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UCRL-14238 S^^^^5_JC0PY NO. 4 B
University of California
Ernest O Lawrence FASTER Radiation Laboratory
Livermore, California B/ 8800
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UNIVERSITY OF CALIFORNIA
Lawrence Radiation Laboratory
Livermorej California
July 27, 1965
TO: Distribution
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RE: UCRL-14238 by J. Hadley, T. Stubbs, M. Janssen, and L Simons.
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UCRL-14238
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UNIVERSITY OF CALIFORNIA
Lawrence Radiation Labora tory
L i v e r m o r e , California
AEC Contract No. W-7405-eng-48
THE HELIOS PULSED NUCLEAR PROPULSION CONCEPT
(Title: Official Use Only)
J . W. Hadley
T. F . Stubbs
M. A. J a n s s e n
L. A. Simons
June 2, 1965
crr.-^-
L E G A L N O T I C E This report was prepared as an account of Government sponsored work Neither the United States, nor the Commission, nor any person acting on behalf of the Commission.
A. Makes any warranty or representation, expressed or Implied, with respect to the accuracy, completeness, or usefulness of the information contained In this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights, or
B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed In this report.
As used in the above, "person acting on behalf of the Commission" includes any employee or contractor of the Commission, or employee of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contractor.
Tlul'llftm.ment contains pfe«*ffcited data as def ineo?%l^e^pja*fS Energy Act of 1954. Its t r a S i i i i ^ or the d isc losure of i ts coQ^*?fsin any*'!Hafl^er to an un-
5ed pe r son is prohibuS
- 1 - 1 1 - tfOCUMENT IS OMITEtJ
n
THE HELIOS PULSED NUCLEAR PROPULSION CONCEPT
J. W. Hadley", T. F. Stubbs"*", M. A. Janssen" ,
and L. A. Simons'.
Lawrence Radiation Labora tory , Universi ty of California
L i v e r m o r e , California
June 2, 1965
I. SUMMARY AND CONCLUSIONS
During the past year or so, a scheme for pulsed propulsion by means of 1 2 3 contained explosions has been r e -examined with some care . ' ' The occasion
of f i rs t suggestion of the Helios concept is not known to us, but it has been a 4
recognized possibi l i ty for many y e a r s and was briefly discussed in r e p o r t s 5
published at LRL in 1957 by Robert Fox. This paper will summar ize cha rac t e r i s t i c fea tures of the concept,- especia l ly with reference to sys tem optimization studies and analys is of heat t r ans fe r to the engine s t ruc tu re .
The goal of the p resen t study has been to gain an understanding in
principle of the contained nuclear pulse engine — to identify the major problems,
to define optimum sys t ems , and to make an overal l appraisa l of i ts value.
A Hel ios-propel led vehicle would make use of a containment vesse l —
perhaps 30 feet in d iameter — in which smal l nuclear explosive charges
would be placed, together with a few hundred pounds of hydrogen. F i r ing of
a charge would r e su l t ' i n . bringing the mix ture of charge res idue and hydrogen
to a high t e m p e r a t u r e — say 5000 or 6000°K — and subsequent r e l ea se through
a nozzle would provide propulsive th rus t . This p rocess would be repeated
severa l thousand t imes , at in terva ls l ikely to be 10 seconds or longer. High
impulsive acce le ra t ion loads would be modera ted by a sys tem of shock
abso rbe r s between the engine and vehicle.
Considerable effort has been put into sys tem studies seeking, for stated
miss ions , the engine configuration and operating p a r a m e t e r s which give mini
mum initial weight and number of pu lses . Analytical studies of detailed
Nuclear Propuls ion Division Leader
^ Physic is t , Nuclear Propuls ion Division
technical problems which seem most fundamental to the Helios scheme have also been pursued, with main areas of investigation being in explosion hydrodynamics, heat transfer to the vessel walls, pulsed propellant discharge behavior, nozzle design, and in engine-vehicle coupling problems. An experimental study of layered structural materials with possible suitability for the pressure vessel walls is in progress.
Conclusions drawn from the studies to date may be summarized as follows: No fundamental fault in the concept has been found. Many processes involved are not well understood, and in fact could not be without application of extensive and painstaking theoretical and experimental efforts, but there is no reason to doubt that such efforts would result in a working engine.
The practicality of this propulsion concept would depend critically on the availability of a suitable nuclear charge, especially with respect to reliability, since an accidental excess yield or fragmented explosion could destroy the pressure vessel. Achievement of low charge mass is also of great importance.
