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RiSO-M-2662
Comparisons Among Various Possible
Ways to Accelerate High-Speed Pellets
Under Constant Base Pressure
Ming-Lun Xue
Rise National Laboratory, DK-4000 Roskilde, Denmark
September 1987
RISØ-M-2662
COMPARISONS AMONG VARIOUS POSSIBLE WAYS TO ACCELERATE
HIGH-SPEED PELLETS UNDER CONSTANT BASE PRESSURE
Ming-Lun Xue*
* On leave of absence from Institute of Mechanics, Chinese Academy
of Sciences, Beijing, China
Abstract. Various possibilities to accelerate a frozen hydrogenic pellet by means of a constant base pressure behind the
pellet are examined.
September 1987
Association EURATOM - Risø National Laboratory
Risø National Laboratory, DK-4000 Roskilde, Denmark
ISBN 87-550-1347-3
ISSN 0418-6435
Grafisk Service Risø 1987
CONTENTS
Page
1. INTRODUCTION
5
2. STATIC PRESSURE PROGRAMMING AT THE ENTRANCE OF
THE BARREL
v
7
3. CONSTANT THRUST BY "ROCKET EFFECT*
10
4. THE ADDITION OF GAS MASS ALONG THE BARREL
13
5. CONCLUSIONS
15
6. NOMENCLATURE
17
7. REFERENCES
18
8. FIGURES
19
- 5 -
1.
INTRODUCTION
Various possible ways to accelerate high-speed frosen pellets
under constant base pressure are examined. If the goal is to
attain a pellet speed of 5 ka/s we find the following: 1) we
must take account of gas-wall friction. Static pressure programming at the entrance should be placed in operation at a high
temperature (Tb > 2000°K) to enable the pellet to survive;
otherwise the pressure at the barrel entrance will be too high
to be tolerable. 2) Constant reaction thrust, i.a. a "rocket
effect", seems to be no basic obstacle if beam technology (a
laser is preferred) is available, though the energy deposition
problem should be carefully studied. 3) Mass addition programming along the barrel through the slots offers an interesting
possibility for using a low-temperature gas, since the gas
speed is able to be separated from the pellet speed here.
It appears that refuelling by pellet injection into the fusion
plasma will offer a most promising way to sustain quasi-steady
thermonuclear reactions, extrapolation of existing theoretical
and experimental results about ablation of a pellet that passes
through a high-temperature plasma indicated that a much higher
speed is probably needed for post-break-even applications [l,2j.
A pellet speed of 5 km/sec seems to be a reasonable choice here
to examine possible improvements in pellet injection technologies.
The dynamic properties of frozen deuterium are yet to be determined. This is clearly the best way to fully exploite the potential of pellet acceleration that is based on constant base pressure acceleration (CBPA) with the maximum allowable stress to
enable the pellet to survive nearly all along the way.
- 6 -
Three different types of CBPA are examined:
1) Static pressure programming with due consideration taken of
friction and heat transfer effects.
2)
'
Constant reaction thrust acceleration by the exhaust from an
ablating pellet ("rocket effect").
3)
Programmed mass addition of the driving gas during acceleration of the pellet in the barrel.
- 7 -
STATIC PRESSURE PROGRAMMING AT THE ENTRANCE OF THE BARREL
(PIG. 1)
The basic philosophy for this kind of acceleration is that all
the gas following the pellet is to be kept at the sane speed
as that of the pellet. Since a pressure gradient is needed to accelerate the gas, the pressure gets higher and higher at the entrance during the acceleration process. It was shown [2] that
friction can be neglected between the pellet and barrel but must
be considered between the gas and wall, especially when a high
pellet speed is desired. The motion of the pell« in the barrel
is described by
dvp
r
Pb Ap
(1-1)
dt
If base pressure remains constant, acceleration
dv p
a« »
P
pb A p
»
<*t
pb
»
Mp
* const.
