This article is intended to acquaint the nontechnical or
non-specialized person with the basics of compressible flow in rocket nozzles, and how they are sized for rocket engines and
rocket vehicles. Scope here is limited
to only conventional bell nozzles.
The author had two 20-year careers, the first in aerospace/defense new product
development engineering. He is
qualified! The second was mostly
teaching at all levels from high school to university, but with some civil engineering and aviation
work, as well.
Nontechnical and non-specialized people have difficulty with
this topic, because the behavior of
supersonic compressible gases is quite foreign to their experience. This paper attempts to address that, as simply as is possible, so that the behavior is not so foreign as to
obscure what is needed to do the figuring.
A familiarity with high school-level algebra is the only
math required! There are spreadsheet
tools to do this kind of figuring, but
it really helps for the user to understand what the spreadsheet is actually
doing for him or her. That’s how to
detect input problems.
The converging-diverging passage of a rocket nozzle is quite
unlike the garden hose sprayer gun or spray nozzle that most people are
familiar with! On the subsonic side of
the throat, behavior is familiar, because the flow accelerates in speed (and the
pressure drops) as the passage narrows.
It is the supersonic side, downstream of the min-area throat, where behavior is quite unlike common
experience. The supersonic flow
continues to accelerate in speed (with further drop in pressure) as the
passage grows larger! Other
than that, the nozzle works to expel a
fast jet that creates thrust, by means
of the large pressure drop from upstream of the nozzle out to ambient
conditions. That is grossly the same as
the garden hose sprayer experience,
actually. Just the
supersonic-side details are different!
The gases cool off as they accelerate in speed, because the sum of the heat energy and the
flow kinetic energy, anywhere in the
nozzle, is a constant that pretty much
matches the heat energy of the almost-stationary gases upstream of the
nozzle. Energy conservation is not all
that unfamiliar a concept, even for
nontechnical people.
Everyone is familiar with the reaction “thrust” of a garden
hose, or more especially that of a small
fire hose. That thrust is the momentum
of the ejected stream of water. The
faster it moves, the bigger the thrust. The more water is ejected, the bigger the thrust. Simple!
In the compressible nozzle, the momentum of the ejected stream of gas is also part of the reaction thrust, but there are pressure forces that are also part of the thrust, unlike the water hose. The pressure of the gas just as it leaves the supersonic nozzle exit can be quite different from the surrounding atmospheric pressure! The thrust of a rocket nozzle is the sum of the exit momentum and the pressure forces at the exit plane. See Figure 1.
Figure 1 – Fundamentals of Compressible Flow Nozzles
That difference in pressure between the exiting gas stream
and the surrounding atmosphere leads to some behaviors of such nozzles, that would otherwise look incomprehensible to
the nontechnical or non-specialized person.
There is a force associated with the exiting gas pressure that adds to
thrust, and a force associated with the
surrounding atmospheric pressure that subtracts from thrust. Both of these pressures act upon the flow
cross section area right at the exit plane.
As shown in Figure 2,
when the exiting gas pressure is greater than atmospheric, we say that the nozzle is
“underexpanded”, since expansion to just
the right size larger exit area, would
reduce that exiting gas pressure to exactly atmospheric, while at the same time increasing its speed still
further. This is the leftmost image in
the figure, corresponding to a high
chamber-to-ambient pressure ratio. The
exiting plume actually spreads out wider after leaving the nozzle exit, because its pressure really is higher than
atmospheric.
For the same geometry,
the “perfect expansion” to atmospheric pressure, at a slightly lower chamber-to-ambient
pressure ratio, is the second image in
the figure. That exiting plume neither
spreads wider, nor does it narrow, after leaving the nozzle exit! The pressure forces add to zero, leaving only the momentum component of
thrust.