Engineering problems in reaching a workable detailed design would be as severe as in most of the currently discussed advanced propulsion schemes.
Most of the detailed technical study has been based on materials and engineering design believed attainable within a decade or so. Under this restraint, performance advantages over other propulsion schemes do not appear to be great for the range of missions currently in the planning and proposal stages. For the manned Mars landing mission, for example, it is not thought likely that a reduction in total system mass by as much as a factor of two below that of a solid-core nuclear rocket system could be achieved. Even this great an improvement would require remarkable achievements in design, especially in suppression of heat deposition in the containment vessel, as will be apparent from the paper of E. Piatt at this conference.
Unpredictable advancements in a number of areas could change this conclusion, especially for very large payloads and mission ideal velocities, which stand most in need of high-specific-impulse systems. Achievement of a very high strength-to-weight ratio containment vessel, of very lightweight nuclear charges, or of striking advances in surface heat protection could bring a sharp rise in the competitive position of the Helios engine.
mm
-2-
'*^^'tw_/ '"'•^^"If^ll -3-
In view of these findings, it appears that development of Helios with a
view even toward detailed design studies would be inappropriate at this time.
The concept, however, is one of possible importance in the future, and should
be clearly understood in principle.
II. SYSTEM PERFORMANCE
A. General Ground Rules
Basic to any discussion of vehicle performance is a knowledge of the t. specific impulse the engine can be expected to deliver and a description of the mission which the vehicle must accomplish. The many sources of degradation of the specific impulse are covered by Piatt ; in the following ' discussion we consider only the effects of pulsing and of a finite nozzle. The missions flown are determined solely by a total ideal velocity increment, AV, through which the vehicle is to be accelerated. This ignores the interplay between the planetary gravitational fields and the rate at which the engine is pulsed, as discussed by Piatt. We further ignore the gains to be realized from staging and report results for a single-stage vehicle.
The study presented below is based on technology believed available to a development program starting within the next decade.
B. Gas Properties
For computational purposes the gas composition is fixed as follows: Primarily, the propellant is considered to be hydrogen; however, a certain amount of charge residue will be present in addition. For want of a better approximation, we have assumed throughout our calculations that the energy release is independent of the mass of the charge, which we have set at 70 lb (this is thought to include expendable positioning gear left in the chamber). The composition is now determined by x. the mass ratio of hydrogen to all of the material in the chamber. In order to arrive at reasonable gas thermodynamic properties we have further assumed that the charge debris, largely
high explosive residue, acts like carbon both from a chemical and molecular 7
weight standpoint. Tables of thermodynamic functions of reacting mixtures of hydrogen and carbon over a wide range of temperature, pressure, and x
are used to determine the change in enthalpy for an isentropic expansion through a finite nozzle.
- 4 -
These tables indicate that so long as the re is dissociated hydrogen in
the gas (between 5 and 95%), the mix tu re behaves like a perfect 7 - l aw gas
with a 7 ~ 1.20. This holds throughout mos t of the t empe ra tu r e and p r e s s u r e
range of in te res t .
C. Masses Explici t ly Calculated
1. Nozzle
The m a s s of the expansion nozzle is based on the assumption that all
of the thrus t is taken at the throat of a conical nozzle of 20° half-angle. An
empir ica l formula which fits a m o r e exact es t imate of the m a s s is used:
M ^ « k ( e ) l / 2 p ^ r ^ 3 ( £ ) ^ . (1)
Here k is a constant, e the a r ea expansion ra t io of the nozzle, p^ the initial
p r e s s u r e within the chamber , r, the rad ius of the nozzle throat , and (p /c)^
the weight/strength ra t io for the nozzle ma te r i a l .
2. P r e s s u r e Vesse l
The minimum m a s s of a spher ica l p r e s s u r e vesse l that can withstand
a steady in ternal p r e s s u r e p without yielding is given by
(3/2)p V I
M , = —^ . (2) s 1 - (3/2) S-
Here V is the cavity volume in the shell , p is the density of the shell
ma te r i a l and a i s i t s yield s t r e s s . As noted in Sec. III. B . 6, we include a
factor of 4 in p to allow for the rapid step to " s teady-s ta te" p r e s s u r e and
the momentum t r ans fe r of the pulse, plus a safety factor of a l i t t le l e s s than 2.
We thus have
^s- 5 ^ - "' 1 - 6 - ^ a s
The var iable t e r m in the denominator is a th ick-shel l cor rec t ion (where the
shell becomes a measu rab le fraction of the radius) .