(1-2)
Pplp
and
v p » apt
(1-3)
L --jrapt2
(1-4)
Prom (1-3) and (1-4)
I «-*2a p
(1-5)
Por a fully developed turbulent flow in the boundary layer, the
flow of the driving gas can be approximately described by
dv q
Pq — *
9
dt
dp«
v« 1
* - fpq—*•—
dL
9
2 d
(1-6)
- 8 Here £ is the friction coefficient [4J. since
dvg
dvp
dt
dt
a D = const, and v„ « v D for CBPA, (1*6) can be
F
*
*
written as
d
Pg
L
dL «
f — dL
(1-6a)
Pg «p
<*
assume the existence of a polytropic relation between the pressure and density as a simplified way to take account of heat
transfer
Pq
Pg n
Tg n/n-1
— » e-2) « C-2)
Pb
Pb
*b
(1-7)
integrate (1-6a)
L2
Pb
n
, Pg»L-o n-1/n
f—. + L «
(—-) [ ( —
)
- 1]
2d
P p a p n-1
p0
(1-8)
or
PgrL«o r
, pp*p K _4
1 _,,, K(n-1).n/n-1
—
• (I + Ut-*-*>)v* + - v*]} -i
i}
(l-8a)
Pb
Pb d 8 P 2 P
n
here v p > Vp/Cb. C b - (KRTj,)1/2, k-specific heat ratio. Without
consideration of friction and heat transfer, f * 0 and n • k, Eq.
(1-8a) reduces to an ideal CBPA for« [sj:
Pg'L-o
K-1 _« K/K-1
t-1
)
"<1+-T-vg>
Pb f«o, n«K
2 *
(1-9)
Typical results of Bq. O-8a) are shown in Fig. (2). It is clear
that in order to keep the maximum pressure at a tolerable level
the gas temperature T& should be kept high enough when high pel'
let speed v p is needed. Higher gas temperature means lower gas
density and the gas then needs a reduced pressure gradient to
accelerate it.
- 9-
So*e numerical results are shown in Table 1.
Table 1.
Bxaaples of Pg,L*o f° r CBPA
5 km/s
5 ka/s
5 Iw/s
Pb
100 bar
(300)*
100 bar
(300)
100 bar
(300)
Tb
1000 K
2000 K
3000 K
2.4*10~ 3
g/c«3
(7.2*10-3)
1.2*10-3
0.802*10*3
(3.6*10-3)
(2.4*10-3)
V
P
p/KT
*b
cb
(KBTb) 1 / 2
2.412*10*
ca/s
3.411*10 5
4.178*10 5
*P
Vp/C b
2.0726
1.4655
1.1967
0.2 g/c« 3
0.2
0.2
Vd
1.0
1.0
1.0
f
0.01
0.01
0.01
n
1.2
1.2
1.2
Pg,L«o
(-2
1
Pb
*"©
11.44
3.825
2.5269
Pg,L«o/Pb
(25.04)
(6.1947)
(3.58)
Pg,L«o
9314 bar
(7513)
1466 bar
(1858)
678.5 bar
(1074)
a
10* cw/s*
( 3*10*)
125 cm
(41.6)
10*
10*
(3*10*)
125 ca
(41.6)
(3*10*)
125 em
(41.6)
P
P
L
(
)* parameter« for Pb • 300 bar
- 10 -
With other parameters fixed, from Eq. (1-8a) an optimal base
pressure (Pb)opt. exists to make Pb,L=o minimum. From
d
<Pg,L«o>
* o
dPb
f K2
p
DLDRTb vo
1 K(n-1) — t
Prom these numerical results it seems reasonable to have a pellet
speed of Vp * 5 km/s.
1) Pressure loss due to friction is always the dominant part of
the pressure gradient.
2) Pellet base gas temperature below 1000 K will make the static
pressure (as well as stagnation pressure) at the entrance too
high to be tolerable.
3) The pressure programming for pellet base gas temperature
above 2000 K is probably tolerable. In this case single-shot
injection will be possible only if the pellet could sustain
the high pressure. Multi-injection will be very difficult.
3. CONSTANT THRUST BY "ROCKET EFFECT"
If some kind of beam energy could be absorbed by the rear part of
the pellet, part of the mass of the pellet will be vaporized,
heated and expanded as rocket exhaust. A constant thrust will be
applied to the pellet to produce a constant base pressure acceleration if the mass flux and exhaust speed are maintained constant. This is expressed by
- 11 -
integrate Eq. (2-1)
_
vp
Mpo
1
v » -Z- - I n — — * In — 3
v
M
ex
p
1-t
(2-2)
Here
_
p
t
T
p *po vex
Pb
Integrate Eq. (2-2) to get the launch length of the barrel L
L -—
= 1 - (1-t)[l - ln(l-t)]
(2-3)
with
L.
2
* vex
p
p*po
Pb
The pellet speed v, launch length L and pellet mass M p as functions of time in dimensionless forms are shown in Pig. 3.
Typical numerical results are shown in Table 2. (The left one is
a D2 pellet, and right one is a hybrid D£ + H2 pellet in which
the H2 part is to be burned).
