Below is the text paper form of the presentation I gave at the Carbon Society meeting at NC State (see "Presenter at Workshop" posted this site 4-23-2024). It was one of 3 presentations I brought to the meeting. The others I may also post here in the future.
This presentation was really about using the old, approximate by-hand design analyses as a fast and inexpensive means to screen concepts. That allows using the brainstorm process to generate many concepts, raising the probability of eventual success, but without breaking the budget and wrecking the schedule.
Today's design analyses with software packages require much more than a notional design to support putting together much more detailed models for analysis. That is inappropriate at the concept screening stage. But, today, few young engineers fresh out of school can do the old by-hand analyses any more.
The example used in this presentation was rapid estimates of atmospheric entry, by means of a 1953-vintage by-hand analysis used then for warhead entry. The analysis is very simple, and was rendered extremely rapid by embodying it in a spreadsheet to semi-automate design iterations.
The spreadsheet I used for developing this presentation, and a recent upgrade to it, are available for free at the Mars Society's New Mars forums site newmars.com/forums/.
For the original, scroll down to the Acheron Labs section and select the Interplanetary Transportation topic. Scroll down a page or two to the thread titled Orbital Mechanics Traditional, and scroll down to post # 16 regarding lesson 7. There is a link there for the spreadsheet file https://www.dropbox.com/scl/fi/9xjaw3m6 … d03mf&dl=0 and for the user’s manual that goes with it https://www.dropbox.com/scl/fi/rdqawq8z … y27y8&dl=0.
For the upgraded spreadsheet, go to newmars.com/forums/ and the Meta New Mars section. Go to the topic titled GW Johnson Postings and exrocketman1 You Tube videos. Select page 12 of that thread and scroll down to post 287. The link to the updated spreadsheet is there as https://www.dropbox.com/scl/fi/9nqdv47z … f6avf&dl=0. The link to the user's manual is there as https://www.dropbox.com/scl/fi/dw1o8end … 0c208&dl=0.
The original had worksheets representing the atmospheres of Earth, Mars, and Titan. It only estimated convective stagnation heating. The upgrade adds a worksheet representing the atmosphere of Venus, and it has added a stagnation heating model for plasma radiation heating, important at entry speeds exceeding about 9 or 10 km/s.
Here is the text of the presentation paper:
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Early Ballpark Analysis: Entry
G.
W. Johnson, PE(ret.), PhD 2 Jan 2024
Abstract
New product development often starts with the brainstorming
of several candidate concepts and ideas.
Full modern engineering design analyses mainly use finite element
computer software. These are expensive
in terms of effort and schedule time,
because the computer models require very large data sets, which in turn have to be verified before
use. There is rarely enough budget and
schedule time to do this for all the concepts and ideas.
Guessing the “winner” among them is low-probability, as is getting a larger budget and longer
schedule in order to enable doing all of them “right”. What is needed are simple, quick analyses that are “ballpark
correct”, for screening the candidates
down to the winner. The old
by-hand, pencil-and-paper analyses used
long ago, but updated and automated with
modern spreadsheet software, provide the
necessary screening tools.
The examples
presented deal with entry analysis as it is used to design heat shields.
Background
Many new product development efforts are going to start with
some sort of brainstorming to generate several possible design concepts or
ideas. However, there will usually only be enough budget and
schedule time for a team to do real engineering design analysis on one (or
maybe two) of them. Just arbitrarily
selecting, or guessing, the “winner” from among those candidates has
a rather low probability of success, as
does begging for more budget and schedule time to enable “doing this right”.
What you really need up front in this process are some
ballpark-correct, but small, fast, and
simple design analyses with which to screen your candidate ideas and concepts. You use these to screen down to the one or
two “winners” that are actually worth the significant investment of
effort, resources, and time,
that real engineering design analyses inherently consume. See Figure 1.
Figure 1 – Approximate Analysis for Screening, Rigorous Analyses for “Winner”
That inherent high cost of real design analysis is because today
those are typically done with multiple sophisticated finite-element computer
models of several different kinds. These
typically have large input data-gathering requirements, and there is significant debugging time
associated with those massive sets of inputs.
However, experience clearly shows
the results are more reliable, by far, when done right.
As it turns out, the
best candidates for such ballpark-correct fast-but-simple analyses are what
engineers used to do long ago before the advent of desktop computers, with pencil-and-paper, by-hand methods. (Back then,
the only computers were mainframes,
using card batch inputs, with
long turnaround times.) Unfortunately, the career training at most engineering
schools now greatly emphasizes the use of computer programs over any
significant training in the old by-hand methods. Note also that, assisted by modern technology such as
spreadsheets, those old methods now
require far less effort and time to execute than they did back then.