The third image shows what happens when the exiting gas
pressure is lower than atmospheric, but
not by too much. We call this “overexpanded”, because at this pressure ratio, we would need less expansion of the nozzle
passage than we have, to bring the exit
pressure back up to equal to atmospheric. The plume actually does contract
some, after leaving the nozzle exit! Under certain circumstances, the indicated oblique shock waves from the
exit lip can actually be seen, often as the lead in a series of “shock
diamonds”.
Figure 2 – Nozzle Behavior as Chamber to Ambient Pressure
Ratio Reduces
Where one gets into trouble is illustrated in the 4th
and 5th images in the figure,
where the chamber-to-ambient pressure ratio is too low for proper
operation. The oblique shocks first
coalesce into a normal shock wave at the exit plane, then move a bit upstream, causing flow to separate-off of the inner
wall of the nozzle! The lower the
chamber-to-ambient pressure ratio, the
further upstream this shock-separation phenomena moves! Flow downstream of a normal shock is always
subsonic (meaning very low speed), so
there is very little thrust, once the
shock is inside the nozzle and separating the flow from the wall.
The “trouble” one gets into is called “shock-impingement
heating”. Where the shock wave hits the
nozzle wall (causing flow separation),
there is a large but very local amplification of the rate at which heat
is transferred from the hot gas to the cool wall. The nozzle can actually burn through and
fail, in a matter of only several
seconds, when this happens!
The last (rightmost) image in the figure shows what happens
when the atmospheric backpressure exceeds about 50-some percent of the chamber
pressure. The throat “unchokes” (goes
subsonic), and flow throughout the
nozzle is subsonic. There is no useful
thrust when this happens. There is
almost no useful thrust even when choked,
if shock-induced flow separation occurs.
There is none when unchoked.
Most rocket engines have a set of turbopumps, with pre-burners of one sort or another to
create modestly-hot gases at high pressure,
which then get used to drive those turbopumps. How this is done varies, and is what we call the “cycle” of the
engine.
Those details do not matter to the functioning of the
rocket nozzle! All that stuff up to
the chamber right before the nozzle entrance is just a “hot gas generator” that
feeds the nozzle. It is the nozzle
that creates the thrust and its associated performance with that hot gas.
The only effect of the engine “cycle” upon that nozzle
behavior is whether-or-not any of the turbopump drive gas gets dumped
overboard, without going through that
nozzle! That does reduce the
performance , even at the same thrust! This is indicated in Figure 3, among several other things.
Figure 3 – How the Engine and Nozzle Work Together to Create
Thrust
There are a couple of nozzle efficiency factors that depend
upon the exact geometry going through the nozzle. The effective flow area of the throat is
slightly smaller than its geometric area,
because of “boundary layer displacement” effects. This can be held to a minimal difference, by using a smooth profile curve through that
throat, from ahead to downstream. Effectively,
you just need the profile radius of curvature to be about the same as
the throat diameter, in order to get a
good, high discharge coefficient CD.
The shape of the supersonic expansion passage, called the nozzle “bell”, influences something called the nozzle
kinetic energy efficiency factor ηKE. Curved bells,
like that illustrated, have half-angles
that are large near the throat, and
smaller near the exit lip, which need to
be averaged. Simple conical bells have only
the one half-angle. Curved bells
require careful design using a computer program that does something called
“method of characteristics” analysis.
Conical bells of half-angle equal to the average curved-bell half
angle, have exactly the same kinetic
energy efficiency, but are only somewhat
longer than the curved bell. They
require no complicated analysis in order to lay out a design!
This ηKE factor measures the effect of having
many of the exiting streamlines oriented not exactly aft. There is a very simple empirical estimate of
this efficiency, computed with the bell
average half angle, as shown in the
figure. It applies to the momentum
component of thrust, but not to the
pressure-forces component of thrust.
