The use of discrete ordinates and Monte Carlo techniques to solve radiation transport problems is discussed. A general discussion of two possible coupling schemes is given for the two methods. The calculation of the reactor radiation scattered from a docked service and command module is used as an example of coupling discrete ordinates (DOT) and Monte Carlo (MORSE) calculations

Childs, R. L.

Emmett, M. B.

Straker, E. A.

English

E.A. Straker

R.L. Childs

M.B. Emmett

Crossref

APPLICATION OF DOT-MORSE COUPLING TO THE ANALYSIS OF THREE-DIMENSIONAL SNAP SHIELDING PROBLEMS.

Many radiation transport problems can best be solved by using both discrete ordinates and Monte Carlo techniques with a coupling between the two techniques occurring at a geometry interface. A general discussion of two possible coupling schemes is given. The calculation of the reactor radiation scattered from a docked service and command module as an example of coupling discrete ordinates (DOT) and Monte Carlo (MORSE) calculations

UNT (University of North Texas) Digital Library

Application of Dot-Morse Coupling to the Analysis of Three-Dimensional Snap Shielding Problems.

NASA Technical Reports Server

Application of DOT-MORSE Coupling to the Analysis of
Three-Dimensional SNAP Shielding Problems *
E. A. Straker, R. L. Childs, and M. B. Emmett
Oak Ridge National Laboratory
Many radiation transport problems can best be solved by using both discrete
ordinates and Monte Carlo techniques with a coupling between the two techniques
occurring at a geometry interface. A general discussion of two possible coupling
schemes is given. The calculation of the reactor radiation scattered from a
docked service and command module is used as an example of coupling discrete
ordinates (DOT) and Monte Carlo (MORSE) calculations.
Many SNAP shielding problems involve determin-
ing neutron and gamma-ray transport in three-dimen-
sional geometries. The solution of these problems
by the application of Monte Carlo techniques alone
frequently requires considerable computer time;
however, many parts of the transport problem are
only one- or two-dimensional and can easily be
solved with discrete ordinates techniques. A con-
venient approach to solving the three-dimenslonal
problem is to combine the discrete ordinates and
Monte Carlo techniques, using Monte Carlo only for
the transport that is three dimensional.
In coupling discrete ordinates with Monte Car-
lo, there is generally some difficulty associated
with the compatibility of cross-section representa-
tion. This is due to the use of multlgroup cross
sections in the discrete ordinates codes and point
cross sections in the Monte Carlo calculations.
Additional problems in performing coupled neutron-
gamma-ray calculations in the Monte Carlopart also
arise in some cases. The use of the multlgroup
Monte Carlo code MORSE (ref. 1) alleviates both of
these problems in that the same multlgroup cross
sections can be used in both the discrete ordinates
and the Monte Carlo calculations, and the coupled
neutron-gamma-ray transport can be solved as one
problem.
In using the results of discrete ordinates cal-
culations as a source term for a Monte Carlo calcu-
lation, it is desirable to use as much as possible
of the detail that is available, or else determine
the effect of ignoring the detail. In other words,
the assumption of separation of energy, angular,
and spatial variation of the radiation should be
avoided.
Research sponsored by the U. S. Atomic Energy
Commission under contract with the Union Carbide
Corporation.
In general, the source term for the Monte Carle
calculation may be distributed over a volume or may
Be a boundary or surface source. Both types of
coupling will be discussed, and then an example will
be given to illustrate a specific application.
