Accelerator Driven Systems (ADS) are subcritical nuclear reactor cores driven by spallation neutron source.

These spallation neutrons are provided from the bombardment of a liquid metal flowing in a central region of

the core, by a proton beam coming from a linear accelerator. Since neutron source dominates the neutronic of

the ADS, its control usually is not related to the delayed neutron fractions as in the critical systems. In this way,

ADS kinetics diverges from the former ones. This work presents some results of transients in ADS. Since ADS

is to work in a pulsed regime, with time duration of pulse, a parametric result is showed to point out that

different pulse durations should be studied in conjunction with pulse intensity, to maintain the ADS power

stable. In addition, before generating these results, some transients were used to validate the CINESP-ADS

code. It solves numerically the kinetic equations based on the multigroup diffusion theory, in One- or Twodimensional,

in the Cartesian or Cylindrical geometries, and for any energy and delayed neutron groups. Threedimensional

simulation is possible using a reactor transversal buckling. The numerical solution is obtained via

finite differences for the spatial discretization, using the so-called box integration, and with the use of

Alternating Direction Explicit (ADE) methods, for the time discretization. The code application to some

transients due to source neutron variations demonstrates its efficiency and accuracy, when compared with

analytical techniques, such as one that uses expansion in series of Helmholtz eigenfunctions

Santos, Rubens Souza dos

Carpe dIEN

2009 International Nuclear Atlantic Conference - INAC 2009 
Rio de Janeiro,RJ, Brazil, September27 to October 2, 2009 
ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN 
ISBN: 978-85-99141-03-8 
 
STARTUP TRANSIENTS IN ACCELERATOR DRIVEN SYSTEMS 
USING CINESP-ADS CODE 
 
Rubens Souza dos Santos 1,2 
 
1
 Instituto de Engenharia Nuclear (CNEN/IEN) 
Rua Hélio de Almeida, 75 – Cidade Universitária – Ilha do Fundão – 21945-970 
Rio de Janeiro - Brazil 
rsantos@ien.gov.br 
 
2
 Instituto Nacional de Ciência e Tecnologia de Reatores Nucleares Inovadores / CNPq 
 
 
 
ABSTRACT 
 
Accelerator Driven Systems (ADS) are subcritical nuclear reactor cores driven by spallation neutron source. 
These spallation neutrons are provided from the bombardment of a liquid metal flowing in a central region of 
the core, by a proton beam coming from a linear accelerator. Since neutron source dominates the neutronic of 
the ADS, its control usually is not related to the delayed neutron fractions as in the critical systems. In this way, 
ADS kinetics diverges from the former ones. This work presents some results of transients in ADS. Since ADS 
is to work in a pulsed regime, with time duration of pulse, a parametric result is showed to point out that 
different pulse durations should be studied in conjunction with pulse intensity, to maintain the ADS power 
stable. In addition, before generating these results, some transients were used to validate the CINESP-ADS 
code. It solves numerically the kinetic equations based on the multigroup diffusion theory, in One- or Two-
dimensional, in the Cartesian or Cylindrical geometries, and for any energy and delayed neutron groups. Three-
dimensional simulation is possible using a reactor transversal buckling. The numerical solution is obtained via 
finite differences for the spatial discretization, using the so-called box integration, and with the use of 
Alternating Direction Explicit (ADE) methods, for the time discretization. The code application to some 
transients due to source neutron variations demonstrates its efficiency and accuracy, when compared with 
analytical techniques, such as one that uses expansion in series of Helmholtz eigenfunctions.  
 
Key Words: ADS, Reactor Kinetics, Multigroup Diffusion Theory, Numerical Methods 
 
 
1. INTRODUCTION 
 
Accelerator Driven Systems (ADS) are subcritical nuclear reactor cores driven by spallation 
neutron sources. Theses promising devices must be used not only as dedicated burners of 
transuranic elements but also as energy producers. Since the fuel materials to be used in these 
reactors must contain fertile-free fuels, it introduces some drawbacks in its operation. That 
derives from the fact that: most of these materials present low delayed neutron fraction; 
shorter neutron generation time than other reactor types; incipient Doppler Effect, among 
other things. In this way, the control of these systems is over the proton beam, which 
produces the source neutrons [1,2,3].  
 