One does need to address the subsonic contraction area ratio
from chamber to throat! If this is not
large enough, the flow Mach number at
the nozzle entrance may be too high to use the measured chamber pressure as if
it were the “total” or “stagnation” pressure for the nozzle flow. There is a simple correction factor to
increase measured chamber pressure slightly,
in order to have exactly the right “total” pressure for the nozzle
thrust analysis.
Note in the figure that there is a nozzle massflow, that depends upon both throat
geometric area and its discharge coefficient. That massflow may not be the massflow
actually drawn from propellant tankage,
if there is dumped bleed from the turbopump drives! It is the massflow drawn from tankage that
affects rocket vehicle masses, so
for “rocket equation estimates” of vehicle performance, the specific impulse needs to be computed
from thrust using that total massflow,
not just the nozzle massflow!
The other factor affecting calculation of the nozzle
massflow is the “chamber characteristic velocity”, usually denoted as “c*”. That will be discussed below. Just be aware that experimental values are far
more reliable than theoretical thermochemical estimates.
In figuring all these things out, one needs to be aware that there are two
different design applications, each
with its own sizing methods. Those are
“vacuum design”, for use outside the
atmosphere, and “atmospheric
design”, for use down in the
atmosphere. They are done differently
using the same basic math, just in a
different sequence and with different constraints. See Figure 4.
We start by determining the “right” nozzle bell area
expansion ratio. For the vacuum
case, this number is assumed from the
outset! For the atmospheric design
case, this is determined by the pressure
ratio at the exit, in one fashion or
another. There are actually 3 distinct
options to do atmospheric design sizing.
Be aware that the very same math will analyze the chamber to
throat contraction for us, determining
whether we need to factor-up the chamber pressure measurement.
Figure 4 – Both Streamtube and Ratio Analyses Get Used First
Vacuum sizing is done to a presumed max expansion ratio,
limited only by having the engine (or engines) actually fit behind the
stage. There is simply no such thing
as a “vacuum-optimized” design!
Everything about it is constraint-driven, and constrained even more if gimballing is
needed for thrust vector control. See Figure
5.
Figure 5 – Essentials of Vacuum Nozzle Sizing
Atmospheric nozzle sizing is done in one of three distinct
ways, starting with appropriate ratios
of expanded pressure to chamber total.
These all use the same math, just
not in quite the same ways. This is
shown in Figure 6.
Figure 6 – Three Options for Atmospheric Nozzle Sizing
The option on the left in the figure is “standard” sea level
perfect-expansion sizing. One knows a
suitable chamber pressure. One
assumes the expanded exit plane pressure to be exactly equal to sea level
atmospheric pressure. That sets
the ratio of expanded pressure to total.
From that comes the exit Mach number, and from that, the expansion area ratio. These three items (and a nozzle kinetic
energy efficiency) are needed to get a thrust coefficient, in turn a way to book-keep where the thrust
comes from.
Sea level nozzles have good thrust at sea level, but their thrust does not increase much, as you climb to higher altitudes. Which in turn means the specific impulse does
not increase very much with altitude. They
typically have rather low area expansion ratios.
If you want to average a higher specific impulse as you
climb to much higher altitude, you can
obtain it by sizing the expansion ratio to a higher altitude’s ambient
pressure (top right), or by sizing
the nozzle to incipient separation at sea level (bottom right). The penalty you pay for that higher average
specific impulse during ascent, is lower
thrust right at sea level, at
liftoff, when weight is largest! So,
the design point selection is a tradeoff! These do have somewhat larger area expansion
ratios.
Clearly the compressible streamtube analysis math is crucial
to running numbers for a nozzle. This streamtube
math is illustrated in Figure 7.
Figure 7 – Compressible Streamtube Analysis
This is the analog to V1A1 = V2A2
in incompressible flow, that many people
have actually heard of, or actually even
used. The equation is different, but it is the same fundamental idea! However,
everything is figured relative to the choked min area at the throat.
One case of interest is finding the area ratio from a known
Mach number. That is a direct
solution. Just fill in the formula
items, starting with the gamma
constants.
