neutron transport in molten salt reactors€¦ · neutron transport in molten salt reactors imre...
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Chalmers University of Technology
Neutron transport in Molten Salt Reactors
Imre Pázsit Chalmers University of Technology Department of Nuclear Engineering
ICTT-22
September 12-16, 2011 • Portland, Oregon
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Chalmers University of Technology
Neutronics in systems with moving fuel
• Molten salt systems (MSR), containing liquid fuel in motion, have both static and dynamic properties different from those in traditional reactors
• Solutions in simple models give insight into the physics of such systems
• In this talk closed form analytical solutions are derived for both the static and the dynamic equations
• The results for the dynamic case show the effect of stronger neutronic coupling and a larger domain of validity of the point behaviour.
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Chalmers University of Technology
The Molten Salt Reactor
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Chalmers University of Technology
Footnote to history: Weinberg’s Molten Salt Reactor (HRE-11)
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Chalmers University of Technology
H L
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Chalmers University of Technology
A one-dimensional model of MSR
Fuel velocity = u Core height: H core transit time External loop: L; loop transit time Total length: T = H + L; total tr. time
! =
Tu
!
l=
Lu
!
c=
H
u
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Chalmers University of Technology
Time dependent diffusion equations
1v!!(z,t)!t
= D"2!(z,t)+ "#f (1$#)$#a(z,t)%&'
()*!(z,t)+$C(z,t)
!C(z,t)
!t+ u !C(z,t)
!z= "#$ f%(z,t)& 'C(z,t)
7
Boundary conditions:
!(z = 0, t) = !0(z = H , t) = 0
C(0,t) = C(H ,t !L /u)e!!
L
u = C(H ,t ! "l)e!!"l
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Chalmers University of Technology
Static equations
D!2!
0(z) + ""f (1##)#"a[ ]!0
(z) + $C0(z) = 0
u!C0(z)!z " !"#
f#
0(z) + $C
0(z) = 0
!o(z = 0) = !0(z = H ) = 0
Boundary conditions:
C(0) = C(H )e!!
L
u = C(H )e!!"l
Delayed neutron precursors do not disappear from the static equations.
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Chalmers University of Technology
The equation for the neutron noise
D!2!"(z,#)+ $"f (1#%)#"a#i#v
$
%&&
'
()) !"(z,#)+&e
#& #( )
uz %$"
f
u
e#& #( )'
1#e#& #( )' e
& #( )u
z
!0
H
* "(z,#)dz + e
& #( )u
z '
!0
z
* "(z ',#)dz '+,--
.--
/0--
1--
= !"a(z,#)"
0(z)2 S(z,#)
9
where
!(") = ! + i"
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Chalmers University of Technology
The Greens function (response to a localised perturbation)
!2!"(z,#)+ B2(#)!"(z,#)+$e
"$ #( )
uz %&#
f
Du$
$e"$ #( )'
1"e"$ #( )' e
$ #( )u
z
!0
H
% "(z,#)dz + e
$ #( )u
z '
!0
z
% "(z ',#)dz '&'((
)((
*+((
,((
= !(z "z0)
10
B2(!) = B
02 1!
i!""#!#
$
%&&&&
'
(
))))); $(!) = $+ i!
with
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Chalmers University of Technology
Solution of the static equations Eliminating the precursors by quadrature, one obtains the integro-differential equation
!2!0(z)+ B
02!
0(z)+
+e
"z"u "#$#f
Du
e""%
1"e""%ez '"u
0
H
$ !0(z ')dz '+ ez '"u!0
0
z
$ (z ')dz '
%
&
''''''''
(
)
*********= 0
11
Solution in form of expansion in the eigenfunctions of the traditional problem (Sandra Dulla).
B
0
2 =!!
f(1"")"!
a
D<#
2a
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Chalmers University of Technology
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Chalmers University of Technology
Flux and precursor distributions
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Chalmers University of Technology
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Chalmers University of Technology
Criticality, as a function of circulation speed:
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Chalmers University of Technology
Simplification: infinite fuel speed
!2!0(z)+ B
02!
0(z)
+e
"z"u "#$#f
Du
e""%
1"e""%ez '"u
0
H
$ !0(z ')dz '+ ez '"u
0
z
$ !0(z ')dz '
%
&
''''''''
(
)
*********= 0
!2!(z)+ B0
2!(z)+"#"
f
DT!(z ')dz '
0
H
# = 0
For u = : ! (! = 0)
Analytical solutions exist for both the static and the dynamic problem
16
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Chalmers University of Technology
Static equation
Solution: Criticality equation
17
!2!
0(x)+ B
02!
0(x)+
"0
T!
0"a
a
# (x ')dx ' = 0
B
0
2 =!!
f(1"")"!
a
D<#
2a
#
$%%%%
&
'((((
2
; $0
=!!
f"
D
!0(x) = A[cosB
0x ! cosB
0a ]
B
02 cosB
0a +
2a!0
TcosB
0a!
