neutron transport in molten salt reactors€¦ · neutron transport in molten salt reactors imre...

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


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.

2


The Molten Salt Reactor

3


Footnote to history: Weinberg’s Molten Salt Reactor (HRE-11)

4


H L

5


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

6


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


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.

8


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"


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


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


12


Flux and precursor distributions

13


14


Criticality, as a function of circulation speed:

15


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


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


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

"


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

#$%%%

&%%%


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

!


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


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


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

%

&''''

(

)

*****


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:


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


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


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,")


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


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.


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= #


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 ]}


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


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( )

#

$%%%%

&

'((((


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)


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


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


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


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


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

neutron transport in molten salt reactors€¦ · neutron transport in molten salt reactors imre...

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