The first case to be discussed is the coupling
of results of discrete ordinates calculations [in
this case, DOT (ref. 2) results] to Monte Carlo
with a volume or region being unique to both calcu-
lations. The results of a DOT calculation of the
fluence _(r,z,E,8,_), a flve-dimensional array de_
fined for r,z within the volume of interest, is
assumed to exist. In general, one would like to
select the source term for subsequent calculations
from a conditional probabilitiy distribution for
particles leaving a collision as a function of each
of the five variables. Because of the large array
of numbers, approximations have been made in the
past in which some of the variables are selected
from non-condltional probabilities that are volume
integrated. For example, the energy may be sel-
ected from the spectrum integrated over the volume.
Likewise, the angular distribution may be assumed
to be the same for all points in the volume, or for
all energies. However, complete detail can be used
with increased data handling and storage require-
ments.
The first step in coupling over a spatial reg-
ion is to convert the fluence % to X, the density
of particles leaving a collision. This is accom-
plished by multiplying the fluence by the group-re-
group cross section:
x(r,z,Eg,8,_)- I I _(r,z,E',8',%)
8' E'
x Es(E'+Eg,8'-_ )
Then, define
676
https://ntrs.nasa.gov/search.jsp?R=19720010041 2020-03-17T03:44:33+00:00Z
T = I 2_ar I az [ X [ wt(_,m)x(r,Z,Eg,e,¢)
v v g _m
r g
where x(r,z,Eg,0,_)2_ArAz is defined as the densi-
ty of particles leaving a collision at r,z with
energy E in group g per unit polar angle 6 and azi-
muthal angle _.
wt(£) = _ wt(£,m) is the solid angle segment
m
for each polar angle 8, and
wt(_,m) = the solid angle of each azimuthal angle
for each polar angle index 4.
In order to select from this distribution, one
must form the five following distributions:
R(r) = 2zrA-------!r[ Az [ [ [ wt(£,m)
T
v g£m
Z
z(z]r)---
x x(r,z,Eg,O,$) ,
2_Az_Ar
TR(r) [ [ I wt(£,m)x(Z,mg,O,_)r) ,
g £ m
E(Eiz,r) = Az2_rAr
rZ(zl_')R(r } I_ _ wt(£,m)
x X(Eg,e,_lz,r ) ,
az2_r&r
P(01Eg,Z,r) = TE(Eg]z,'r)Z(zlr)R(r) [ wt(t,m)
m
× X_,_]Eg,z,r) ,
wt(_,m)_z2_rAr x($1e,E,z,r)
A(¢I_,Eg z,r) =
' TP(O ]Eg,z,r)E(Egl z,r)Z (ZIr)R(r) •
From these conditional probability distributions
one may select values for the source coordinates.
With the assumption of linear variation between
spatial points and angular directions the discrete
values are not carried forth in the Monte Carlo
calculation, but instead a continuous distribution
of coordinates is selected. Care must be exercised
in utilizing this type of coupling so that the
transport in the overlap region is not included
twice. This may be accomplished using a pure ab-
sorber in the overlap region in the Monte Carlo cal-
culation.
Perhaps a more useful coupling technique is
the coupling at a boundary of the system. For this
case, similar conditional probabilities must also
be formed, with some minor exceptions. Consider
the coupling surface to be the outer boundary of a
cylinder.
The leakage fluence, ¢(r,Z,Eg,6,_), at the top,
bottom, and curved surface of a cylinder is deter-
mined as a function of energy, polar, and azimuthal
angles, as well as the spatial variation on the sur-
faces. The leakage fluence on these surfaces desig-
nated as Ro_ ZT, and ZB.must be converted to current
for the selection of the source term of the Monte
Carlo calculation. Thus, there are three spatial
terms - the radial variation at the top, the radial
variation at the bottom, and the height variation
at the outer surface of the cylinder. Define T, B,
and S for the integrated leakage for the top, bottom,
and side of the cylinder.
[ 2_Ar [ [ cos(Og)[ wt(t,m)
O,R ° Eg cos0>0 m
r _
x ¢(r,ZT,Eg,O,¢ ) ,
O,R ° Eg cos6<0
_ wt(g,m)¢(r,ZB,Eg,O,qb) ,
m
S = 2_R _ Az _ [ sin(6£)
o ZB,Z T Eg £ cos_>O