UP^ ?-
3. Fixed Mass of Payload M. ,
0f.j^_i\i'
This i s an input number hiding a multi tude of s ins . Included in this
m a s s a re such things as the nozzle th roa t valve, shock abso rbe r s and shadow
shie lds . Calculat ions indicate that the total m a s s of these th ree is probably
l e s s than 10 tons. Also assumed to be included as payload a re life support
and observat ional equipment, t a rge t planet excurs ion modules, and Ear th
a tmospher ic r e - e n t r y vehicle.
4. Fu r the r Mass Allowances
The propellant (hydrogen) will have to be s tored as a liquid. This will
r equ i r e not only tank m a s s but also a cer ta in amount of insulation to prevent
l o s s by evaporat ion during the long coasting per iods of the orbit . Prel iminary,
calculat ions have indicated that a m a s s of about 8% of the useful propellant
will have to be c a r r i e d along. This includes, in addition to the tanks and
insulation, the amount of hydrogen used to keep the tanks p res su r i zed and
the amount that evapora tes during flight.
Associa ted with each nuclear charge will be a m a s s for s torage and
handling. We es t imate this to be about 5 lb per charge. As stated ea r l i e r ,
we have assumed that each charge will weigh around 70 lb.
5. Total Mass of Vehicle, Hydrogen, and Charges
Each pulse will exhaust a m a s s of gas composed of the charge debr is
6M and the propellant 6MTT. These m a s s e s a re re la ted through the variable
"x" as indicated e a r l i e r :
6M„ - ^ ^ ^ - 6 M . (4) H 1 - X c
Let us say that it t akes N pulses to acce le ra te the vehicle through the miss ion
velocity increment AV. Then the total amount of effluent required is
N 6M N(6M„ + 6M ) = —. ~ .
H c 1 - X
Descr ibed in Sec. 4 above a r e the tankage allowances for both the charges
and the hydrogen. We define a as the ra t io of the hydrogen tankage m a s s to
the useful hydrogen m a s s and j3 is the ra t io of the charge s torage and handling
m a s s to the total m a s s of the cha rges .
SLAJKO -6-
The total initial vehicle m a s s is then given by
M„ = M,, + M + M ^ + (1 + Of) N 6M„ + (1 + ^) N 6M 0 N s F H ^ c
N 6M = M,. + M + M „ + -.
N s F 1 -X +
3 +X{a - 13) 1 -X
N 6M . c (5)
The rocket equation now gives us another equation in t e r m s of these var iab les ,
. the sf
AV/gl
the miss ion ideal velocity AV, and the specific impulse I :
M 0 N 6M
= /Li = e sp (6)
M, 0 1 -X
F r o m Eqs . (4) and (5) we obtain
iu(M„ + M + M^) M = N s F
and
l-(ju - 1) [ + x (o - 3) ]
N 6 M ^ = ( 1 - X ) p ^ MQ ,
N 6M,, - xl - ^ I M^.
(7)
H M
6. Quantit ies Derived from Thermodynamic Input
Immediate ly obtainable from the input values of t empera tu re Tp.,
p r e s s u r e p„, and composit ion x a r e such p rope r t i e s as gas density, specific
energy, specific enthalpy, and specific entropy. When coupled with the total
m a s s of gas in the cavity ( indirect ly obtained from x and 6M ) the density
yields the total volume in the cavity and thus the m a s s of the p r e s s u r e vesse l
from Eq. (2); and the specific energy gives the total nuclear yield requ i red to
produce these conditions. The expansion ra t io , e, when used in conjunction
with the initial enthalpy, produces the specific impulse for a finite nozzle;
and a t ime constant for exhaust, T (see Sec. III.B.3), fixes the nozzle th roa t
a r ea and thus the nozzle m a s s .
D. Resul t s
As a convenient figure of m e r i t we have devised a cost pa r ame te r ($)
which depends upon the total vehicle m a s s and the number of pulses . It has
been es t imated that in the future, the freight charge for putting mate r ia l into
a low ear th orbit will be around $200 per pound. We further es t imate that
nuclear ma te r i a l will be re la t ive ly plentiful and the cost of the nuclear devices 4
could be around $5 x 10 per charge . Such a cost analysis obviously neglects
development charges . The cost p a r a m e t e r thus defined is art if icial , but does
yield useful information.
Resul t s computed on the bas i s outlined above a r e shown in Fig. 1, where
we plot curves of constant cost on the N vs r plane, r being vesse l rad ius .
