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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
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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The Molten Salt Reactor
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Footnote to history: Weinberg’s Molten Salt Reactor (HRE-11)
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H L
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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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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)
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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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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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The equation for the neutron noise
D!2!"(z,#)+ $"f (1#%)#"a#i#v
$
%&&
'
()) !"(z,#)+&e
#& #( )
uz %$"
f
u
e#& #( )'
1#e#& #( )' e
& #( )u
z
!0
H
* "(z,#)dz + e
& #( )u
z '
!0
z
* "(z ',#)dz '+,--
.--
/0--
1--
= !"a(z,#)"
0(z)2 S(z,#)
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where
!(") = ! + i"
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The Greens function (response to a localised perturbation)
!2!"(z,#)+ B2(#)!"(z,#)+$e
"$ #( )
uz %&#
f
Du$
$e"$ #( )'
1"e"$ #( )' e
$ #( )u
z
!0
H
% "(z,#)dz + e
$ #( )u
z '
!0
z
% "(z ',#)dz '&'((
)((
*+((
,((
= !(z "z0)
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B2(!) = B
02 1!
i!""#!#
$
%&&&&
'
(
))))); $(!) = $+ i!
with
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Solution of the static equations Eliminating the precursors by quadrature, one obtains the integro-differential equation
!2!0(z)+ B
02!
0(z)+
+e
"z"u "#$#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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Flux and precursor distributions
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Criticality, as a function of circulation speed:
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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
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Static equation
Solution: Criticality equation
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!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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Dynamic equation: traditional system
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!2!"(x,#)+ B2(#)!"(x,#) =
!"a(x,#)"
0(x)
D# S(x,#)
where
B2(!) = B
02 1!
1""G
0(!)
#
$%%%%
&
'(((((; B
0=#2a
Solution: Green’s function
!2G(x,x
0,!)+ B2(!)G(x,x
0,!) = "(x "x
0)
!"(x,#) = G(x,x
0,#)S(x
0,#)dx
0!a
a
"
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Solution
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G(x,x0,!) =
!1
B(!)sin 2B(!)a
sinB(!)(a + x)sinB(!)(a!x0) x " x
0
sinB(!)(a!x)sinB(!)(a + x0) x > x
0
#$%%%
&%%%
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Simplification to u =
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!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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Solution
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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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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
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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
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!
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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Comparison with the traditional case
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!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
% $0(z)d #t
In the stationary (time-independent) case:
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Moving precursors: infinite slab
25
C
0(z) =
!"!f
ue"#u(z" #z )
"$
z
% $0( #z )d #z
Neutrons generated at time were born at t ' < t
z ' = z ! u t! t '( )
dt ' = dz '/u; t! t ' =
z !z 'u
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Moving precursors: finite slab
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C
0(z) =
!"!f
ue"#u(z" #z )"n#$
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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The dynamic case
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!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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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
$n=1
%
& !&( #z ,t"z " #z
u"n%)d #z
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The approximation of no recirculation (long recirculation time)
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Now the first integral is neglected besides the second, leading to
C
0(z) = e
!!u
z "#"f
u
e!!$
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 $z = 0
This equation has a closed form analytical solution.
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Solution
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Characteristic equation:
!
0'''(z)+
"u!
0''(z)+ B
02!
0' (z)+
"u
(B02 +#$!
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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Solution
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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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Criticality equation
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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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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( )
#
$%%%%
&
'((((
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Reverting to the case of infinite velocity
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Then the full solution will revert to that obtained before
k 3 +
!u
k 2 + B02k +!u
(B02 +"#!
f
D) = 0
! k 3 + B02k = 0;
!0(x) = A{e"x cos(#x)!e!("!$)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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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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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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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
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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
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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
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