x wt(£,m)cos¢ ¢(Ro,Z,Eg,O,¢) ,
with total leakage TL = T + B + S.
For each of the above terms, a four-dimensional
conditional probability must be determined. For
example, consider the bottom surface leakage:
_(r) 2_Ar= --f-[ [ Icos(o)ll
E cos6<0 m
g
× wt(£,m)¢(r,ZB,Eg,e, _) ,
EB(Eglr ) = 2_a_.__.._rBR(r) [ ]cos(e)ll _t(_,m)
cos0<O m
x ¢(r,ZB,Eg,O,¢ ) ,
675
2E_Arlcos(O)I
PB(01Eg,r)= BR(r)EBCEglr) _ wt(£,m)
x ¢(r,ZB,Eg,e, _) ,
AB(_IO,Eg,r ) = 2_ArlcosOlwt(£,m)
BR(r)EB_Eglr)PB(OlEg,r)
x ¢(r,ZB,Eg,O, _) •
The results for the top surface leakage and side
leakage are similar.
With probability B/TL and T/TL the source
would be selected from the bottom and top surface,
respectively, and with probability S/TL the source
would be selected from the side of the cylinder.
Then, given the spatial coordinates, an energy
group is chosen from %, and then the direction
cosines are chosen for PB and A B.
The selection from the probability distribution
given above is sometimes hindered by negative fluxes
in the DOT-calculated source term. If there are no
negative values, then the selection of a value x is
selected from the distribution by choosing a ran-
dom number R and finding J such that
k=O k=O
then
[R - P(Xj_l)]
x P(xj)-P(xj 1) (xj -xj_I) +xj_ 1
However, if some values of the distribution
P(xj) are negative, then an alternate scheme is
required. The random number is scaled by the sum
of the absolute values of the probabilities and a
selection is made. A weight correction is required
which can result in negative particle weights.
These negative weights in general will not severely
affect the solution, but they do, of course, affect
the variance of the result.
J  l,(xk)1<R,SuM<_ i <xk>l
k=0 k-0
where
SUM =
Nk
[IP (Xk) I
k=0
and
wate = wate* _*SUM .
If there is a large probability of selecting a neg-
ative weight, then the DOT source term should be
recalculated.
To illustrate DOT-MORSE coupling, we have con-
sidered a problem in which the coupling is at a
boundary surface. More specifically, the problem
consists of calculating the radiation scattered
from a docked command and service module due to the
leakage radiation from a 600-kWt ZrH reactor (with
its associated shielding). The problem is illus-
trated in figure i. The problem which consists of
determining the scattered neutron and gamma-ray
dose at a detector plane due to neutrons and gamma
rays leaking from the reactor power assembly has
previously been investigated at Atomics Internation-
al (refs. 3 and 4). A drawing of the reactor and
shield assembly is shown in figure 2.
i i
i l
J
ORN L-Ow6 _q-_639
REAcToR AND
SHIELD ASSEMBLY
MODUL_
DOSE PLANE
FIGURE l.-Schematic of Reactor Power System and
Docked Service and Command Module.
676
FIGURE2.-ReactorShieldAssembly.
Thereareseveralapproachesthat couldbeused
in determiningthedosedueto radiationscattering
fromthedockedmodule.Forexample,one may use
the boundary leakage flux obtained in a discrete
ordinates calculation of the reactor and shield
assembly as a source term for the Monte Carlo calcu-
lation, transport the radiation to the module,
follow the transport in the service module, and
estimate the dose at the detector plane. A second
approach is to perform an adJoint calculation of the
importance of radiation leaking from the core-shield
assembly by Monte Carlo and then integrate the impor-
tance function over the source of particles leaking
from the reactor assembly which was obtained in a
discrete ordinates calculation. Another approach
is to determine the spatial and energy distribution
of the fluence at a surface surrounding the module
and use this surface as the boundary for coupling
the discrete ordinates results to a Monte Carlo cal-
culation of the scattering in the service module.
This last approach will be discussed.
A two-dimensional calculation of the reactor