What I just said applies in pretty-much all engineering
topic areas and disciplines. The example
illustrating it here is the topic of heat shields for atmospheric entry, to be relevant at this meeting.
Usually what is most important are the influences of peak
surface temperatures and pressures upon choices of heat shield type, and for the selection of specific materials
with which to construct that heat shield.
Generally speaking, in today’s
world, the overall scope of this
includes refractories and ablatives. Exposed
heat sinks long ago proved unpromising,
being too heavy. See Figure 2.
Figure 2 – Application In General and Specific Application
to Entry Heat Shields
Entry is Complicated,
But Is Also The Source of the Design Criteria
Actual atmospheric entry is complicated, and requires analysis methods that are
generally outside the experience of more Earthly topics. The mass, size, and shape of the object, plus its speed and angle-below-horizontal at
start of entry, are what determine its
entry trajectory. That trajectory
results in time histories for speed,
deceleration gees, and the
measures of air friction heating, of
which heating at the stagnation point or zone is the most important. See Figure 3.
Figure 3 – Entry Is the Complicated Source of the Necessary
Temperature and Pressure Loads
One must obtain “from somewhere” estimates of peak
deceleration gees, and of peak
stagnation heating. The peak gees
essentially determine how much aerodynamic pressure gets applied to the heat
shield material. The peak heating goes
into one or another kind of heat balance that determines the peak surface
temperature. It is peak surface
temperature and pressure that determine what materials might be feasible for
any given entry scenario.
It sure would be nice to have some simple, fast means to determine ballpark-correct
values for the peak deceleration gees,
and for the peak stagnation heating,
for any given entry scenario.
Old Simplified Analysis as the Solution
As it turns out,
there really is a ballpark-correct entry analysis tool. It is the old 1953-vintage analysis
attributed to H. Julian Allen, and used
by him and his colleagues to estimate re-entry of ballistic missile warheads
throughout the 1950’s, and into the
early 1960’s. Back then, this was security-classified
information. But by the late 1960’s, it had been declassified, and was taught in aerospace engineering
classes for high speed aerodynamics at the graduate school level.
This analysis was simplified to a form that could produce a
closed-form equation for the variation of object speed with altitude. From that,
it is easy to estimate deceleration gee history. And coupled with some object geometry plus an
old empirical stagnation heating correlation,
it is also easy to estimate stagnation heating history. Analysis starts at an entry interface
altitude, with a speed at
interface, and that angle-below-horizontal.
The analysis is two-dimensional, and formulated in Cartesian coordinates, meaning you must “wrap” its approximate
results around the curve of the Earth’s surface. It uses a simplified exponential distribution
of density with altitude. It assumes a
constant trajectory angle below horizontal,
which you must “wrap” around the curve of the Earth to represent a
constant angle below local horizontal. It
presumes the object has a constant mass,
and that its hypersonic drag coefficient is a constant value.
The results are only ballpark-correct, but it is astonishing just how correct they
usually are. Atmosphere models now exist
for this analysis tool, beyond just the original
Earth’s atmosphere model.
If you add to this a couple more items, this becomes a tool good enough to use in entry
concept screening analyses. You need a
model for radiative heating from the glowing plasma sheath about the
vehicle, something that gets important
for entry speeds at interface above about 10 km/s. You also need some rule-of-thumb means to
estimate heating items at other locations than the stagnation zone.
Those usually just scale from the stagnation value in a
simple way, as indicated in Figure 4.
Figure 4 – How the Old By-Hand Analysis Worked
In the old days, this
was all done by hand,
pencil-and-paper, with slide
rules. Pocket calculators speed this up
only a little bit. But it is the
spreadsheet technology available today, that really drastically speeds this up! Once you have such a worksheet formulated, and every cell calculation verified, then all you need for running many
variations, are sets of only 4 inputs at
any one planetary body with an atmosphere.
Those are the object’s ballistic coefficient and “nose” radius, and its speed and angle at entry
interface. The other advantage of using
spreadsheets is the ability to do easy data plotting.
How Best Done In a Spreadsheet
Figure 5 shows an annotated image of the Earth
worksheet in an entry spreadsheet that the author set up and debugged a few
years ago. The things highlighted yellow
are the user inputs, with the object’s 4
data items circled and noted. The other
yellow inputs relate to the model of the Earth’s atmosphere. Other worksheets in this same spreadsheet
file are set up for Mars, and for
Saturn’s moon Titan.