A Co-ordinated Research Project (CRP) on Analytical and Experimental Benchmark 
Analysis Driven System is ongoing, supported by IAEA [4]. This CRP aims to assist various 
national programs by providing collaborative research and development in the ADS 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
technology, addressing all major physic phenomena of the spallation source and its coupling 
to the sub-critical system, such as YALINA-Booster in Belarus [5].  
 
In this work we present numerical solutions for some transients of localized pulsed sources 
using CINESP-ADS computational code [6]. The objective is to qualify CINESP-ADS to be 
used for Yalina system in the scope CRP on Analytical and Experimental Benchmark 
Analysis Driven of the IAEA.  
 
 
2. CINESP-ADS KINETIC MODEL 
 
The physical model of the core reactor is dealt with considering the multigroup diffusion 
theory. To that we add the external neutrons source, allowing one- and two-dimensional 
calculations, in the Cartesian )0( =α  and Cylindrical )1( =α  geometries. With that we have 
[6]: 
 
+∑−−
∂
∂
∂
∂
+
∂
∂
∂
∂
=
∂
∂
gRgggg
g
g
g
g
g
g
BD
y
D
yx
Dx
xxt
ϕϕ
ϕβϕϕ
ϑ
α
α
211
                                        
        g
I
k
kkagk
G
g
ggfgpg
G
gg
gggs SC ++∑−+∑+ ∑∑∑
==′
′′′
≠′
′′
11
)1( λχϕνβχϕ ,   Gg ,...,2,1= ,               (1) 
kk
G
g
ggfgk
k C
t
C λϕνβ −∑=
∂
∂
∑
=′
′′′
1
, Ik ,...,2,1= ,                                    (2) 
where, ),,( tyxgg ϕϕ = , ),,( tyxCC kk = , and ),,( tyxSS gg = . For boundary conditions at 
bx  and by we have: 
 
0),,( =tbybxgϕ .                                                         (3) 
In case of 1=α  we have: 
0
),,0(
=
∂
∂
x
ty
D gg
ϕ
                                                      (4)                                       
We point out here that the existence of the term ggg BD ϕ2 in Eq. (1) permits to simulate until 
3D problems. 
 
 
3. NUMERICAL DISCRETIZATIONS 
 
Equation (1) and (2) are discretized in space using the box integration, with mesh centered in 
the interface. The symbolic integrations are given by: 
 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
),,(),,()(
2
2
2
2
tyxHtyxdxPxdytT gg
y
y
y
y
x
x
x
x
gij
j
j
i
i
∫ ∫
∆
+
∆
−
∆
+
∆
−
=
α
,                                        (5) 
 
),,(),,()(
2
2
2
2
tyxVtyxdxPxdytU kk
y
y
y
y
x
x
x
x
kij
j
j
i
i
∫ ∫
∆
+
∆
−
∆
+
∆
−
=
α
,                                        (6) 
 
where )(tTgij and )(tU kij are generic terms, only time dependent, associated with the variables 
),,( tyxH g and ),,( tyxVk , respectively, as given in Eq. (1) and Eq.(2). When be the case, 
parameters ),,( tyxPP gg =  and ),,( tyxPP kk = can represent mean properties, like wgΣ , gD  
etc., where w  means the reaction types. The integrations given by Eq. (5) and Eq. (6) leads to 
the following matrix equation: 
 
SM
dt
d
+= ψψ ,                                                                     (7) 
 
where )(tψψ = and )(tSS = are vectors given by: 
 { }ILNkijGLNgij CCCCCCcolt ,...,...,,,...,,,...,,...,,...,)( 122112121111122112121111 ϕϕϕϕϕϕψ = ,     (8) 
 { }0,0,...,0,...,0,0,,...,,...,,...,)( 122112121111 GLNgij SSSSSScoltS = ,                 (9) 
 
and M is a matrix associated with the spatial discretizations of Eq. (1) and Eq. (2), which 
structure is shown in Fig. 1, considering: 2D, a grid of 4=L  and 3=N , two energy and one-
precursor groups. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 1.  Block structures of matrix M . 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
Before discretizing Eq. (7) in time we use the fact of matrix M be able of splitting in such 
way: 
 
EDM += ,                                                                 (10) 
 
where matrices D and E have their structures given by Fig. 2 and Fig. 3. 
 