The other case of finding Mach number from a known area
ratio has no direct solution, because
the equation is what they call “transcendental” in Mach number! It is impossible to isolate Mach number in
the equation, because it appears in two
places under very different mathematical circumstances (different exponents and
functional forms).
For that case, there
is only the iterative (trial-and-error) solution. Keep guessing Mach numbers until the equation
result is the area ratio you really want.
That is where spreadsheet-assist is so useful: it makes such iteration very easy, boiling down to just inputting the guesses in
one cell and looking at the result in another cell.
The other piece of this math is the set of compressible flow
ratios, static vs total (or
stagnation). Those are shown in Figure
8. These are “reversible” in the
sense that a known Mach number gets you all the ratios, and a known pressure ratio can be solved
directly for a Mach number. The basic
math here is based on total/static ratios,
but their inverses are what we need for thrust coefficient and flow
separation. Those inverses are included.
Figure 8 – The Compressible Flow Ratios
We use the thrust coefficient form of this math to
separate the variables, allowing
expansion ratio to be determined before actually sizing dimensions to meet a
thrust requirement. You cannot do
that, working directly in the primitive
variables! That thrust-sizing math based
on thrust coefficient is shown in Figure 9.
Thrust coefficient has two components, the vacuum thrust coefficient, and a correction term that reduces it
somewhat to the thrust coefficient down in the atmosphere. The vacuum thrust coefficient is actually
independent of the specific value of chamber pressure! The correction term depends directly upon
chamber pressure and ambient pressure,
so that the down-in-the-atmosphere thrust coefficient is also dependent
explicitly upon them, as well.
Once you know the thrust coefficient, you can use it, your intended chamber total pressure, and a thrust requirement, to find the geometric throat area. Knowing the expansion (and contraction) area
ratios, lets you define those chamber
and exit areas from that throat area!
Very simple, actually.
Once you have a throat area,
you can compute nozzle massflow,
adjust it to total, and compute
specific impulse. That is discussed
below.
Figure 9 – Thrust Coefficient Math Equations
The math for thrust-based sizing is a bit more complicated than simple performance of an already-sized configuration. This is shown in Figure 10.
Figure 10 – Thrust Requirement-Based Sizing of Dimensions
and Flow Rates
By definition of the thrust coefficient, thrust is the product of thrust
coefficient, chamber total
pressure, and geometric throat area! Once you have a thrust coefficient
defined, you can size throat area from a
required thrust value and your chamber pressure. The contraction and expansion ratios then
size those areas from your sized throat.
You can use the sized throat area, your chamber total pressure, the c* model for your propellant at that
pressure, and your throat discharge
coefficient, to size the nozzle
massflow. That and the dumped bleed
fraction define your total massflow drawn from tankage. In turn,
that and your sizepoint thrust define your sizepoint specific
impulse.
Computing performance of an already-sized system is even
easier, as is also shown in the
figure. You compute the thrust from
thrust coefficient at that altitude and your chamber pressure, and you already know the total flowrate at
that pressure. Thrust at altitude divided
by total massflow rate is specific impulse at that altitude. Very simple indeed!
Be aware that all of these estimates are computed
assuming there is no shock-separation going on in the nozzle bell! So, you
must check for that! If it
occurs, your calculated performance data
are no good! Do not use them!
The math predicting flow separation is an old correlation
from designing tactical missile rocket nozzles.
It is slightly conservative. The
math is given in Figure 11.
Figure 11 – Math for Dealing With Flow Separation
The use of this empirical correlation is quite
straightforward when computing performance vs altitude at any given throttle
setting. The nozzle expansion has a
fixed ratio of exit plane pressure to chamber total pressure. From that,
the correlation determines the ratio of separation backpressure to
chamber total pressure. That ratio and
your operating chamber total pressure,
give you the value of backpressure that will risk inducing flow
separation.