2!0
TB0
sinB0a = 0
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Chalmers University of Technology
Dynamic equation: traditional system
18
!2!"(x,#)+ B2(#)!"(x,#) =
!"a(x,#)"
0(x)
D# S(x,#)
where
B2(!) = B
02 1!
1""G
0(!)
#
$%%%%
&
'(((((; B
0=#2a
Solution: Green’s function
!2G(x,x
0,!)+ B2(!)G(x,x
0,!) = "(x "x
0)
!"(x,#) = G(x,x
0,#)S(x
0,#)dx
0!a
a
"
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Chalmers University of Technology
Solution
19
G(x,x0,!) =
!1
B(!)sin 2B(!)a
sinB(!)(a + x)sinB(!)(a!x0) x " x
0
sinB(!)(a!x)sinB(!)(a + x0) x > x
0
#$%%%
&%%%
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Chalmers University of Technology
Simplification to u =
20
!2G(x,x0,!)+ B2(!)G(x,x
0,!)
+"(!)T
G"a
a
# (x,x0,!)d $x = #(x "x
0)
with
B2(!) = B
02 1!
i!""#!#
$
%&&&&
'
(
))))); B
02 <$2a
; %(!) =&
&+ i!%
0
!
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Chalmers University of Technology
Solution
21
G(x,x0,!) =
"(!)#0(x,!)#
0(x
0,!)
A2TK(!)B(!)2 cosB(!)a
!1
B(!)sin 2B(!)a
sinB(!)(a!x0)sinB(!)(a + x) x < x
0
sinB(!)(a + x0)sinB(!)(a!x) x > x
0
"#$$$
%$$$
with
!0(x,") = A[cosB(")x ! cosB(")a ]
K(!) = B2(!)cosB(!)a +
2a"(!)T
cosB(!)a!2"(!)TB(!)
sinB(!)a
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Chalmers University of Technology
New developments • The case of infinite fuel velocity was used
- to investigate the point kinetic behaviour at low frequencies (ANS Annual Meeting 2011 Florida) - the neutronic response to various perturbations (PHYSOR 2010 Pittsburgh)
• However, the case of infinite fuel velocity does not allow to study the effect of varying fuel velocity
• Hence further approximations were searched for
• It then turned out that the full problem has a compact analytical solution
22
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Chalmers University of Technology
Physical meaning of u= and of the integral terms
!2!0(z)+ B
02!
0(z)
+e
"z"u "#$#f
Du
e""%
1"e""%ez '"u
0
H
$ !0(z ')dz '+ ez '"u
0
z
$ !0(z ')dz '
%
&
''''''''
(
)
*********= 0
23
!
C
0(z) = e
!!u
z "#"f
u
e!!$
1!e!!$
e
!u
#z
0
H
$ %0( #z )d #z + e
!u
#z
0
z
$ %0( #z )d #z
%
&''''
(
)
*****
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Chalmers University of Technology
Comparison with the traditional case
24
!C(z,t)!t
= !""f#(z,t)#$C(z,t)
C(z,t) = !"!
fe"#(t" #t )
"$
t
% $(z,t)d #t
C
0(z) = !"!
fe"#(t" #t )
"$
t
% $0(z)d #t
In the stationary (time-independent) case:
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Chalmers University of Technology
Moving precursors: infinite slab
25
C
0(z) =
!"!f
ue"#u(z" #z )
"$
z
% $0( #z )d #z
Neutrons generated at time were born at t ' < t
z ' = z ! u t! t '( )
dt ' = dz '/u; t! t ' =
z !z 'u
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Chalmers University of Technology
Moving precursors: finite slab
26
C
0(z) =
!"!f
ue"#u(z" #z )"n#$
0
H
$n=1
%
& %0( #z )d #z + e
"#u(z" #z )
0
z
$ %0( #z )d #z
'
())))
*
+
,,,,,
e!!"
1!e!!"= e!!" +e!2!" +e!3!" ...
C
0(z) = e
!!u
z "#"f
u
e!!$
1!e!!$
e
!u
#z
0
H
$ %0( #z )d #z + e
!u
#z
0
z
$ %0( #z )d #z
%
&''''
(
)
*****
The different terms in the sum correspond to the once, twice, three times recirculated precursors
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Chalmers University of Technology
The dynamic case
27
!2!"(z,#)+ B2(#)!"(z,#)+$e
"$ #( )
uz %&#
f
Du$
$e"$ #( )'
1"e"$ #( )' e
$ #( )u
z
!0
H
% "(z,#)dz + e
$ #( )u
z '
!0
z
% "(z ',#)dz '&'((
)((
*+((
,((
= !#a(z,#)"
0(z)- S(z,#)
!C(z,") = e!