and shield assembly, as shown in figure 2, was made
with the DOT (ref. 2) code. An S8 quadrature was
used to determine the energy spectra in 21 neutron
and 18 gamm-ray groups. The radiation field at 15
radial locations along a plane at the top of the
service module was obtained with SPACETRAN (ref. 2)
using the leakage boundary fluence. SPACETRAN per-
forms a ray-tracing calculation from each spatial
mesh point to the detector point. The intensity
and energy distribution of both neutrons and gamma
rays was determined for each of these 15 locations.
See Table I for energy group structure and fluence-
to-dose conversion factors. The angular distribu-
tion of the source was determined by assuming that
all the radiation was coming from the center of the
reactor core located approximately 90 feet away.
(The problem is not expected to be very sensitive
to the small angular variation of the radiation
incident on the module.) This radiation field as
a function of space and energy was used as the source
term for Monte Carlo.
Table I. Energy Group Structure and
Fluence-To-Uose Conversion Factors
Fluence-to-Dose
Conversion Factors
Upper (mrem/hr)/
Group Ener 8y (eV) (par t/cm2/see)
Neutrons:
1 1.4918(+7)* 1.5000(-1)
2 1.0000(+7) 1.5000(-1)
6.7032(+6) 1.3700(-i)
4.4933(+6) 1.3200(-1)
5 3.0119(+6) 1.3100(-1)
6 2,0190(+6) 1.2500(-1)
7 1.3534(+6) 1.1600(-1)
8 9.0718(+5) 1.0600(-1)
9 5.5023(+5) 7.5700(-2)
I0 3.3373(+5) 5.5100(-2)
Ii 2.0242(+5) 4.0100(-2)
12 1.2277(+5) 2.4500(-2)
13 4.0867(+4) 8.5000(-3)
14 1.1709(+4) 5.0000(-3)
15 3.3546(+3) 5.0000(-3)
16 7.6852(+2) 5.0000(-3)
17 1.6702(+2) 5.0000(-3)
18 3.7266(+1) 5.0000(-3)
19 8.3153(0) 5.0000(-3)
20 1.8554(0) 5.0000(-3)
21 4.1399(-1) 3.7500(-3)
Gamma Rays:
22 1.0000(+7) 9.8000(-3)
23 8.0000(+6) 8.5000(-3)
24 7.0000(+6) 7.6000(-3)
25 6.0000(+6) 6.7000(-3)
26 5.0000(+6) 5.8000(-3)
27 4.0000(+6) 5.0000(-3)
28 3.5000(+6) 4.5000(-3)
29 3.0000(+6) 4.0000(-3)
30 2.5000(+6) 3.5000(-3)
31 2.0000(+6) 3.0000(-3)
32 1.6000(+6) 2.4000(-3)
33 1.2000(+6) 2.0000(-3)
34 9.0000(+5) 1.5000(-3)
35 6.0000(+5) 1.0500(-3)
36 4.0000(+5) 6.0000(-4)
37 2.1000(+5) 2.8000(-4)
38 1.2000(+5) 1.4000(-4)
39 7.0000(+4) 4.0000(-4)
*Read as 1.4918 x 10 7.
At a plane just above the module, the intensity
variation was determined to be that shown in figure
3. The energy distribution, or the number in each
of the 21 neutron groups and 18 gamma-ray groups,
is shown as a cumulative distribution for three
different radii in figure 4. These curves illus-
trate the distributions used for the selection of
the source in the MORSE Monte Carlo calculation.
By using MORSE, it was possible to determine
the contribution to the scattered dose of the
incident neutrons and gamma rays as well as the
secondary gamma rays produced in the module.
677
uJ
p-
I07
io6 _
ioS
10_ LO0*
I I I I I I
700 800
RADIUS fCMI
I000 II00 12(](]
FIGURE 3.-Spatial Distribution of Radiation at
Source Plane Adjacent to the Service and Command
Module•
s
2
i
io-i
E:
u_1o';
io-:
I
. ...,--'"
1190 cm
• ...oe o•'• °
• 820 ci
• ° • " • " 335 ei
°
°
°
----I,0
GI_OLIP
FIGURE 4.-Cumulatlve Probability for Energy
Distribution of Neutrons and Gamma Rays at
Three Locations Along Source Plane.
The geometry employed in the calculation is
illustrated in figure 5, which shows boundary cross-
ing events in the transport of neutrons in the mod-
ule. The radial and longitudinal location of the
crossing events is shown. This collision density
pl0t is an output from MORSE and gives the boundary
crossings for approximately i000 source particles•
The typed numbers give the density of the homogen-
ized parts of the service module, and Table II
gives the material composition for each region•
Figure 6 shows the density of collisions of neutrons
in one particular problem• The rotational symmetry
of the module is taken into account in the plot,