If your ambient atmospheric pressure is less than, or just equal to, the separation pressure, no separation occurs and your thrust and performance
estimates are good. If your ambient
atmospheric pressure exceeds the separation pressure, shock-separation will occur, and your thrust and performance estimates are
no good! Simple as that!
When sizing an atmospheric nozzle for incipient separation
at sea level (as discussed above), you
use the empirical correlation in reverse (which is also shown in the
figure). You know a suitable value of
your chamber total pressure, and you
literally set the separation pressure equal to sea level atmospheric
pressure. Their ratio determines the
expansion pressure ratio exit-to-chamber for your design process. That gets you a Mach number, and from that, the expansion area ratio. From them,
thrust coefficient is easy to find.
You have to think about your rocket vehicle and where it is
flying, to determine suitable thrust
requirements. Some items typical of
launch vehicles are given in Figure 12.
Cases do vary, though!
Figure 12 – Typical Considerations for Thrust Requirements
For launch vehicles, you need to accelerate the vehicle at half a
standard gee or more, above the
retarding effects of drag and the pathwise weight component. Such would apply at vehicle masses
appropriate to stage ignitions, where
vehicle weight is high. The half-gee
figure is only a rule-of-thumb minimum.
If you achieve lower, you will definitely
“dawdle around” at low speeds near the launch pad burning off lots of
propellant, without it actually buying
you very much in the way of flight speed.
Higher gee is better, but that
requires more thrust, and the engines
might not fit behind the stage. It’s a
tradeoff!
Vehicle acceleration also has max values, especially if crewed, but a lot of potential payloads have similar
acceleration limits. Those limits might
be roughly in the 4 to 6 gee range. You
can always turn off some engines while throttling others, to stay within such limits. They would occur when vehicle masses are
low, near stage burnout.
The nozzle massflow equation uses characteristic velocity c*
as the denominator. You must have a
suitable model for this value,
consistent with your propellant combination and design chamber
pressure. In the real world, c* is weakly dependent upon chamber pressure
as a power function. This is shown in Figure
13.
Figure 13 – About Chamber Characteristic Velocity (c*)
You must run a thermochemical code (computer program) on
your propellant combination at your intended chamber pressure, and your intended oxidizer-to-fuel
ratio, to determine the resulting
combusted chamber temperature and gas properties. These are theoretical values, and from them a theoretical c* can be
computed with the equation shown. It
will have a very weak power-function dependence upon chamber pressure.
In the real world,
delivered test c* is always a little less than the theoretical
value, by a factor we call the “c*
efficiency”. This factor also typically
has a weak power-law dependence on chamber pressure. Therefore,
the actual experimental delivered c* is best modeled as a weak power-law
dependence upon chamber pressure, with
an exponent that is usually crudely in the vicinity of 0.01. All of this is
shown in the figure.
In Figure 14 is a table of values for PR = Pt/Pc vs
Mach number, including values for Ac/At, created with the usual factors, for gamma = 1.2 as “typical”. Plotting Pt/Pc vs Ac/At reveals the
importance of allowing for non-zero Mach number at station c (the aft “chamber”, right before the nozzle entrance).
Pc is always measured on real engines as a simple static
pressure tap. This is the total pressure
fed to the nozzle only if the Mach number of the flow in the chamber is trivially
close to zero. For the recommended
and often-observed Ac/At ratios, this
Mach number is simply not trivial, so
the Pt/Pc ratios are not trivially close to 1!
The error incurred by using Pc as Pt would seem to range from about 1%
to 6%. Pt = Pc * (Pt/Pc for the Ac/At
ratio).
Figure 14 – Why Accounting for the Contraction Ratio Is
Important
For an engine with a maximum nominal chamber pressure of
around 3000-4000 psig in test, one might
select a pressure transducer of nominal 5000 psig capability at the very
least, which might have an accuracy of
0.25% of full scale. That would be an
expected error of 12.5 psi. That is the
inherent uncertainty, within which one simply
cannot distinguish measurements.