 
 
 
 
 
 
 
 
 
 
Now, using the ADE methods [7], the time derivative in Eq. (7) is approximated in two steps: 
 
                                
nnnn
nn
SSMM
t
+++≈
∆
− ++
+
2/1
2
2/1
1
2/1
2/
ψψψψ ,                                    (11)                                 
 
                            
2/112/1
4
1
3
2/11
2/
++++
++
+++≈
∆
− nnnn
nn
SSMM
t
ψψψψ ,                                (12)                                  
 
where, t∆  is the time interval, ttt nn ∆+= −1 , and, )( nn tψψ = . Matrices 1M , 2M  3M and 4M  
are splitting of M , as the Alternating Direction Explicit Methods [6], given by Fig. 4 until 
Fig. 7. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Defining I , in this context, as the identity matrix, Eq. (11) and Eq. (12) can be put in the 
following forms: 
Figure 2. Structure of matrix D .  Figure 3. Structure of matrix E .  
Figure 4. Structure of matrix 1M .  Figure 5. Structure of matrix 2M .  
Figure 6. Structure of matrix 3M .  Figure 7. Structure of matrix 4M .  
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
                          [ ] [ ] ( )nnnn SSttMItMI +∆+∆+=∆− ++ 2/122/112 ψψ ,                                  (13)                                 
 
                       [ ] [ ] ( )2/112/14132 ++++ +∆+∆+=∆− nnnn SSttMItMI ψψ .                                (14)                                 
 
The right side of Eq. (13 and Eq. (14) are vectors, easily obtained by simple operations of 
multiplication. Fast algorithms can be improved, from matrix structures of the equations. 
That is possible because each block is dictated by the ordering given by Eq. (8). The spatial 
grid is marched in alternated directions, using Eq. (13) and Eq. (14). For each step, the 
equation system is solved simultaneously, in an explicit point wise way [6].  
 
4. STEADY STATE CONDITION 
 
The steady-state condition, considering the nominal condition, is obtained setting derivatives 
as zero, in Eq. (7). In this way, we have: 
 
)0()0( 1SM −=ψ ,                                                                (15) 
 
As M is a sparse matrix, the iterative method Successive Over-Relaxation is used to obtain 
)0(ψ [8]. Indeed, as external source neutron it is which sets neutron fluxes, first all, a 
subcritical core is searched, for a given 1<effκ . Done that, the nominal power is defined 
using the following equation: 
 
                                          ∑∫∫
=
Σ=
G
g
gfg yxyxdxdyP
1
)0,,()0,,()0( ϕ .                                      (16) 
 
5. RESULTS 
5.1. CINESP-ADS Validation 
 
First of all, CINESP-ADS was validated using data for a 1D uniform slab, from reference [9]. 
Table I and Table II show multigroup constants, considering 3 energy groups. With that, a 
search of sub criticality was made in order to find a state where 9858184.0=effK , which 
implied a slab of 23.064cm. In this calculation, 031 =Σ →s  because the version of CINESP for 
criticality search was developed for direct coupling for scattering cross-sections, gsg →−Σ 1 . 
 