Superimposed upon this is a gr id of p„ vs x. To produce these curves we have
fixed the t e m p e r a t u r e at 6000°K, the expansion ra t io at 200, AV at 60,000 ft/sec,
and the fixed m a s s at 100 tons. The curves close on the left because the m a s s
of the p r e s s u r e vesse l i n c r e a s e s rapidly with p r e s s u r e — p r imar i l y due to the
th ick-she l l cor rec t ion to Eq. (2) — and on the right because the available
enthalpy, and thus the specific impulse , d e c r e a s e s with decreas ing p r e s s u r e
for fixed expansion ra t io and init ial t e m p e r a t u r e .
F igure 2 shows in a different way the effects of the gas composition x
and the expansion ra t io e. That the re should be a (weak) minimum with r e
spect to e is eas i ly seen. As the expansion ra t io i nc r ea se s we t rans form
more and more of the stagnation enthalpy into specific impulse, thus d e c r e a s
ing the total m a s s via Eq. (6). However, the nozzle m a s s is increas ing with
e according to Eq. (1) and soon the m a s s inc rease from this source over
shadows the dec rea se from the improvement in I , This i s a smal l effect '^ sp
since the nozzle m a s s i s , in any account, a small fraction of the engine m a s s
( less than 20%).
The s imple model we have employed for these data is inadequate to
show the exis tence of minima in $ with r e spec t to initial t empera tu re ,
d ischarge t ime constant T , and pulse r a t e . These come about as the resu l t
of consider ing the effect of t e m p e r a t u r e on the strength of the meta l s of the
p r e s s u r e vesse l and nozzle and the effect of the pulse ra te on the effective
AV due to the influence of the gravi ta t ional field. Piat t has shown" that the re
is a p rac t i ca l upper l imit to the specific impulse that the engine may deliver,
based upon the t empe ra tu r e that the ma te r i a l of the p r e s s u r e vgssel^r^jt^^s.^^,,^ 1 <' ^ , J;
mfr*n
This leads to the existence of a minimum in $ with respect to initial gas
temperature. The heat load in the vessel walls is a function of T , as will
be shown in Sec. III.
Figure 3 shows the effect of varying the mission AV and payload on the
total initial system mass .
III. HEAT TRANSFER
A crucial element in determining the feasibility of the Helios concept (and that of many other advanced propulsion schemes) is the heat load that the material of the engine must withstand. Toward this end, it is necessary to understand the various processes which take place within the engine during and following the release of nuclear energy.
A. Description of Firing Cycle
Chronologically, the events taking place prior to the nuclear detonation are as follows: A charge is positioned within the vessel and the nozzle throat is sealed. Next, the cavity is filled with propellant (hydrogen) which has
removed from the vessel walls the energy deposited there by the previous
charge. The high explosive (H.E.) which tr iggers the nuclear device is set
off and the debris of the chemical reaction expands radially into the hydrogen toward the vessel wall. The chemical explosion is followed very shortly by the nuclear explosion.
In order more precisely to explain the events which now occur, we
refer to a particular numerical example computed via a one-dimensional hydrodynamics and radiation-diffusion code. The postulated conditions are as follows:
Pressure vessel radius 15 ft Charge mass (H.E. and nuclear) 50 lb Mass of hydrogen 150 lb
9 Nuclear energy release (yield) 5 X 10 cal
The post-nuclear events are listed below:
1) The central charge mass rapidly expands into the surrounding H.E. By the time the expanding masses reach a radius of 150 cm, approximately 90% of the yield has been converted into kinetic energy in H.E.
2) As the charge m a s s e s expand far ther , the m a s s of hydrogen affected
by the p rog res s ing disturbance begins to become significant. The kinetic
energy begins to r eappea r as in ternal energy in the hydrogen as it is com
p r e s s e d and heated. The shock front, init ially at a p r e s s u r e of roughly
3 kbars , gradual ly drops to about 0.5 kbar as it approaches the wall. As the
shock wave contacts the wall, 0.15 m s e c after "detonation," the yield energy
Y is divided as follows:
Total kinetic energy 0.33 Y
Internal energy, H.E. 0.10 Y
Internal energy, hydrogen 0.57 Y
3) At the initial instant of reflection, the vesse l wall is subjected to a
peak p r e s s u r e of 5 kbars , and a t e m p e r a t u r e in the adjacent hydrogen of
14,000°K. These values subsequently decay by a factor of 2 in t imes on the
o rder of 10 /jsec and 20 jusec, respec t ive ly .
