Why Know This Stuff?
(1 (1) It provides a more efficient way to expend limited
resources (see Figure 1).
(2) It is integral to
brainstorming, raising probability of
success with multiple ideas.
(3) No organization can afford to
do real design work on all candidates!
(4) However, it requires “real” pencil-&-paper
engineering training.
(5) Those so-capable can spot bad results
coming from computer codes!
Figure 1 – Knowing “Pencil-and-Paper Engineering” Is the
Efficient Way to Use Resources
The example here is making entry estimates that include both
dynamics and heating. The basic by-hand entry
model is old, simple stuff used for
warhead entry work back in the early 1950’s.
It is usually attributed to H. Julian Allen, although both he and A. J. Eggers published
it in a NACA report, once this method
was declassified. See Figure 2.
This kind of analysis is now best done in spreadsheet, for
fast changes, that automate any
iterative explorations. This analysis
only handles straight-in entries: no
skips, no multi-pass trajectories. It is
fundamentally 2-D Cartesian, so one must
“wrap” the range-related results around the central body.
Figure 2 – How the Old Entry Model of Allen and Eggers
Actually Works
The inputs divide into 3 groups: (1) the atmosphere scale-height density model
and entry interface altitude, (2) the
entry speed and direction information,
and (3) the vehicle model. That
last includes both ballistic coefficient as well as an effective “nose” radius
for heating estimates. Ballistic
coefficient requires mass and dimensional information, plus an estimate of the very-nearly-constant
hypersonic drag coefficient.
Where to obtain such information for the inputs is also summarized
in Figure 3. The Justus and Braun
reference has atmosphere models for using this kind of analysis at a variety of
places around the solar system. The
author’s spreadsheet file has separate worksheets corresponding to the scale height
models and entry interface altitudes for Earth,
Mars, Titan, and Venus,
all from Justus and Braun.
The author also has another spreadsheet file that does the
classical 2-body orbital mechanics of elliptical orbits. This is the best kind of source for speed at
entry interface. Technically, evaluating slopes at the entry interface
location will get you the entry angle below local horizontal, but a default guess of 2 degrees is rather
representative for spacecraft items.
Some warheads come in steeper,
but if so, usually slower, too,
because they are fundamentally suborbital.
Masses and dimensions for many craft can be found on the
internet. The author has found the old
Hoerner “drag bible” reference a good source for drag coefficients.
Figure 3 – Typical Sources of Data
Ballistic coefficient β = M/(CD A) is a measure
of how well the vehicle penetrates through the air while decelerating. If the hypersonic CD is a constant, then the hypersonic beta will be
constant, which is what the Allen and
Eggers model assumes. That assumption is
at least approximately true all the way down to about local Mach 3 for blunt
shapes.
The dimensions and shape enable calculating a volume
corresponding to the outer shape envelope.
Dividing mass-at-entry by that volume gets you an “effective density”
for the craft. Not all craft will have
the same “effective density”: manned
craft will compute lower because of the interior volumes required to be open
space in which the astronauts can live.
Unmanned craft will typically have higher “effective densities”, because things can be packed as tightly as
possible.
As indicated in Figure 4, ballistic coefficient β ~ eff. density *
dimension3/dimension2
= eff. density * dimension, for
any given shape, since volume is
proportional to dimension cubed, while
area is proportional to dimension squared. That means for the same shape and
density, ballistic coefficient scales as
the cube root of mass at entry. The
same shape corresponds to the same blockage area-basis drag coefficient.
Figure 4 – How Ballistic Coefficient Varies With Mass, Density,
and Dimensions
The author’s entry analysis spreadsheet is depicted in Figure
5 below. This particular one is for
the Earth atmosphere model, for an
Apollo coming back from the moon.
Highlighted in yellow near the top of the worksheet are 3
groups of inputs. Of these, the user need only worry about 2! The leftmost group has the atmosphere
model, and there is one worksheet for
each different atmosphere model.
Currently, there are worksheets
for Earth, Mars, Titan,
and Venus. The atmosphere model
has ρ0 and hscale for the exponential density variation
model, plus the entry interface
altitude. It also has the upper and
lower values of altitude, between which
the scale height density approximation best matches reality. There is an input with the name of the world the
worksheet models.
The center yellow input group is the user-input entry
conditions: speed at entry interface Vatm, and angle below horizontal Ɵ. There is an input denoting what the mission
is about.
The vehicle model is the rightmost yellow input group near
the top. It has values for ballistic
coefficient β and the effective nose radius Rn.
There is also an input for the name of the vehicle.
The heating model constants are also given for convection
and for plasma radiation. These are not
yellow, and are not user inputs. They are as meant, for each worksheet.