 
 
 
 
 
 
 
 
 
Fig. (8) shows the slab with its source location. Throughout applications was used a spatial 
grid with 39 meshes and one very low delayed neutron fraction )0( ≈β . Also, spallation 
Table I. Group parameters. Table II. Group scattering cross-sections. 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
neutrons were considered all being born in-group 1 )0( 32 == SS . Target is located at region 
II, but I, II and III ones, have the same properties. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. (9) exhibits the normalized flux distribution in the slab. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
CINESP-ADS solution was compared with an analytical approach, based on Helmholtz 
eigenfunctions, disregarding delayed neutrons [9]. For that, use was made of 150 harmonics 
to reduce the error due to the series truncation. The transient starts with zero flux, when a 
pulse of sµ10 duration and intensity 11 =S ( 032 == SS ). After that the system was shuts 
down. Fig. 10 and Fig 11 show power variation, based on CINESP-ADS and analytical 
solutions. Fig. 12 and Fig. 13 exhibits variations between 5  and sµ25 . We can note good 
agreement between CNESP_ADS and analytical solution. 
 
 
0 5 10 15 20 25
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
Groups
Nominal Condition
 1
 2
 3
N
or
m
a
liz
e
d 
Fl
u
xe
s 
x(cm)
 
      Figure 9. Flux distribution. 
2.306cm 
I II III 
10.379cm
m 
10.379cm
m 
Figure 8. Slab dimensions and source location. 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
5.2. Power transient 
 
CINESP-ADS was applied to calculate a power transient, simulating a power elevation, 
followed a shutdown. The transient is defined by: 1)0(1 =S ; during sµτ 101 = , 5)(1 =tS ; 
after st µ10> , 0)(1 =tS . In this transient, the time-step was st 710−=∆ . The grid was the 
same above. The duration of the transient was sµ100 . Fig. 14 shows CINESP-ADS and 
Analytical solution [10], where we can see the excellent agreement between the methods. 
Once more CINESP-ADS was validated. Fig. 15 shows the normalized flux variations. 
 
0,00 0,02 0,04 0,06 0,08 0,10
0
20
40
60
80
100
Po
w
e
r 
(A
rb
itr
a
ry
 
Un
it)
t(ms)
 
 
Figure 10. CNESP-ADS solution - 1 pulse. Figure 11. Analytical solution - 1 pulse. 
0,02
80
100
Po
w
er
 
(A
rb
itr
ar
y 
Un
it)
t(ms)
Figure 12. Zoom: CNESP-ADS solution. 
 
 
Figure 13. Zoom: Analytical solution. 
 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
0,00000 0,00002 0,00004 0,00006 0,00008 0,00010
0,7
0,8
0,9
1,0
1,1
1,2
1,3
 CINESP-ADS
 Analytical Solution
N
o
rm
al
ize
d 
Po
w
e
r
t(s)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
5.3. Startup transient 
 
The startup transient consists in bringing the ADS from zero power to nominal power. For 
that, a pulse train initiates the transient. Each pulse has a duration time sµτ 251 = and the 
same time of turn off, sµτ 250 = , representing a period sT µττ 5010 =+= . The startup is 
Figure 14: Normalized power variation. 
0,00000 0,00002 0,00004 0,00006 0,00008 0,00010
0,4
0,6
0,8
1,0
1,2
t(s)
Groups
G
ro
u
ps
 
g=
1,
2
 1
 2
0,00045
0,00050
0,00055
0,00060
0,00065
G
ro
up
 g
=3
 3
Figure 15. Normalized flux variation. 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
observed during s60 , using a time-step st µ5.12=∆ . The source intensity was setup as 
21 =S ( 032 == SS ). The intensity 2 is necessary because half of transient the source is 
turned off. Step transients were also calculated in order to compare the asymptotic behavior. 
In this case, the source intensity was set as 1, obviously. Fig 16 shows the normalized power 
for both transients. 
0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0
 Step
 PulsedN
o
rm
a
liz
e
d 
Po
w
e
r
t(s)
 
5.4. Pulse duration effects 
 
Fig. 17 shows the differences due to the pulse duration. We can see that the ADS proton 
beam bunches must be synchronized with the beam intensity to maintain the ADS power 
stabilized. 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
Figure 16.  Power due to source variation in step and pulses. 
t(s) 
0,00000 0,00002 0,00004 0,00006 0,00008 0,00010
0,85
0,90
0,95
1,00
1,05
1,10
1,15
 τ0<τ1
 τ0>τ1
 τ0=τ1
N
or
m
a
liz
ed
 
Po
w
er
t(s)
Figure 17. Pulse duration effects on the power. 
 