4) Also at this t ime, near ly all of the m a s s contained within the vesse l
is compacted against the wall in a shell l e s s than 7 5 cm thick. The r e tu rn
ing shock wave rapidly t r a v e r s e s this region and d isappears into the center ,
leaving the gases essent ia l ly at r e s t . The energy at this point is then
divided as :
Total kinetic energy 5%
Internal energy, H.E. 18%
Internal energy, hydrogen 77%
5) The g a s e s then implode toward the center of the vesse l , reaching a
maximum kinetic energy of 0.19 Y.
A subsequent shock wave, much sma l l e r than the f irst , is formed at
a rad ius of about 200 cm as the m a s s e s concentrate in the center of the
vesse l . This produces a second s e r i e s of events s imi la r in nature to the
f irst , but reduced by near ly an o rde r of magnitude as measured by the
sys tem kinetic energy. The second shock front a r r i v e s at 0.67 msec and
has a p r e s s u r e on the o rde r of 0.15 kbar, producing a p r e s s u r e peak at the
wall of 0.6 kbar . After the second pulse at the wall, the kinetic energy
osci l la t ions in the vesse l decay rapidly from a maximum of about 0.0 5 Y.
The energy is finally divided as :
Internal energy, hydrogen 0,72 Y
Internal energy, H.E. 0.28 Y ^^^^^^^^^te«. .__^
The opening of the throat valve is triggered by the arrival of the first
shock wave and proceeds slowly in comparison to the events following the
detonation. Indeed, we expect that the kinetic energy induced by the detona
tion within the gas will have all but disappeared by the time any appreciable
fraction of the gas has been released. The hot gases now exhaust through
the nozzle, completing a full firing cycle of the Helios engine.
B. Sources of Energy
1. Prompt Nuclear and Atomic
The nuclear reaction itself will irradiate the walls of the vessel with neutrons and 7 rays. The energy associated with these represents about 6% of the total energy released. A portion of this radiation will be contained within the gases, and some will pass through the vessel walls and escape into space. Although no quantitative calculation can be made without an actual design, we believe that the total amount of energy of this form trapped within the material of the vessel walls can be held to less than 1% of the total nuclear yield. This energy will be fairly uniformly deposited throughout the material of the pressure vessel.
A first guess at the total energy associated with early x radiation from the nuclear charge is around 1/2% of the total yield. Proper design of the nuclear device can probably trap most of the energy within the confines of the fireball.
2. Early Thermal Radiation
As stated earlier, the numerical techniques by which we arrive at the dispersal of energy (excluding the above two forms) include only hydrodynamics and radiation diffusion. It may be inferred from the discussion of Sec. III. A that at least until the initial shock wave has reached the vessel wall, the gas will contain regions wherein there exist large thermal gradients and relatively cold hydrogen which is optically transparent. The diffusion approximation to the radiation transport equation is evidently invalid over certain regions of the gas at early times in the cycle. However, the temperatures are low enough that essentially all energy transport is through the hydrodynamic work done on and by the fluid. The energy radiated in the hot gas is so small that its neglect does not alter the hydrodynamic behavior of the g9,s^ Formal'c«»iiputer results agree well with rough calculations in terms of the general behavior described below.
Until the hydrogen near i t s interface with the charge res idue (H.E.) is
heated, the ex t r eme ly hot f i reball will r ad ia te toward the cavity wall. Ionizing
radiat ion — including the x r a y s mentioned above — emanating from this
f ireball will be s t rongly attenuated in the H.E. res idue . Once the hydrogen
near the H.E. begins to heat, i ts opacity i n c r e a s e s , allowing it to absorb
radiat ion and heat s t i l l more rapidly. Quite soon the opacity of the hydrogen
has r i s en to a point where it completely blocks further radiation from the
cent ra l r eg ions . The t ime sca le for th is i s negligibly short when measu red
in t e r m s of the total radiant energy reaching the cavity wall. The hydrogen
t e m p e r a t u r e nea r the H.E. interface is now between 1 and 3 eV and has a
l a rge negative gradient with r ad ius . Since the opacity of hydrogen is an
ex t remely s t rong function of t e m p e r a t u r e , it drops precipi tously with
increas ing rad ius , creat ing what appears to the cavity wall to be a well
defined surface in the hydrogen radia t ing toward the wall at a t empera tu re of
about 1 eV. This surface p e r s i s t s and is inoved about by the passage of the
shock wave for about the t ime it t akes the shock to t ravel th ree cavity d iam
e t e r s . At this t ime the total energy r e l ea sed has been distr ibuted throughout
the cavity and although the centra l ga ses a r e sti l l quite hot, they a r e highly
expanded and contain only a smal l fraction of the total energy. Also by this
t ime the gas will begin to feel the effects of the opening throat valve and will
s ta r t falling in t empe ra tu r e due to i sen t ropic expansion. It is also expected
that a cer ta in amount of mixing will have taken place, tending to make the
gases in the cavity i so the rmal at a t empe ra tu r e of about 6000°K,
To find an upper l imit to the energy incident on the vesse l wall from
this source , we a s sume that the radiat ion is blackbody at 10,000°K. Thus
CT T^ = 1.4 X 10^ cal-cm~^-sec~-^. (8)
The radiat ing surface will pe r s i s t for probably no longer than 1.5 msec and
will be moved about by the shock wave so that i ts average a rea is about one-_2
half that of the vesse l wall. Thus, about 10 cal*cm in radiant energy from
the fireball impinges upon the cavity wall. Fo r the example discussed in
Sec. A, this is about 0.53% of the total energy re leased .