The main calculation block starts in the left column with a
list of altitudes highlighted green,
that starts at its top with the input entry interface altitude. The user may freely adjust that list to get
points denser in distribution where speeds,
gees, and heating rates are
changing rapidly. It ends with a yellow
highlighted user input altitude to find exactly the altitude that corresponds
to the intended end-of entry speed. Mach
3 on Earth is typically right at 1.0 km/s.
Mach 3 on Mars is typically close to 0.7 km/s.
Figure 5 – Appearance of the Author’s Entry Spreadsheet
Worksheets
Typical spreadsheet results are
shown in Figure 6 just below.
These plots are generated automatically by the worksheet. The user can see where the points need to be
denser when adjusting his altitudes list.
Then when done, he can copy these
plots and paste them into a “Paint 2-D” png file. It is recommended to read values out of the
worksheet calculation block, and
annotate the resulting plots with them,
once they are in the png file.
Figure 6 – Image of Png File Containing Annotated Worksheet
Plots
The convective and radiative heating models currently
embodied in the spreadsheet file’s worksheets are illustrated in Figure 7
just below. The original Allen and
Eggers model had only the convective stagnation heating model. The author found one for plasma sheath
stagnation radiation heating in the SAE Aerospace Applied Thermodynamics Manual
(1969), modified it slightly, and incorporated it into the
spreadsheet.
The figure also has the old entry engineer’s “rule of thumb”
that says the effective temperature in degrees K, of the plasma sheath near stagnation, is numerically equal to vehicle speed in
meters/second. This is rather
crude, being only about 10%
accurate, but it is “in the ballpark”.
The figure also includes the author’s wild guesses for how
to rescale the stagnation heating rates to other locations on the vehicle. There are regions of attached flow that
feature severe flow “scrubbing” of the surface,
and separated-wake locations that do not. The plasma radiation heating rescales
differently than the convective heating.
For regions where flow is still attached, the plasma sheath is still crudely as close
to the surface as it is at stagnation,
implying radiation heating rates still very near stagnation, unlike convective. In the wake,
the plasma sheath is remote, but
still “shining upon” the surfaces, so
the author does not rescale it down as far as he does the convective.
Figure 7 – Stagnation Heating Models Currently In the
Spreadsheet, Plus Scaling Elsewhere
Complicating Factors:
tumble-home angle vs angle-of attack for capsule shapes
Most capsule shapes have what is called a “tumble-home angle”
of the lateral walls inward. Flow
usually accelerates sub-sonically,
radially outward behind the bow shock,
to a sonic line that is usually at the very rim of the heat shield. Flow usually separates at the rim, just downstream of the sonic line, leaving the lateral walls in separated wake
flow, if the capsule flies straight with
no angle of attack.
A modest angle of attack to create a lateral lift force has
been used for a long time (since Gemini in the 1960’s) to better “fine-tune”
the entry trajectory. One just rolls the
capsule to point that lift vector in the desired direction. This has the effect of reducing the angle
between the lateral wall and the separated flow on the side where the
stagnation point is closest to the rim. On
Apollo, this had the effect of flow
staying attached to the lateral wall (with higher heating) in a localized swatch
of surface, on that side. This sort of thing is depicted in Figure 8
just below.
The simple entry model does not handle such subtle
differences, it just pulls the capsule straight
in, along a straight line in Cartesian
coordinates, and it only estimates
stagnation heating. The user has to
allow for this possibility, when
rescaling stagnation heating rates to lateral walls where flow might actually
be attached!
Figure 8 – Effects of Modest Angle of Attack Upon Heating
for Capsule-Type Shapes
As an example of this angle of attack effect, consider the data the model predicts for
Apollo coming back from the moon, in Figure
6 above. Stagnation convective was
144 W/cm2, and radiative was
236 W/cm2, for a stagnation
total of 380 W/cm2. Those
numbers scale for attached flow locations to 48 W/cm2 convective, and 236 W/cm2 radiative, for a total of 284 W/cm2. For separated wake zones, those same numbers rescale to 14.4 W/cm2
convective, 78.7 W/cm2
radiative, for a total of only 93.1 W/cm2.
Note that the rim of the heat shield would definitely be a
region of attached flow, at total
heating 284 W/cm2, some 74.7%
of that at the stagnation point! At some
angle of attack causing flow attachment for a swatch along only one lateral
side, the same high heating at something
like 284 W/cm2 would exist!
The rest of the lateral sides are all in separated flow, at a heating rate only in the neighborhood of
93.1 W/cm2, only 24.5% of
stagnation.
