Update 4-8-2024:
Should any readers want to learn how to do what I do (estimating
performance of launch rockets or other space vehicles), be aware that I have created a series of
short courses in how to go about these analyses, complete with effective tools for actually
carrying it out. These course materials are
available for free from a drop box that can be accessed from the Mars Society’s
“New Mars” forums, located at http://newmars.com/forums/, in the “Acheron labs” section, “interplanetary transportation” topic, and conversation thread titled “orbital
mechanics class traditional”. You may
have scroll down past all the “sticky notes”.
The first posting in that thread has a list of the classes
available, and these go far beyond just the
two-body elementary orbital mechanics of ellipses. There are the empirical corrections for
losses to be covered, approaches to use
for estimating entry descent and landing on bodies with atmospheres, and spreadsheet-based tools for estimating
the performance of rocket engines and rocket vehicles. The same thread has links to all the materials
in the drop box.
The New Mars forums would also welcome your
participation. Send an email to newmarsmember@gmail.com to find out
how to join up.
A lot of the same information from those short courses is
available scattered among the postings here.
There is a sort of “technical catalog” article that I try to main
current. It is titled “Lists of Some
Articles by Topic Area”, posted 21
October 2021. There are categories for
ramjet and closely-related,
aerothermodynamics and heat transfer,
rocket ballistics and rocket vehicle performance articles (of
specific interest here), asteroid
defense articles, space suits and
atmospheres articles, radiation hazard
articles, pulsejet articles, articles about ethanol and ethanol blends in
vehicles, automotive care articles, articles related to cactus eradication, and articles related to towed decoys. All of these are things that I really
did.
To access quickly any article on this site, use the blog archive tool on the left. All you need is the posting date and the
title. Click on the year, then click on the month, then click on the title if need be (such as if
multiple articles were posted that month).
Visit the catalog article and just jot down those you want to go see.
Within any article,
you can see the figures enlarged, by the expedient of just clicking on a
figure. You can scroll through all the
figures at greatest resolution in an article that way, although the figure numbers and titles are
lacking. There is an “X-out” top right
that takes you right back to the article itself.
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I updated the “compressible.xlsx” spreadsheet file as “liquid rockets.xlsx”, and deleted the extraneous worksheets. I added a convenient block of relevant outputs that requires no new inputs other than a name for the propellant combination. I developed a “Paintbrush” file “engine sizing report.png” on which to copy and paste the convenient outputs block in one fell swoop. You need only adjust the name text above the engine diagram. See Figure 1 for what this looks like.
Figure 1 – Image of What the “Engine Sizing Report” Format
Looks Like
This is all you really need, to understand what the engine can do, except that you must look at the
green-highlighted separation limits data,
and understand that the design shown is unseparated at sea level for
full throttle, and part-throttle
settings (in this case 80% Pc). This
example has a backpressure-induced flow separation in the bell at min
throttle, below very near 12.6 kft.
You still have the two performance vs altitude plots already
made by the “r noz alt” worksheet. If
you want to use them, I recommend you copy and paste them to a
“Paintbrush” png file, then annotate
them for separation. Such is illustrated
in Figure 2 below. These are
located just to the right of the altitude performance calculation block.
When evaluating flow separation in any of the calculation
blocks, note that the pressure in the
standard atmosphere was modeled, for
purposes of quick and easy estimates of the altitudes below which to expect
separation. That model was reversed to altitude
as a function of Pa, plotted, and a 4th-degree polynomial trend
line developed with the spreadsheet software.
The quality of the fit was excellent.
But, because of the nature of the
fitted curve shape, using this on
pressures above sea level standard 14.696 psia produces nonsensical
results. See Figure 3 below for
why.
Figure 2 – Example Plots,
Showing How to Annotate for Flow Separation
Figure 3 – When and How to Use the Estimates of Separation
Altitude
Using
the spreadsheet
To obtain such results quickly and conveniently, I added an automated determination of the
expansion-design value of Pe appropriate to compromise-ascent design, once the design Pc has been selected and
input. Just copy and paste-123 the
design Pe value into the indicated input cell for it. I am recommending that you use 80% Pc for
this purpose, and that you size for
thrust at sea level, using the sea level
CF for that input.