^ ^ r 3. Late The rma l Radiation
As indicated above, the gas should be roughly i so thermal within a
mil l isecond or so and will have begun to exhaust through the nozzle. As g a s ^
•11-
l eaves trre chamber , that which r e m a i n s expands isentropical ly , dropping in 5
t e m p e r a t u r e . R. Fox h a s shown that the t ime var ia t ion of the t empe ra tu r e
of the gas in the chamber (assumed i so thermal a c r o s s the chamber) is
T(t) = T(0) 1 + (9)
where T(0) is the gas t e m p e r a t u r e within the cavity after it comes to an
i so the rmal state and pr io r to any appreciable amount being expelled. Fo r
a perfect 7- law gas , T is d i rec t ly proport ional to the cavity volume and
inverse ly proport ional to the product of the effective throat a r e a and the
speed of sound in the gas at t ime t = 0.
If the gas within the cavity r ad ia t e s like a blackbody, the total radiant
energy str iking the wall is the t ime integral of the instantaneous flux, i .e. .
00
T^ dt = a T(O)'*
00
J d . t / r ) ^ d t
0 0
- a T ( 0 ) ^ ( T / 7 ) .
F o r the reasonable values of T(0) = 6000°K, and r
(10)
0.2 sec we have
00
0 a T^ dt == 50.1 cal-cm ^. (11)
In t e r m s of the example of Sec. III. A the fraction of the energy incident on
the wall is 2.6 5% of the total energy r e l eased .
4. Conduction
Since the gas i s in int imate contact with the cavity wall, both conductive
and convective heat t r ans fe r may take place. The fo rmer is but a lowest-
magnitude special case of the l a t t e r wherein the only gas motion allowed is
a rad ia l expansion or contract ion. An upper bound to the total conductive
heat t r ans fe r can be obtained by postulating that the t e m p e r a t u r e of the wall
be kept at ze ro .
S'
• 1 3 -
Carrying out a calculation which takes into account the time-varying
tenaperature and density of the gas (see Ref, 3), we find that the heat con
ducted to the wall is given by
Ai
Q 1 ^ 1 ° ° k ^ d t = kQ T(0) [p (0 )T]^ / l (12)
where the constant k_ is a function of the gas properties and T(0), p(0) are
the gas temperature and pressure at the beginning of the expansion.
For the case quoted in the section above (r =0.2 and T(0) = 6000° K) we _2
have, for a pressure of 100 atm, Q =2.8 cal- cm . The fraction of energy
transferred to the wall by pure conduction is thus 0.15% of the total energy
released. 5. Convection
Pure conductive heat transfer presupposes a stable thermal gradient in the gas which tends to protect the wall from the hot material within the cavity. Turbulent motion will, however, destroy this cold layer and continuously supply the wall with hot gases from the interior. The heat flux, abetted by convection, is expected to be considerably larger than that due to conduction along. Near the nozzle, the flow of gas leaving the chamber will provide strong convective motion at the wall, and it is in this region that the greatest amount of convective heat transfer is expected.