From these numerical results/ it is shown that nearly 20% of the
initial D2 pellet mass could remain after a speed of 5 krn/s by
the reaction force is reached. If a laser is used the beam energy
and power rate needed is already well understood, apart from
some uncertainty in the absorbing properties of a pellet on the
impact of laser energy.
- 12 Table 2.
Examples of constant thrust by rocket effect
V
5 km/s
5 km/s
Pellet
*>2
D2 + H2 (H2 for
propellent)
Final size
tp 0.5 cm
0.5
dp 0.5 cm
0.5
Tex
3000 K
3000 K
v
» (KRT^)1^
4.178x10 s
P
ex
2.954*105 cm/s
Pb
100 bar
100 bar
Mp
0.0196 g
0.0196 g
Mp©
V
P
M p exp(-—) = 0. 1065 g
v
ex
0.0648 g
0.184
0.3021
2.72 cm
3.877 cm
66.47 g/s
47 g/s
Mp
Mp«,
*po
dMp
~"dt"
t
A
V
dMp
• 1.,3*10~ 3 s
0.96X10" 3 s
" dt
L
234 cm
192.5 cm
Energy
input
E
Iff
2465 J.
1.9x106 w
2650 J.
2.76x106 W
02
burned
2.22 cm
H2
burned
2.877 cm
- 13 -
4. THE ADDITION OF GAS MASS ALONG THE BARREL (FIG. 4)
The basic idea in adding gas in the barrel is that when the pellet moves forward inside, a correct amount of gas enteres the
barrel along the side wall to barely fill the volume evaculated
by the pellet and maintain the barrel pressure constant. In that
way the pellet will continuously be acted upon by a constant
base pressure.
Since
dvp
Mp —
" PbAp
with p D * constant.
<Jvp
aD »
F
PbAp
»
dt
Mp
Pb
» — — • const.
Pp*p
v p - apt
L(t) - i a p t 2
The total mass that should be filled is
M - PbAbMt)
Gas flow rate
dH
dL
0 • — • PbAp — • Pb*p»P
• Pbty>pt
- 14 -
Here the gas and pellet speeds can be separated* which is not
the case in single- or two-stage injectors where the gas and
pellet speeds are closely connected.
In our case the gas axial speed is very close to zero so that
only a much slower radial speed is needed to permit the gas to
fill the barrel. In this event,even room temperature gas can
be used.
In actual practice« the slot along the barrel should be opened
discretely in order to feed in the gas. We examine the most
serious case (latest period of launch):
Vp
pD
Tb
db
»
»
=
5 km/s
100 bar
300 K
0.5 cm
For the last 10 cm length of the barrel, the pellet speed is
nearly 5 km/s. The transit time to enable the pellet to pass
through is
10
AL
At »
»
vp
,
« 20x10-6 sec » 20 Ms.
5*105
The gas mass should fill the volume swept by the pellet during
each increment of time, so that
At
***•*• J- " pbAbVp*t
subscript s refers to the parameters at the slot, where the gas
flow is at the sonic speed. Por a simple slot
V
S " C s " ( K R T , ) 1 ^ • (KRTb) 1 / 2
- (1.4 *
8.314*107
.„
* 300)1/2 • 1.32*105 c»/»
and the density P» is the same as P&
- 15
»S
— »1
Pb
p v
*
o
5»105
2
A. « 2Aw
- 2 * - (0.5) *
^
^ PSVS
4
1.32x105
b p
« 1.487 c«2
The length of slot L s is equal to 10 cm, so the width of the slot
< 8 - 0.15 cm » 1.5 mm.
The stagnation pressure for the feeding gas
*
Ps " Ps*
1 +
k-1 *A-1
~T>
"
2
U9X
P s
* 190 bar (k * 1.4)
CONCLUSIONS
If the aim is to have a pellet speed of 5 km/s, we concludes
1) For static pressure programming at the entrance, the pressure
loss due to friction between gas and wall will be the dominant part of the total pressure gradient. Any gas temperature T D below 1000 K will have a pressure that is too high
to be tolerable; if T D > 2000 K it will be possible to make
a single shot. Multi-injection will be extremely difficult,
because the lower the pressure the higher will be the temperature.
- 16 2) If beam technology is available (B ~ 2,500J w ~ 2*10*11) and
energy deposition is not a problem then propulsion by
"rocket" reaction force seems to be no basic obstacle.
Furthermore, since the exhaust gas fro« the pellet moves
in the opposite direction of the motion of the pellet«
friction between gas and wall is even helpful.