The lesson is quite clear:
lateral sides that might see attached flow at angle of attack, require thicker heat protection than those
that do not! That increased thickness
requirement is at least similar to the thickness near the rim of the base heat
shield!
Max pressure on the heat shield is important for choice of
an adequate material, as some can be
crushed. You have a mass at entry, and a blockage area, in order to set up your calculation of
ballistic coefficient β. The entry model
spreadsheet gives you an estimate of the max deceleration gees. Mass * max gees * gc equals the decelerating force F acting upon
the vehicle. Max deceleration occurs
high enough up, that backside pressures
on the aft surfaces are essentially zero.
So, the average pressure on the
heat shield is simply that deceleration F divided by the blockage area. The sonic pressure near the rim is roughly
half the stagnation pressure, so the
average pressure is roughly ¾ of the stagnation pressure. Reversing that leads to Pstagn =
(4/3)*Pavg, as indicated in Figure
9.
Figure 9 – Approximate Stagnation Pressure Estimate
Heat shield materials have definite operating limits. Ablatives are usually rated to max heating
rates per unit area, and max pressure
exposure, as shown in Figure 10 just
below. Transpiration surfaces would likely be
similarly rated, although that
technology has yet to fly (but it might soon).
Refractories are usually rated somewhat differently, being rated directly in terms of a max
service temperature, although there is
still a max pressure rating. The user
should be aware that these max rating values recommended for ablatives will
vary from source to source.
Looking at the Apollo lunar return example above, the exposures and the ratings for its Avcoat
5026-39 heat shield compare as follows:
Item…………….exposure…….rating…….remarks
Q/A, W/cm2…..380…………….600……….OK
Max P, atm……0.56……………0.50………barely not OK, but it worked
Figure 10 – Max Rating Values for a Few Ablatives (Values in
Other Sources Vary)
The variation in ratings from source to source can be seen
comparing Avcoat 5026 in Figure 10 above to “Avcoat” for Apollo and
Orion in Figure 11 just below.
Note particularly the manufacturing difference between Avcoat for Orion
EFT-1 versus Orion as flown in Artemis.
Artemis leaves out the reinforcing hex, to get bonded tiles instead of hand-gunned
honeycomb cells. Most such sources leave
out sufficient clarifying details!
Figure 11 – Many Ablative Applications and Rating Data (from
a different source)
Ratings for some refractory ceramic materials are shown in Figure
12. The first 3 in the figure were
used on the space shuttle. The windward
tiles were colored black to increase their thermal emissivity, where heating was larger. The leeward tiles were white where high
emissivity was not required, but solar
reflectivity was required, for passive
thermal balance control.
These were very low density aluminosilicate materials, whose max service temperatures were not
limited by melting, but by a solid phase
change causing shrinkage and fatal embrittlement. That last is exactly why Coleman gasoline
lantern mantles were so fragile!
The ceramic blankets were more sharply temperature limited, and were only used on leeside surfaces
immersed in separated flow.
Tufroc is not a single material, but two layers of different ceramic materials
mechanically coupled together. These are
usually set up as two-part tiles bonded to the surfaces they protect. The outer surface layer is a denser, more thermally conductive ceramic that is
rated to a higher temperature than aluminosilicates, and also quite a bit stronger than the
shuttle tile material. The inner layer
is somewhat similar to shuttle tile,
being low density, not as
strong, and very low thermal conductivity. It is rated to a bit-higher temperature.
Figure 12 – Some Data on Refractory Ceramic Materials
Exposed metals are possible,
but only if the heating rate is low enough to permit a survivable equilibrium
temperature, with a hot strength that is
still acceptable. This was done on Mercury
and Gemini, which returned only from low
circular Earth orbit where the heating rates were far lower. This could not be done with Apollo, which returned from the moon at very near
escape speed, with very much higher
heating rates. It is being done again by
SpaceX with its “Starship” leeside surfaces,
but only in separated flow zones,
and only from low circular Earth orbit speeds (at least so far). See Figure 13 for materials data.
Figure 13 – Some Data on Exposed Metals as Refractory
Candidates
For ablatives,
refractories (ceramics and metals),
and transpiration-cooled designs,
the heat balance concepts, as
simplified, are shown in Figure 14
below. These are couched in terms of
heat flux format, that being heat flow
rate per unit of exposed surface area. That matches the output data from the entry
spreadsheet model.
For the ablative scenario,
there is both ablation and re-radiation cooling available to establish
equilibrium, but no adequate way to
determine how much of each! For the refractory
scenario, there is only re-radiation
cooling, and an equilibrium temperature
is easily determined iteratively. For
the transpiration scenario, equilibrium
surface temperature is constrained by coolant vaporization at an acceptable
coolant pressure. Thus, an actual coolant flow rate is determined
from that acceptable temperature.