INAC 2009, Rio de Janeiro, RJ, Brazil. 
 
3. CONCLUSIONS  
 
Two important transients were calculated using CINESP-ADS. In both was demonstrated the 
necessity to observe the ADS operation since the startup. The frequency of pulses plays an 
important role in the modulation of the intensity. Before these analyses, in addition, 
comparing numerical results with analytical ones, we were able of validating CINESP-ADS. 
The application of Code to pulsed source transient showed that it is able to provide consistent 
results of a step source. Even though, the application was circumspect to one-dimensional 
problems, CNESP-ADS can be applied to analysis the Benchmark problem proposed for 
YALINA-Booster, a typical Three-dimensional problem in Cartesian geometry. 
 
ACKNOWLEDGMENTS 
 
The author would like to thank the National Council for Scientific and Technological 
Development (CNPq) of Brazil, for the financial support.  
 
REFERENCES 
 
1. C. Rubbia et al, Conceptual Design of a Fast Neutron Operated High Power Energy 
Amplifier, CERN/AT/95-44 (ET), 29th September (1995). 
2. M., Eriksson, J.E., Cahalan, W.S., Yang, On the Performance of Point Kinetics for the 
Analysis Accelerator-driven Systems, Nuclear Science and Engineering, 149, 298-311 
(2005).  
3. H. A. Abderrahim et al, MYRRHA: A multipurpose accelerator driven system for 
research & development, Nuclear Instruments and Methods in Physics Research A 
463,487-994, (2001). 
4. H.A., Abderrahim, A., Stanculescu, IAEA Coordinated Research Project on Analytical 
and Experimental Benchmark Analyses of Accelerator Driven Systems, Proceedings of 
ANS Topical Meeting on Reactor Physics, PHYSOR 2006, Vancouver, (2006). 
5.  V., Bournos, C., Routkovskaia, I., Serafimovich, H.I., Kiyavitskaya, Yu., Fokov, A., 
Fokov, S., Mazanik, A., Adamovich, T., Korbut, A., Koulikovskaya, A., Khilmanovich, 
B., Martsinkevich, S., Chigrinov, Experimental Investigations on ADS at Sub-Critical 
Facilities of Joint Institute for Power and Nuclear Research-Sosny of the National 
Academy of Sciences of Belarus, Meeting of the CRP Analytical and Experimental 
Benchmark Analyses of Accelerator Driven Systems, Minsk, 5-9 December (2005). 
6. R.S., Santos, Dynamics of Nuclear Core Based on One- and Two-dimensional Multigroup 
Diffusion Theory, X ENFIR, (1995). 
7. Wm.H., Reed and K.F., Hansen, Alternating Direction Methods for the Reactor Kinetics 
Equations, Nucl. Sci, Eng., 41, 431, (1970). 
8. M., Clark Jr. and K.F., Hansen, Numerical Methods of Reactor Analysis, Academic Press 
Inc., New York, (1964). 
9. S., Dulla, P., Ravetto, P., Picca, D., Tomatis, J.R., Maiorino, T., Carluccio, A., Antunes, 
A., Santos, F., Oliveira, R.S., Santos, Analytical Benchmarks for the Kinetics of 
Accelerator-Driven Systems, Joint International Topical Meeting on Mathematics & 
Computation and Supercomputing in Nuclear Applications (M&C+SNA 2007), 
Monterrey, California, (2007). 
10. F.L., Oliveira, Solução Analítica da Cinética Espacial do Modelo de Difusão para 
Sistemas Homogêneos Subcríticos Acionados por Fonte Externa, Master Thesis - IPEN-
USP, SP, Brazil, (2008). 


Startup transients in accelerator driven systems using cinesp-ads code

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Startup transients in accelerator driven systems using cinesp-ads code

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