The items highlighted brown are for calculations that the
author found were incorrect, probably due
to units conversion errors, so he did
not use them. He substituted numerical
integration and differentiation for what those formulas were supposed to
do. Actually, as it turns out, the closed-form peak gee and peak heat rate
estimates actually work. It is the peak
heat absorbed that had serious problems.
Figure 5 – Image of the Author’s Earth Entry Worksheet In
His Entry Spreadsheet File
The leftmost column contains altitudes z, in km,
starting from entry interface altitude at the top. Altitude is a true independent variable in
this analysis. You may have any numbers
you like in these cells, as long as they
steadily decrease down the column. One
may revise this list to get lots of data points where any of the other things
are changing rapidly. Changes happen
very slowly early in the descent.
You will probably want to pick one altitude for iteration to
a specific endpoint velocity. That
entire row in this example has been highlighted green. The specific velocity you want is for local
Mach 3, the lowest speed considered to
be hypersonic for a blunt object. Below
that, the analysis assumptions are
violated, because drag coefficient, and the corresponding ballistic
coefficient, are no longer closely
constant. For Earth, this speed is 1 km/s. It’s closer to 0.7 km/s on Mars.
The author wants to call your attention to seven columns for
plotting. Those are the altitude (z,
km), the range (R, km), the time from interface (t, sec), the deceleration gees (decel gees), the speed (V, km/s), the stagnation heating flux (q,
W/sq.cm), and the stagnation heat flux
integral with time (Q, KJ/cm2).
You will want to make 4 plots. Plot
only the data down to your end-of-hypersonics local Mach 3 speed row. Leave the rest out, they are incorrect, meaningless,
and only serve to screw up the graph scales.
Figure 6 shows a single sheet presentation of the
four recommended data plots for a scenario.
Figure 6 – Presentation of An Entry Scenario’s Results As
Four Data Plots
The first is range vs altitude, which verifies “no problems” when it plots as
a straight line, whose slope is the
tangent of the entry angle. The second
is time vs altitude, which relates those
two variables graphically for you. It
goes significantly nonlinear at the end of the trajectory, as the velocity sharply drops. The third is speed and deceleration gees vs
time, which give you plots of those
histories. The fourth is stagnation heat
flux and its integral vs time, which
gives you plots of those histories. Note that the peak heating always occurs at
a time earlier than peak deceleration gees.
Those two peaks are not simultaneous,
contrary to most preconceptions!
The four plots just described are easily made in the
author’s version of the spreadsheet,
which was done in Microsoft’s “Excel” software. These plots can be copied and pasted into
Windows “Paintbrush” or something similar,
to create a single-sheet representation of everything anybody would want
to know about any one entry scenario calculation. As you change one vehicle input data variable
at a time, you can create a new plot
sheet for each such scenario, and you
can record the pertinent peak data and endpoint altitude data, into a running table, from which you can also plot sensitivity trends.
Accumulated Results: Trends
Figure 7 is what plotting trends of peak gee, peak heating,
and end-of-hypersonics altitude look like, versus speed at interface, entry angle,
and object ballistic coefficient.
The only direct influence of object nose radius is upon peak
heating, which was not included in these
plots. Heating varies in inverse
proportion to the square root of that nose radius. It also affects hypersonic drag
coefficient, but only slightly, for fairly significant changes in radius.
Figure 7 – Obtaining Sensitivity Trends By Plotting
Accumulated Data
Max Pressure and Surface Temperature Estimates
Getting a ballpark max pressure upon the heat shield is
actually quite easy. All you need are
the peak gees from the entry scenario,
and the object mass and blockage area that went into its ballistic
coefficient. These combine as shown into
a very accurate figure for the effective average pressure upon the heat shield, since the ambient atmospheric pressure is
quite low at entry altitudes. Since
pressure is high at stagnation, and low
at the edge of the heat shield, doubling
the average is a decent ballpark estimate for the max value of pressure acting
at stagnation. Any material that
can take your scenario’s max pressure is thus a feasible candidate. See Figure 8 below.
It is a bit more complicated doing the heat balance
necessary to find the surface temperature on the heat shield at the stagnation
zone, where it is usually highest, or at any other locations. This depends upon whether the shield is
refractory or ablative, and upon whether
speed at entry is high enough to make the plasma sheath radiation
significant, and also opaque to
re-radiated infrared.
There are 5 significant heat fluxes to balance: convective,
radiation, re-radiation, backside conduction, and ablative.
Usually, they don’t all
simultaneously play a role. With
significant low-density insulation behind the heat shield, backside conduction can be made trivial. Otherwise,
backside conduction, and thermal
re-radiation, are explicitly functions
of surface temperature. Convective
heating comes from the entry analysis,
and plasma radiation heating from that model at each speed. Plasma sheath opaqueness will zero any
thermal re-radiation. Crudely, that happens at about the same speed as the
speed where plasma radiation heating becomes dominant. See Figure 9 below.