(#+i")u
z $%"f
u
e!(#+i")&
1!e!(#+i")&
e
(#+i")u
#z
0
H
$ !'( #z ,")d #z%
&''''
+ e
(#+i")u
#z
0
z
$ !'( #z ,")d #z(
)
*****+ !C
1(z,")+ !C
2(z,")
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Chalmers University of Technology
The integral terms in the time domain
28
!C
2(z,") =
#$!f
ue"
i"u
(z" #z )e"%u(z" #z )
0
z
$ !&( #z ,")d #z
!C
2(z,t) =
"#!f
ue"$u(z" #z )
0
z
$ !%( #z ,t"z " #z
u)d #z
After inverse Fourier transform:
Similarly, for the first integral one obtains
!C
1(z,t) =
"#!f
ue"$u(z" #z )"n$%
0
H
$n=1
%
& !&( #z ,t"z " #z
u"n%)d #z
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Chalmers University of Technology
The approximation of no recirculation (long recirculation time)
29
Now the first integral is neglected besides the second, leading to
C
0(z) = e
!!u
z "#"f
u
e!!$
1!e!!$
e
!u
#z
0
H
$ %0( #z )d #z + e
!u
#z
0
z
$ %0( #z )d #z
%
&''''
(
)
*****
!2!
0(z)+ B
02!
0(z)+
"#$"f
Due#"u(z# $z )
0
z
% !0( $z )d $z = 0
This equation has a closed form analytical solution.
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Chalmers University of Technology
Solution
30
Characteristic equation:
!
0'''(z)+
"u!
0''(z)+ B
02!
0' (z)+
"u
(B02 +#$!
f
D)!
0(z) = 0
On physical grounds we expect
k 3 +
!u
k 2 + B02k +!u
(B02 +"#!
f
D) = 0
k
1,2= !± i"; k
3= #
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Chalmers University of Technology
Solution
31
Two coefficients can be eliminated by the boundary conditions:
!0(z) = A
1e"z sin(#z)+ A
2e"z cos(#z)+ A
3e$z
Or, in the x-coordinate system, in the reactor centre:
!0(z) = A[e"z sin#z(e$H !e
"H cos#H )
!e"H sin#H(e$z !e
"z cos#z)]
!0(x) = A{e"x cos(#x)!e!("!$)a cos(#a)e$x !
! cot(#a)tanh[("! $)a ][e"x sin(#x)+e!(a!$)a sin(#a)e$x ]}
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Chalmers University of Technology
Criticality equation
32
Substituting back to the orignal equation gives the criticality condition. This can be written symbolically as
0 =
An
!n
+" /un=1
3
!
In reality this is much more complicated, because the relationship between the has to be used explicitly.
An
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Chalmers University of Technology
Solution of the full equation
33
The full integro-differential equation leads to exactly the same differential equation as the one with no recirculation. Hence it has the same characteristic equation and same form of the solution, only the criticality condition is different (because it is determined by the full original integro-differential equation):
0 =
An
!n
+" /un=1
3
! "1+1
e"# "1
e(!
n+"/u)H "1( )
#
$%%%%
&
'((((
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Chalmers University of Technology
Reverting to the case of infinite velocity
34
Then the full solution will revert to that obtained before
k 3 +
!u
k 2 + B02k +!u
(B02 +"#!
f
D) = 0
! k 3 + B02k = 0;
!0(x) = A{e"x cos(#x)!e!("!$)a cos(#a)e$x !
! cot(#a)tanh[("! $)a ][e"x sin(#x)+e!(a!$)a sin(#a)e$x ]}
! = " = 0; # = B0
! !0(x) = A(cosB
0x " cosB
0a)
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Chalmers University of Technology
Static solutions for zero flux and logarithmic boundary conditions
For the general case, the two solutions coincide
35 50 100 150 200 250 300
0.001
0.002
0.003
0.004
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Chalmers University of Technology
Static solutions for zero flux and logarithmic boundary conditions
For the no recirulation case, and with high beta-eff, the two solutions differ significantly
36 10 20 30 40 50
0.005
0.010
0.015
0.020
0.025
0.030
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Chalmers University of Technology
Dynamic behaviour: The Green’s function The point kinetic behaviour is retained up to higher frequencies (or system sizes) as in an equivalent traditional system
Imre Pázsit
37
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Chalmers University of Technology
Dynamic behaviour: The Green’s function(cont)
Imre Pázsit
The physical reason is the spatial coupling, represented by the moving precursors and the smaller value of beta-eff
38
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Chalmers University of Technology
Conclusions • The MSR equations in one-group diffusion theory
have a closed form analytic solution for both the static and the dynamic case
• The infinite fuel velocity (short recirculation time) and no recirculation are limiting cases with eve simpler anlytical solutions, which are useful conceptual models for analytical investigations
39