For 3000 psig Pc,
that 12.5 psi is 0.42% error, and
for 4000 psig, it is 0.31% error. Most of the Pt/Pc error factors in the figure
are very much larger than that, so
correcting for Pt/Pc before doing a nozzle analysis, really is crucial for getting accurate
results!
To Sum Up
Everything shown here is basically pencil-and-paper
calculation stuff, using the algebra
equations given in the figures.
However, this is better done with
spreadsheet software, to make iteration
much easier! In particular, the computation of exit Mach number from the
nozzle area expansion ratio is inherently iterative.
The latest and best version of my own spreadsheet for this
is the Excel spreadsheet file “liquid rockets.xlsx”. It has 3 worksheets, one the nozzle-sizing work space, one has a compressible flow streamtube tool
for relating Mach number and expansion ratio,
and there is one that is a propellant data library. There is a “.PNG” file that goes with
it, as the template for your results
report. You just copy-and-paste your
results into a copy of it.
Figure 15 is an overall view of the nozzle-sizing
worksheet, where it is too small to read
things in this view. The main working
area is top left across to the top center,
the results to be copied are top right,
and there are tables and plots of altitude performance across the
bottom.
Figure 16 shows the compressible flow tool worksheet
that supports this. You just iterate
your Mach number until you get the desired area ratio, then copy the pressure ratio for pasting into
the main working space. Figure 17 shows the propellant data
library worksheet. You just copy what
you need, and paste it in where it goes,
in the main workspace space.
Figure 18 shows just that portion of the nozzle
sizing worksheet where you actually do your sizing. It is large enough to read easily. Figure 19 shows just that portion of
the nozzle sizing worksheet where your results are summarized. This is what you copy, and then paste it into the results
report.
Figure 20 shows an image of the “.PNG” file that you
make a copy of, and then paste your
results into it, and finally do some
minor final edits as needed. This “.PNG”
file was drawn in the old 2-D Windows “Paint” software, which is where I do my copying and pasting
and editing.
Figure 15 – Overall View of the Main Nozzle-Sizing Worksheet
in the Spreadsheet File
Figure 16 – Image of the Compressible Flow Tool Worksheet
Figure 17 – Image of the Propellant Library Worksheet
Figure 18 – Image of the Main Working Area of the
Nozzle-Sizing Worksheet
Figure 19 – Image of the Results Summary Block on the
Nozzle-Sizing Worksheet
Figure 20 – Image of the “.PNG file” Results Report Into
Which Results Get Pasted
Such spreadsheet tools already exist and are freely
available to interested persons. In
particular, a good spreadsheet embodying
the rocket nozzle math calculations, is
available as part of the course materials included with the “orbits+” course
materials on the Mars Society’s “New Mars forums” website. These are available to anyone for free
download. That rocket spreadsheet is same
“liquid rockets.xlsx” that was just described.
You go to the forums website newmars.com/forums/. Go to the “interplanetary transportation”
topic and select the “orbital mechanics traditional” thread. The links to all sorts of lessons and multiple
spreadsheets are in those postings.
These go way beyond just rocket nozzle sizing and performance, to include orbital mechanics, and even entry, descent,
and landing. All these materials
are located in an online dropbox accessed by those links.
The author has other materials and courses available
directly from him. Contacting him by
email is preferred at gwj5886@gmail.com. He has a blog site with all sorts of stuff
posted, much of it technical. That is http://exrocketman.blogspot.com. You may copy anything you like from that blog
site. He also has a presence on
LinkedIn, and another on Youtube under
the name “exrocketman1”.
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Appendix – Where the Thrust Coefficient Comes From
The boundary layer displacement factors are all very close
to 1 and so divide-out of all the Ae/At ratios.
It appears explicitly only in the nozzle massflow equation as CD, not in CF.