Figure 8 – Obtaining a Max Pressure Estimate Using Peak
Entry Gees
Figure 9 -- Basics Of
the Heat Balance
Now, the details can
vary greatly within the heat balance framework,
as indicated in Figure 10.
For a refractory ceramic heat shield,
there is no ablation. There
cannot be plasma radiation heating, or
else the plasma would also be opaque to the thermal re-radiation. Refractories cannot work at all in such
circumstances. Otherwise you simply
balance the convection against the conduction and radiation, by varying the surface temperature until
balance is achieved. That is
really easy to do in a spreadsheet, whether
or not the conduction is made trivial by insulation.
Ablation is the odd one.
It may or may not be a function of surface temperature. If the carbonaceous char erodes away as fast
as it forms, the surface temperature may
in fact actually be that value at which material pyrolysis is completed! In that case,
any conduction and re-radiation heat fluxes are fixed by that value, and the ablation heat flux is “whatever is
left” directly from the heat balance, as
shown.
That ablative flux drives the shield “erosion” rate, using shield material density and heat of
ablation values. However, it is really just the speed of the pyrolysis
front. If the char is relatively weak
enough to be scrubbed away as fast as it forms,
then it is also the surface regression rate. Otherwise,
not.
By using convection and radiation heat fluxes scaled to
other locations than stagnation, you can
do these same kinds of heat balance analyses at other typical locations around
the vehicle. The lateral or leeside
surfaces typically need less protection,
and in some cases under 8 km/s entry speeds, bare re-radiating metals are feasible. Such was true with the old Mercury and Gemini
capsules in the 1960’s.
Figure 10 – Heat Balance Variations For Refractories and
Ablatives
Concluding Remarks
All of the above is how one gets “into the ballpark” with
entry calculations, but with little
effort and time expended upon each of many candidate concepts and ideas. The better answers come from the full
engineering design analyses using finite element software packages. Those require much more time and effort, but need only be done to the winning concept, thereby generating data you can trust for
detailed design. That is how you
stay on schedule and within budget for a new development!
References
The author used two sources for his spreadsheet formulation
of the old H. Julian Allen entry analysis.
One was a second edition of the “SAE Aerospace Applied Thermodynamics
Manual”, published by the Society of
Automotive Engineers (SAE), originally
in 1960, and the second edition in
1969. This had good information for the
scaling factors of heating away from the stagnation zone, and a version of the simple stagnation
heating correlation in US customary units,
similar to Allen’s original.
The second source was “Atmospheric Environments for Entry
Descent and Landing (EDL)” by C. G. Justus and R. D. Braun, a conference paper presented June 2007 at the
“5th International Planetary Probes Workshop and Short Course”, in Bordeaux,
France. This described Allen’s
analysis in sufficient detail to duplicate it,
although they had converted his original US customary units to
metric.
The author believes the huge errors he saw in in the closed
form equation for the time integral of heat flux may be due to an incorrect
units conversion. That is why he
resorted to finite-difference integrations and differentiations in his
spreadsheet. While initially
suspect, the peak heating and peak gee
closed-form equations ultimately proved correct.
That same second source also has the basic atmosphere models
in the same basic scale height format that Allen’s original analysis used, for all the planets and moons with
significant atmospheres. Those would be
an entry interface altitude, a
scale-height type of exponential density distribution vs altitude model, and the altitude limits for the density model. From those many models in the Justus and
Braun paper, the author created
worksheets in his spreadsheet file for Earth,
Mars, and Titan. He did not create worksheets for Venus, Jupiter,
or Saturn, although that could
easily be done.
The original H. Julian Allen publication after its declassification
is somewhat hard to acquire. It was NACA
TN-4047 “A Study of the Motion
and Aerodynamic Heating of Missiles Entering the Earth's Atmosphere at High
Supersonic Speeds”, dated October 1957, with Allen and A. J. Eggers, Jr., listed as authors.
About the author
The author had a 20 year career in aerospace defense doing
new product development design,
analysis, test, and evaluation. He first entered the workforce in the slide
rule days, transitioning to
then-expensive pocket calculators, with
desktop computers still several years away.
This first career was mostly (but not entirely) in rocket
and ramjet missile propulsion. That
ended with a plant shutdown and layoff,
just when the industry was shrinking drastically. His second 20 year career was mostly in
teaching (at all levels from high school to university), plus a little civil engineering and aviation
work. He is now retired.
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