In the absence of detailed knowledge of the three-dimenional hydro-dynamic behavior of the gas we have made an estimate of this mode of energy transfer by assuming that the cavity wall behaves like a cylindrical duct carrying gas at the same mass flow rate and temperature as the gas actually leaving the ve area in this duct to be
3 actually leaving the vessel. We have shown the heat transferred per unit
{^J s'^^nofrf'' i^Y'' (TO)'''' -1--"'. (13) where D (in feet) is the postulated duct diameter. The heat transfer rate
starts quite high and falls off very rapidly [it goes as (1 + t / r ) for a
7-law gas with 7 = 1.2] and thus most of the heat transfer is over within
about 0.02 sec. Considering conduction within the material of the vessel
wall, the t e m p e r a t u r e of the surface will roughly be given by
T « Q(kp C ^ t ) ' ^ / ^ (14)
With -2
Q - 140 ca l ' cm , C = 0.1 ca l .g~^ .°K"\
P _i _x -k = 0.1 cal-cm -sec '"K
_3 p = 7 . 8 g.cm ,
t = 0.02 sec ,
and
we have
T ~ 4000°K s
which is high enough to vapor ize i ron. It thus appears that a protect ive
coating must be provided within the chamber in the region of the nozzle exit.
6. Energy of Vibration
The sudden r i s e in in ternal p r e s s u r e accompanied by the initial shock
wave will set the p r e s s u r e vesse l into vibrat ion. This vibrat ion will damp
out, and the associa ted energy will mainly be t r a n s f e r r e d into heat deposited
throughout the ma te r i a l of the ves se l . An es t imate of the energy involved can
be obtained by assuming that only the fundamental mode of oscil lat ion is
excited. If the gas p r e s s u r e were to inc rease in a single step to i ts final
value, the maximum s t r e s s in the ves se l would be just twice that which would
r ema in after all osci l la t ions had died out. The f i rs t shock wave and each
reflect ion adds a smal l momentum impulse and so adds to the maximum s t r e s s .
We have es t imated that for the example of Sec. A the inc rease in s t r e s s due
to the shock waves is about 15% of maximum s t r e s s in the absence of shocks.
Thus the vesse l must be designed for a p r e s s u r e 2.3 t i m e s the steady value.
This allows no safety factor, and so we have used an overal l factor of 4 in
determining the requ i red vesse l wall th ickness 6:
PnP 6 = 4 ^ . (15)
s
II. ,'"'- ^ 15-
Here p„ is the final p r e s s u r e at t ime t = 0, r is the radius of the cavity,
a is the yield s t r e s s , and 6 is the th ickness of the shell . The energy of
s t r e s s per unit m a s s is given by
2
* ) (16)
where p is the shell density, v is Po i s son ' s ra t io , and E is Young's
modulus for the shell m a t e r i a l . Thus the maximum total energy of s t r e s s to
be dissipated in the vesse l wall is given by
Q = ^ s ^M 47rr
P Q T
2CT
^^ ^^ PO ^ s I ^ (17)
Assuming
we obtain
P( CT
V
E
r
1.01 X 10 dynes-cm"
2.13 X 10^° dynes-cm"^
0.3
2.08 X 10^ dynes-cm"^
15 ft = 457.2 cm.
Qg = 1,78 X 10 cal .
This is 0.36% of the total energy re l eased .
A shock wave pass ing through the vesse l wall leaves energy in i ts
wake. We m.ay make an upper es t imate by re fe r r ing to Eq. (16)! The
maximum intensity of the shock wave of Sec. A is around 5 kbars . Using the
physical p rope r t i e s quoted above we see that the shock wave can leave only 5
about 4.7 X 10 cal in the p r e s s u r e vesse l . This almost a factor of 50
below the energy of g r o s s vibrat ion and is thus negligible.
7. Summary
Nuclear radiat ion and vibrat ion will deposit heat uniformly throughout
the ma te r i a l of the p r e s s u r e vesse l , while the other sources c a r r y e3fea^gy to
the inner surface of the vesse l only. Convective t rans fe r , especial ly near
•16-
the nozzle entrance, would result in such a large deposition of heat that it seems clear some protection against it must be provided. This protection would also, of course, reduce the other modes of surface heat deposition. The estimates made here must thus be interpreted as only a general indication of heat transfer amplitudes.
Table 1 summarizes the results in this spirit, with the convective contribution in parentheses since it must be presumed absent or reduced in a workable engine.
No functional dependence is given for the deposition of nuclear radiation; it depends upon the charge design and vessel geometry in a complicated manner and the 1% value of charge yield fraction given is meant only as a rough upper limit.