and there are few collisions in the center. The use
of this type of on-line plot has been useful in
determining where the events are occurring and the
effects of importance sampling.
'% !
............................ .7 ....... _..... ,', " ............. " ..... '
FIGURE 5.-Boundary Crossing Events From a MORSE
Calculation. The density of the three material
regions is given•
Table II. Material Composition for the
Service and Command Module
_edium
i
2
3
Weight
(ibs)
13,500
20,800
200
Density
(_Icc)
0.612
0.193
1.36
Composition (wt %)
H o l li TiI
3.3 46.7 20•0 17.0
3.3 I 46•7 20.0 17.0
F
t
Fe
13.0
13.0
I00.0
678
,ant ......... • ......... t ......... t ......... i ......... * ......... * ......... _ ......... •.........
L
Z
l
a
J
I1
L
1
u
i
FIGURE 6.-Collislon Density Piot for Neutrons
Scattering in the Service and Command Module.
Numbers 1-9 include the number of collisions in
the intervals with 0 indicating i0 or more
collisions.
The contribution of scattered neutrons and
gala rays to the dose at the detector plane (117
ft from the reactor) is shown in Table III (about
5% of the gamma-ray dose is due to secondary gamma
rays produced in the service module). The total
scattered dose of 0.34 mrem/hr should be compared
to the direct dose of 0.70 mrem/hr (0.66 mrem/hr
gamma rays and 0.04 mrem/hr neutrons). These results
are compared with values calculated at Atomics
International for various locations on the detector
plane. There is surprisingly good agreement be-
tween the Monte Carlo and single-scattering results.
Table III. Scattered Dose at the Detector Pla=e
(mrem/hr)
Neutron Dose Gamma-Ray Dose Total
Detector
Location (ft)
Along
_is _Axio
0 0
11 0
0 ll
*With self-absorl
Single Monte
Scattering *I Carlo
0,228 0.29
0.73 0.31
0.27 0.39
tion.
S_ngle Monte
Scat_erin 8 Carlo
0.063 0.05
0.27 0.08
0.09 0.04
Single
Scattering
0.30
1.0
0.36
Monte
Carlo
0.34
0.39
0.43
679
As a check on the adequacy of the SPACETRAN cal-
culated spatial distribution at the service module
location, a DOT calculation with a 100-angle quad-
rature biased in the downward direction was made.
Figures 7 and 8 show the spatial distribution at
the source plane and the energy distributions.
There are fairly large differences in these results
compared to those in figures 3 and 4 ; however, the
scattered dose contribution does not change more
than 15%.
10?
_J
z
d
I I I I I
o_L-_. 71-2974
i I I
ii....
I°2 300 _oo I I ] [ _ ] I500 1500 700 800 O0 1000 !IO0 1200
RADIUS _CMJ
FIGURE 7.-Spatial Distribution of Radiation at
Source Plane Adjacent to the Service and
Command Module Calculated With a 100-angle
Asymmetric Quadrature.
i
• . " 820 cm • . • •
335 cm
ZO-_
_lo "_ --
c,flouP
FIGURE 8.-Cumulative Probability for Energy
Distribution of Neutrons and Gamma Rays at
Three Locations Along Source Plane Calculated
With a 100-angle Asymmetric Quadrature.
REFERENCES
i. STRAKER, E. A., et al.: The MORSE Code - A Multigroup Neutron and Gamma-Ray
Monte Carlo Transport Code. USAEC Report ORNL-4585, Oak Ridge National
Laboratory, 1970.
2. MYNATT, F. R.; MUCKENTHALER, F. J.; and STEVENS, P. N.: Development of TVo-
Dimensional Discrete Ordinates Transport Theory for Radiation Shielding.
USAEC Report CTC-INF-952, Union Carbide Corporation, 1969.
3. KESHISHIAN, V.: Shielding Analysis for NASA's Manned Orbital Space _tation
Using the Zirconium Hydride Reaotor Power System. Atomics International
Report TI-696-23-014, 1969.
4. THOMSON, R. J.: Neutron and Photon Scattering Off Command and Service
Module. Atomics International Report TI-696-23-053, 1970.
680


Application of DOT-MORSE coupling to the analysis of three-dimensional SNAP shielding problems

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Application of DOT-MORSE coupling to the analysis of three-dimensional SNAP shielding problems

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