kft design). All are quite comparable near sea level and
up into the stratosphere. The
free-expansion designs are very slightly “better” than the conventional design
in that range of altitudes.

Above about 60 kft and onward to around 200-250 kft, the free-expansion designs are significantly
better than the conventional design.

*Bear in mind that if the very long spikes are not practically achievable, then this advantage cannot be realized!*
Note however that as the altitude heads toward 300 kft = 90
km = “space”, that the free expansion
trends are increasingly downward. Those
trends are quite clear: out in hard
vacuum, the conventional bell gets
better thrust performance with much shorter (and more practical) expansion
structures.

Figure E shows trends of specific impulse (Isp) versus
altitude for all four designs. These
tell exactly the same story as the thrust data.
There is a significant benefit to be had with free-expansion nozzles
between somewhere near 60 kft and somewhere around 200-250 kft. This advantage is ONLY attainable IF you can
practically achieve the VERY LONG expansion spike lengths! No free expansion designs ever tested so far have
ever had anything but very short expansion spikes!

Figure F shows the trends of effective thrust coefficient vs
altitude for all four designs. These
curves tell the very same story as thrust and specific impulse.

*My overall conclusions remain the same as they have been, in both the original article and the first update.*
There is an advantage of free-expansion spike nozzles over
conventional bell nozzles in the higher stratosphere, but ONLY if the

__required very long spikes__are really practically-achievable designs.
If only short spikes are feasible, that advantage goes away, as the 20 kft and 30 kft designs in the
original article so very clearly show.

And,

**. That is because its exit streamline directions are well-collimated, while those of the free-expansion designs inherently cannot be.***out in hard vacuum, the conventional bell is just inherently superior*Figure C – Effective Nozzle Kinetic Energy Efficiency Trends vs Altitude

Figure F – Thrust Coefficient vs Altitude Trends

__Update 1-31-19__:

I updated my spreadsheet file and its user manual to a
second version. These are rocket
nozzle2.xlsx and user2.docx. In the
spreadsheet file, there are fewer
worksheets, one each for the three
nozzle types, and one for comparison
plotting. The twin aerospike nozzle
model was revised to input half angle at design, and use that to solve for the tip spike
spacing Dspike.

With both the twin and axisymmetric aerospikes, the input for Lspike is the analog to design
backpressure with the bell. One varies
spike length to obtain a certain half angle at the desired design
altitude.

With all three models,
a generic study would begin with At = 1 sq.in. Vary that to enforce exactly the sea level
thrust that you desire. If you do adjust
At, then for the aerospikes, go back and readjust Lspike to be
compatible. It varies proportional to
throat diameter.

**Yet Another Study: “Best Long” Designs**

Making use of the previous results, generic nozzles were sized starting from At =
1 sq.in, with Pc = 1800 psia and chamber
c* = 5900 ft/sec. The conventional 15
degree half-angle conical bell was designed at 30 kft to get best compromise
performance from sea level to hard vacuum.
Both aerospikes were designed for effective average half-angle 15
degrees at 100 kft expansions, to get as
good a very high-altitude performance as seemed practical.

**All three had their throat areas At resized slightly, so as to provide exactly 3000 lb thrust at sea level.***These produce quite long expansion spikes compared to the conventional bell length, hence the terminology “long”.**See Figure I below.*

These results gave effective average half-angles versus
altitude as

**. Of course, that trend is constant with the conventional bell. With the two aerospikes, it varies quite strongly, reflecting the perfectly-expanded condition, although with slightly-different curve shapes. These differences reflect the circular plume shape for the axisymmetric design, versus the elliptical plume shape for the twin design. Those half-angle trends produce the nozzle kinetic energy efficiency trends***shown in Figure II***. Again, the aerospike designs show varying efficiencies, versus the constant efficiency for the conventional bell.***shown in Figure III*
The trends with altitude for thrust, specific impulse, and thrust coefficient are

**. They all tell exactly the same story. Not shown is expanded Mach number. This is fixed for the conventional bell, and very variable for both aerospike designs.***given in Figures IV, V, and VI***Conclusions:**

The conventional bell produces slightly better thrust below
the “long” aerospike design altitude of 100 kft, with the aerospike thrusts increasingly worse
than the bell above their design altitude.

The “long” aerospike designs confer a specific impulse
advantage up to about 160 kft. It is
slight at low altitudes, and more
significant near design, then
drops: more advantageous for TSTO than
SSTO.

Figure VI – Thrust Coefficient Trends With Altitude, “Long”

**Still Another Study: “Short” Designs**

For the “long” designs,
the expansion spikes were approximately 4 times longer than the conical
conventional bell. Spikes that long may
not be practical from a thermal or structural standpoint, or even a practical geometric envelope
standpoint.

Accordingly, I took
exactly the same analyses and put in the short spikes for a nominal 30 kft
design altitude. These spikes are pretty
close to the same length, but no longer
than, a conventional bell. These are the “short” designs.

Again, I resized the
generic At = 1.0 sq.in slightly to produce exactly 3000 lb of sea level thrust. The axisymmetric and twin aerospike designs
were then compared to exactly the same 30 kft design conventional bell as
before.

*See Figure VII below.*

These results gave effective average half-angles versus
altitude as

**. Of course, that trend is constant with the conventional bell, as before. With the two aerospikes, it varies quite strongly, reflecting the perfectly-expanded condition, although with slightly-different curve shapes. These differences reflect the circular plume shape for the axisymmetric design, versus the elliptical plume shape for the twin design. They are larger, because the spikes are very much shorter, for the same expanded exit area.***shown in Figure VIII*
Bear in mind that effective half angles above about 60
degrees are essentially “junk”, as the
conical plume expansion presumption breaks down (the geometry just gets
ridiculous). Those half-angle trends
produce the nozzle kinetic energy efficiency trends

**. Again, the aerospike designs show varying efficiencies, versus the constant efficiency for the conventional bell, just not quite as good as the “long” designs, especially at very high altitudes.***shown in Figure IX*
The corresponding trends with altitude for thrust, specific impulse, and thrust coefficient are

**. They all tell exactly the same story. Not shown is expanded exit-plane Mach number. This is fixed for the conventional bell, and very variable for both aerospike designs.***given in Figures X, XI, and XII***Conclusions:**

The conventional bell produces significantly better thrust at
every altitude above sea level than either “short” aerospike design. At
very high altitudes, the twin does
better than the axisymmetric.

The “short” aerospike designs confer a slight specific
impulse advantage up to about 40 kft. It
then drops, the axisymmetric
strongly, the twin less so, but still below the bell.

**Final Observations:**

The
aerospike designs are better-suited to first stages of TSTO designs. The “long” designs are better, but the “short” designs are still almost competitive. The bell nozzle is far superior for the
second stage, because its performance is
superior in hard vacuum. This same
effect says the bell is better for SSTO.

Figure IX – Nozzle Kinetic Energy Efficiency Trends With Altitude, “Short”

Figure XII – Thrust Coefficient Trends With Altitude, “Short”