Bear in mind that transpiration cooling has yet to actually
fly in space. It was supposed to be
investigated with the old X-20 “Dyna-Soar”,
that was cancelled without ever flying.
However, such a thing may well
fly soon. There is at least one “new
space” competitor that wants to use it,
and it was seriously considered by SpaceX, before they went with very slow-ablative tiles
on their “Starship”.
Those notions lead directly to the guidance for
spreadsheet-based heat balances depicted in Figure 15 below. These would likely be self-generated as
custom spreadsheets. This author has
none to offer at this time. The
ablatives scenario must have some other constraint in order to set the point on
the regression rate vs equilibrium temperature trend.
Figure 14 – How the Heat Fluxes Balance for the 3 Scenarios
Figure 15 – Guidance Toward Setting Up Spreadsheet Heat
Balances for the 3 Scenarios
Mars entry is definitely different from Earth entry, as shown in Figure 16 just below. These are the cross-plotted results from a
study run with these tools. The author
“made-up” a small probe, with either a conical
or a blunt heat shield, and ran it for
free direct entries off an interplanetary trajectory at Mars, plus low circular Earth orbit entries, and entries at near escape speed. These data were combined with results from an
earlier Apollo entry study that included entry from low circular orbit and
near-escape returning from the moon.
The 2 left-side plots in the figure basically show the
effect of the very thin Mars atmosphere upon end-of-hypersonics altitude, and upon estimated stagnation pressure on the
heat shield. The surface density on Mars
is numerically the same as density near 35 km altitude at Earth. The plot of stagnation total heating vs speed
at peak heating shows no reliably-discernable trend, except that peak heating speed is higher if
entry interface speed is higher. The
Mars data fall right in the middle of the Earth data, all for comparable entry speed, since direct entry speed at Mars is about the
same as low Earth orbit entry speed.
Figure 16 – Comparison Cross-Plots for Earth vs Mars Entries
Doing these kinds of entry studies using pencil-and-paper
engineering, assisted by modern
spreadsheet software, is actually easier
than most people think. But the engineering
analyst who does this must really know what he/she is doing! This is very heavy into high-speed
compressible flow analysis, and very
high-speed heat transfer techniques!
Plus, in order to
function, the engineering analyst must
know an awful lot about materials, their
properties, and their limitations!
But, there is an
undiscussed advantage if the engineering analyst can really do this
pencil-and-paper engineering stuff! He/she
will have enough experience from running such numbers for many projects, to spot bad results coming from someone else’s
code. Computers process bad inputs and
bad models into bad results, as easily
as they process good inputs and good models into good results. They all look the same, at first glance!
References as indicated above:
#1. H. J. Allen and
A. J. Eggers, “A Study of the Motion and
Aerodynamic Heating of Ballistic Missiles Entering the Earth’s Atmosphere at
High Supersonic Speeds”, NACA Technical
Report 1381, 44th Annual
Report of the NACA 1958, Washington D.C.
1959. (unclassified) – this has the scale height atmosphere model and the
relationship between altitude and velocity,
plus the convective stagnation heating correlation.
#2. C. G. Justus and
R. D. Braun, “Atmospheric Environments
for Entry, Descent, and Landing”,
MSFC-198, June, 2007.
– this has the same Allen and Eggers entry model, and scale height atmosphere model as Allen
and Eggers, but goes beyond just
Earth. Atmospheres for Mars, Titan,
and Venus were obtained from here.
#3. SAE, “Aerospace Applied Thermodynamics
Manual”, 1969. (hardbound) – this had a simple plasma radiation
heating model that was modified and added to the spreadsheet embodying the
Allen and Eggers technique.
#4. Sighard
Hoerner, “Fluid Dynamic Drag”, self-published by the author, 1965.
– drag data for many shapes into the low hypersonic range are in this
reference.
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PS: This
article actually comprises a pretty good user’s manual for my current version
of the entry spreadsheet. This
spreadsheet is available from the New Mars forums as a free download, or you can contact me directly by email. Watch this site for two follow-up articles done
by using this spreadsheet-based analysis technique.
One will be a comparative re-entry study done for typical
Mars probe heat shield shapes and an Apollo capsule shape, all with ablative heat shields, done at both Earth and Mars. It will show how Mars entry is
different, with some indications as to
why.
The other will be a heating distribution study for an Orion
capsule doing free-entry returns from the moon.
Such will be useful for understanding what effects showed up on the
Artemis-1 heat shield versus the Artemis-2 heat shield, and the original Orion EFT-1 test flight’s
heat shield.
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