3) The addition of mass along the barrel through slots could
take place even in room temperature if problems related to
receiving signal« triggering, and gas inlet could be solved.
ACKNOWLEDGEMENTS
The author is very grateful to Dr. V.O. Jensen for an invitation to visit Risø National Laboratory during the spring of 1987
when this work was done. Throughout this period very fruitful
discussions were held with Drs. V.O. Jensen, C.T. Chang,
P. Michelsen, S.A. Andersen, H. Sørensen and other members of
the Risø pellet injector group.
•OMBNCLATURE
a
A
c
d
f
6
*
acceleration
area
sound speed
diaaeter of barrel
friction coefficient
gas flow rate
length
L
N
n
p
R
t
v
length along barrel
mass
polytropic factor
pressure
gas constant
tine
speed
P
density
SUBSCRIPT
b
p
o
ex
base of pellet
pellet
initial state
exhaust
- 18 -
REFERENCES
1. Hilor«, S.L., Journal of Fusion Energy J^, Mo. 1 (1981) 15.
2. Andersen« S.A., Ris#-H-2536, Ris« National Laboratory,
Denmark (1985).
3. Xue, M.L., Considerations of Several Real Effects in
Fneuaatic Pellet Injection Processes, Rise Report (to be
issued). Ris« Rational Laboratory (1987).
4. Schlichting, B., Boundary Layer Theory, Pergaaon Press (19(8)
580.
5. Seigel, A.B., The Theory of Bigh Speed Guns, Agardograph 91,
Hay 1965.
»n
Entrance
Barrel
Pellet
L*
I
Fig. 1.
Schematic diagram of static pressure programming at
the entrance of barrel for CBPA.
- 20 -
100
i
11
1.6 / ; as
K =U
n =U
60 - 1'
V
ff
/f
; i
ff
P = Vp/Cb
Cb= (KRTb)
40
/f'=0
f = tipped)
n
/
/
20
/
/
/
/
/
/
9 10
æ 8
/
/ /
/
/ / /
+
^
1 - ——T~"**T
0Å
02
I
I
i
i I I
0.6 0.8 1.0
2.0
Fig, 2a. Static pracsura prograaning at tha antranca of
barral for C»PA,
4J0
- 21 -
100
60
T—i—l
K =U
n = 1.0
V
40
i i |
1.0 | IQ5/f'=0
P = Vp/Cb
!
Ct= (KRTb)
f = t(Pplp/ppd)
If
!
20 -
'
/
!
I I //
/
/
/
i
7/
li
8
/
4 -
li
i
ii
y
#
4?
j
02
Tig.
OÅ
2b.
i
i
06
• • • i
OB 1J0
2J0
AJO
- 22 -
32
2.8
- 2Å
ZO
- 1.61>
- 1.2
- Q8
- OÅ
02
Fig. 3.
04
06
t
08
Constant thrust by rockst sffsct.
0.0
1.0
w\
Pb
i.
•Ve
L(t)
Flg. 4.
Gaa maaa addition programming along tha b a r r a l .
to
w
Risø National Lahfnratorv
TiMt
Mltor(t)
Rise - M
-
2662
»•«• September 1987
COMPARISONS AMONG VARIOUS POSSIBLE HAYS TO
ACCELERATE HIGH-SPEED PELLETS UNDER CONSTANT
BASE PRESSURE
DtpartiRMt m f * « »
»hysics Department
Gr*«#> •mm r*fi»tr«tt*n
til
Ming-Lun Xue*
•On leave of absence from Insitute of M e c h a n i c s rPM|«it/CMtract
Chinese Academy of Sciences, Beijing, China
M M
2
intimt* 5
nm.
ISSN 8 7 - 5 5 0 - 1 3 4 7 - 3
(Mai. M M cfcar.)
Various possibilities to accelerate a frozen hydrogenic pellet by
ins of a constant base pressure behind the pellet are examined.
Omcripitt - INIS
ABLATION; ACCELERATION;COMPARATIVE EVALUATIONS; DEUTERIUM; FACTION;
PELLET INJECTION; PLASMA GUNS; PRESSURE DEPENDENCE
PJO.Bmm,
mnu,**.im.'iém-.miå,i**m:nnnm
Available on request from.
RIM Library,
Rita National Laboratory, P.O. Box 49,
DK-4999 Roskilde, Denmark
Phone (92) 371212 ext.2292
ISBN 87-530-1347-3
°4i*-«435
,SSN