The r ise of temperature of the vessel as the result of a single pulse is of interest. Generally we find that the ratio of vessel mass to nuclear yield is about 20 tons per ton of H.E. equivalent energy, or about 0.02 gram per calorie. A heat capacity of 0.1 cal-g -°K would result in a 500°K r ise in temperature if the whole yield were deposited and thus a 5°K rise for each unit percent of the yield deposited in the shell.
»-«SSI' - ' • t ^ *
* ^ ^
L f'^^'-.^ **•*
17-
Table 1. Energy Delivered to Pressure Vessel
S o u r c e
P r o m p t n u c l e a r r a d i a t i o n
E a r l y t h e r m a l
L a t e t h e r m a l
Conduc t ion
C o n v e c t i o n
V i b r a t i o n and shock
C h a r g e y i e l d f r a c t i o n
0.0100
0 .0053
0.0267
0 .0015
- -
0.0037
Surface heat (cal-cm~2)
Energy deposition functional dependence
TOTAL 0.047
10
50
3
(140)
63
(-200)
Constant 2 ™ 4
r T T Q
2 rr / xl/2 r T Q ( P Q T ) '
2 Tn r ^ - 2 - (T T ^ D )
D" ^
r p .
0.2
Notation: T
r
^ 0
Po D
discharge time constant
pressure vessel radius
initial chamber temperature
initial chamber pressure equivalent diameter of vessel considered as cylindrical duc:t»
m% « . « « !
Is m 18-
REFERENCES
Helios Quar te r ly Report No. 1, Lawrence Radiation Laboratory, (Livermore)
Rept. UCRL-7675 (1964). SRD 2 Helios Quar te r ly Report No. 2, Lawrence Radiation Labora tory (Livermore)
Rept. UCRL-7939 (1964). SRD 3 Helios P r o g r e s s Report No. 3, Lawrence Radiation Labora tory (Livermore)
Rept. UCRL-12222 (1965). SRD
D. M. Cole, "The Feas ib i l i ty of Propel l ing Vehicles by Contained Nuclear
Explosions," The Mart in Co., January, 1960. P re sen ted at the 6th National
Annual Meeting, Amer ican Ast ronaut ica l Society, New York (preprint
No. 60-49).
Robert H. Fox, "A Study of the Nuclear Gaseous Reactor Rocket,"
Lawrence Radiation Labora to ry (Livermore) Report UCRL-4996 (1957).
E. A. Piat t , "The Effective Specific Imptdse of a Pulsed Rocket Engine,"
Lawrence Radiation Labora to ry (Livermore) Report UCRL-12296 (1965). SRD 7
Hydrogen-carbon thermodynamic tables were kindly made available by
R. E. Duff of this Laboratory.
, . - « # * •
PRESSURE (Atm)
COST (10^$)
15,000
l i j
g 1 0 , 0 0 0 -U-
o £ 7.500 £D
5,000 1.0
GLL-656-959 PLENUM CHAMBER RADIUS
(meters)
Mp= lOOtons = 200
T =6000 **K AV =60,000 ft/sec
i_±J 8 10.0
. 1. Curves of fixed mission cost, showing functional relationship among chamber radius R, number ses N, chamber pressure p, and propellant hydrogen fraction x-
0.70 100
Gli-656-960
wig*"!
200 500 PRESSURE (atm)
1000
Fig. 2. Mission cost vs chamber p r e s s u r e for fixed values of nozzle expansion ra t io e and hydrogen fraction x, and for continuously optimized values of e and x.
I (S3 O
I
10'
100-
-
/
/
o/o/ .p/ oy / c<V
/ / 1
/ / o/ / / o/
/ / o/ / / V
/ / / II 1 , / /
/ /
o/ 07
11
/ /
0 0 /
0 / 0 / v /
/
/
/
/ IDEAL MISSION ' VELOCITY
INCREMENT (ft/sec)
P= 100 atm T= eooo^K €= 2 0 0
1 X CHOSEN SUCH THAT d$/dX =0
1 1
100 10^ 10^ 10^ TOTAL MASS (tons)
GLL-656-961
3. Relationship of payload mass to total initial mass for various mission ideal velocities
m
ffiirsiEo - 22 -
DISTRIBUTION
Series A
LRL Livermore,
J. Hadley
Series B
LRL Livermore, Internal Distribution
LRL Berkeley, Wallace B. Reynolds
Division of Military Application, Brig. Gen. Delmar L. Crowson
San Francisco Operations Office, Ellison C Shute
Los Alamos Scientific Laboratory, Los Alamos Ralph Cooper
LRL Livermore, Internal Distribution
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