If you are doing a traditional sea level-optimized
design, I recommend you use max (100%)
Pc, and the sea level standard 14.696
psia as your design Pe. Again, you can just copy and paste-123 the values
into the cells quickly. I recommend that
you use the sea level CF for sizing to your thrust.
If you are doing a vacuum design, there is some known expansion area ratio A/A*
to which you are designing. I have
retained the “compr flow” worksheet for this purpose. Go to it,
and make sure you have the correct specific heat ratio selected. Then in the indicated input cell, iteratively adjust exit Mach number Me until
you hit exactly the desired value of A/A*.
Read the pressure ratio PR at that Mach,
and go back to the “r noz alt” worksheet and input that PR value, almost
top right.
Input PR in “r noz alt” where shown, and the appropriate design Pe will
appear, to be used with your input
expansion design Pc. Copy and paste-123
that design Pe value into the appropriate input cell. I also recommend that you copy and paste-123
the vacuum CFvac to size vacuum thrust for a vacuum design, which in all probability cannot be
unseparated at sea level, even at full
throttle.
Trade
Study for Throttle Setting to Use for Ascent Compromise Design
For the purpose of determining what throttle-setting
Pc-value to use for ascent-compromise designs,
I ran a trade study. I ran
ascent-compromise designs at Pc values from 60% to 95% max Pc, by increments of 5%. This was for a liquid oxygen-liquid methane
(LOX-LCH4) propellant combination, in an
engine technology characterizable as “low-tech”.
Max Pc was assumed to be only 2000 psia. I ran a constant pressure turndown ratio (P-TDR)
of 2, which set the min Pc to 50% of
max, or 1000 psia. The intermediate throttle setting is what I
varied. I ran this with an 18o-8o
curved bell, a throat discharge
coefficient CD = 0.995, and a
dumped bleed fraction BF = 0.05. All of
this assumes a gas specific heat ratio of 1.20.
For comparison, I
also ran a traditional sea level-optimized design, and two vacuum designs at A/A* = 100 and
300. The results were graphed in 4
different plots, presented as annotated in
Figure 4 below.
Top-left of the figure,
the plot shows how sea level,
ascent-averaged, and vacuum
specific impulse (Isp) vary versus the range of throttle settings
investigated. The ascent-averaged Isp
trend is not linear, and I show a sort
of “aft-tangent” determination of the
rather weak knee in this curve near 80% throttle setting.
Top-right in the figure is the same basic plot, but to a different scale, showing the ascent-compromise trends and the
bounds represented by the sea level and the two vacuum designs. All of the ascent-compromise
ascent-average Isp values beat the sea level design’s ascent-averaged Isp
value. They even beat the sea level
design’s vacuum Isp value! They are not
significantly far below the vacuum Isp value for the A/A* = 100 vacuum
design, which in turn is not far below
the vacuum design for A/A* = 300!
Figure 4 – Trade Study Plots
Bottom left is a plot of estimated expansion bell lengths (Lbell)
vs the throttle setting. These are
crudely estimated as Lbell = (De – Dt)/(2*tan(avg a)), where avg a = 0.5*(a1 + a2). The point here is twofold: (1) the 80% or maybe 85% values for Lbell
are halfway between the sea level and the vac-100 (A/A*=100) designs, and (2) the trend is flat enough that none of
the compromise choices are far from that “halfway-between” point.
Bottom-right is a plot of A/A* vs throttle setting, quite similar to the Lbell
plot. In this case, the 80% point is about a quarter of the way
up between the sea level and vacuum A/A*=100 designs. The trend is flat enough that no power
setting investigated is far from that point.
Recommendations
for sizing engines
For the “traditional sea level” designs, size the expansion between max Pc and Pe = Pa
= 14.696 psia (1 standard atmosphere). That produces a sea level thrust
coefficient CF, which you use
with a sea level thrust requirement to size dimensions and flow rates.
For the “vacuum designs”, there is some max expansion ratio, allowable in terms of fitting the engine into
the available space for it, aboard the
vehicle. Determining that fit may well
be iterative! Size the expansion area
ratio to that max area ratio A/A*, at
max Pc, which produces a vacuum thrust
coefficient CFvac, once you
translate A/A* into a design Pe value in the spreadsheet. Use that
CFvac and max Pc to size the flow rates and dimensions to a vacuum
thrust requirement.
For the ascent compromise designs, determine separately what throttle setting
(percent of max Pc) will be used to size the expansion. The recommendation here is 80%, although variations of 5% up or down from
that make very little difference. I
would not recommend less than 75%, nor
any more than about 85%, though.
Higher setting is higher vacuum Isp but lower sea level
Isp. Lower setting is lower vacuum Isp
but higher sea level Isp. The increase
in ascent-averaged Isp with increased setting is almost negligible, because of the offsetting effects on vacuum
and sea level Isp. But using near-80%
setting gives you more “room for error” in a sea level open-air nozzle
test, where you need to ignite at a low
setting, and then throttle-up rather
quickly above the separation-point setting,
before any damage is done.
What
this spreadsheet does not do
This spreadsheet is for calculating good estimates of
performance for liquid rocket engines of fixed nozzle geometry. It does not do variable geometry
notions such as bell extensions,
excepting as separate estimates for the two geometries as if they were
fixed. It does not do
free-expansion designs at all. Those
would include both coaxial and linear aerospike geometries, expansion-defection designs, or any exit stream with both a free surface
and contact with a physical surface.
While it provides very good performance estimates of fixed
bell geometry designs, it does not
model the “cycle” that powers the turbopumps.
One does not need to do that, to
model thrust and specific impulse, as
long as one has a good estimate of the dumped bleed gas fraction representing
the cycle.
Availability
of the spreadsheet
I would be happy to share this spreadsheet. Simply contact me to make the request. There
is no user’s manual (see #1 Update 4-4-2024), although its basic operation is described in
this article and another on this site. User inputs are highlighted yellow. Significant results are highlighted
blue. Things you need to check or to
iterate are highlighted green. The Excel
“copy” command, and the “paste-123”
command are the best way to transfer numerical data from one cell to another.
#1 Update 4-4-2024: This document was originally written during
12 through 16 March, 2024. There is now a user’s manual, available as a pdf document, along with the spreadsheet file “liquid
rockets.xls”. The “Paintbrush” file
“engine sizing report.png” is also available as a convenient tool for reporting
results. Open it in “Paintbrush”, cut the block of data out of it, and copy and paste the new block from the
spreadsheet into it.
Miscellaneous
things to know about
Be aware that off to the right on the “r noz alt” worksheet
are some other data and plots that I used deciding how to correlate Pa vs
altitude for purposes of determining the altitudes where separation might
occur. These would be of little use to
any user. Just ignore them.
That does bring up the separation backpressure
estimate, which is entirely empirical, and was developed originally for the straight
conical nozzles seen in missile solid rocket motors. It is slightly conservative for curved bells. It takes this form:
Psep/Pc
= (1.5*Pe/Pc)0.8333
The ratio Pe/Psep is a simple function of the nozzle area
expansion ratio. Psep is thus an
easily-computed constant times whatever your operating Pc might be. Whenever the ambient atmospheric pressure Pa
equals Psep, separation is likely. When Pa exceeds Psep to any noticeable
extent, separation is certain!
There are shocks that touch the inside bell surface when
separation occurs. These greatly amplify
the localized heating at the impingement location, leading to burn-throughs and destruction in
only several seconds. That is why
separation is to be avoided. See Figure
5.
Figure 5 – Sketch of Separation Phenomena in a Bell Nozzle
#2 Update 4-4-2024:
There are two very closely related articles on this site that this
document and its partial throttle setting trade study supports. They are
“Bounding Calculations for SSTO Concepts”, dated 4-2-2024
“Bounding Calculations for TSTO”, dated 4-3-2024
Both of these have reference lists of other closely-related
earlier articles on this site, including
two where I investigated free-expansion nozzle design approaches, among other things.
In the performance of the trade study, I sized multiple engines with the “r noz alt”
worksheet, and reported those results (rather
easily and quickly) using the “engine sizing report.png” Paintbrush file. Those sized engine results follow, as a collection of unnumbered and untitled figures.
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