HTR Reactor Physics - Jan Leen Kloosterman · HTR Reactor Physics Slowing down and thermalization...

1

VHTR Eurocourse, Prague, May 21-24, 2013J.L. Kloosterman, Delft University of Technology

HTR Reactor Physics

Slowing down and thermalization of neutrons

Jan Leen KloostermanDelft University of Technology

[email protected]


Reactor Institute Delft / TU-DelftResearch on Energy and Health with Radiation

2



Neutrons

Spin-Echo Small Angle Neutron Scattering (SESANS)

3


Positrons

POSH-Strongest positron beam in the world


Energy• solar cells• batteries• hydrogen storage• nuclear energy

} Materials research

Research Themes (1)

4


Health• radiation and radioactive nuclides for therapy and diagnostics• radiation detection systems for imaging• new production routes for radionuclides• new radionuclides for new applications

Research Themes (2)


OYSTER • Power upgrade from 2 to 3 MW• Higher Density Fuel• Installation of Cold Neutron Source• Installation of New Instruments and Facilities

5


HTR Reactor Physics


Pebble-bed fuel

6


Prismatic fuel


Differences between LWR and HTRLWR HTR

Fuel UO2 pin UO2 sphere

Moderator Water Graphite

Coolant Water Helium

Temperature (oC) 300 900

Enrichment (%) 5 10

Burnup (MWd/kgU) 60 120

Specific power (kW/kgU) 40 80

Power density (kW/l) 100 6

7


Contents of this lecture

• Slowing down of neutrons in graphite moderated reactors

• Resonance shielding and cell weighting procedures in double heterogeneous geometries

• Implications of high temperatures on the thermal spectrum


10-2

100

102

104

106

10-1

100

101

102

103

104

Fis

sion

cro

ss s

ectio

n (b

arn)

Energy (eV)10

-210

010

210

410

610

-8

10-7

10-6

Fis

sion

spe

ctru

m

U-235

Pu-239

U-238

Moderation of neutrons

8


Moderation of neutrons

Ferziger&Zweifel, Theory of neutron slowing down in nuclear reactors, 1966

f

a

10-2

100

102

104

106

0.5

1

1.5

2

2.5

3

3.5

4

U-235

Rep

rodu

ctio

n fa

ctor

10-2

100

102

104

106

10-8

10-7

10-6

Fis

sion

spe

ctru

m

Energy (eV)

10-2

100

102

104

106

0.5

1

1.5

2

2.5

3

3.5

4

U-235

Rep

rodu

ctio

n fa

ctor

10-2

100

102

104

106

10-8

10-7

10-6

Fis

sion

spe

ctru

m

Energy (eV)

U-235


Energy transfer in collisions• Elastic scattering most important• Conservation of energy and

momentum • Large energy transfer in

collisions at light nuclei• Hydrogen same mass as a

neutron* largest E-transfer

2mass nucleus 1

mass neutron 1

M AA

m A

*a neutron is 0.1% heavier. Think over the consequences.

9


Energy transfer in collisions

1 for '

1'

0 elsewhere

' 's s

E E EEp E E

E E E p E E

'p E E

E E

Area=1



Average energy:

1 ' ' ' ' 1

2

Average energy loss:

1 ' 1

2

E

E

E E p E E dE E

E E E E

10



21

Example: Hydrogen 01

1 1Average energy: ' 1

2 2

A

A

E E E

?

1 6

61

Number of collisions to slow down a neutron from

=1 Mev to =1 eV in a hydrogeneous medium:

log 10110 20 collisions

2 log 2

H L

n

L

H

E E

En

E

Two collisions with energy loss of 50%

One collisions with energy loss of 80% and one with 20%


Energy vs Lethargy

Transform to a new variable that changes linearly

in each collisi

Energy

Letharon

lo

gy

g HEu

E

Average lethargy gain per collision:

1 log

1

21 log large A

213

H

H

E

H

HE

Eu dE

E E

A

11


Number of collisionsNow, the number of collisions to increase the

lethargy to corresponding with energy becomes:

log

Numerically the same to the number of collisions

needed to slow down a neut

n

H

average

u E

EE

n

almost

Lamarsh, Introduction to Nuclear Reactor Theory, 1965

ron from to HE E


Number of collisions

Number of collisions to slow down a neutron from

=1 Mev to =1 eV in various media:

log /

H L

H

E E

E En

Element A

H 1 0 1.000 14

D 2 0.111 0.725 19

Be 9 0.640 0.207 67

C 12 0.716 0.158 88

U 238 0.983 0.00838 1649

n

12


Elastic scatter cross section

10-2

100

102

104

106

0

0.5

1

1.5

2

2.5

3

3.5

4

Ela

stic

sca

tter

cro

ss s

ectio

n (c

m-1

)

Energy (eV)

water

graphite

Mean free path 2.5 cm


2

2

large

A good moderator has large

small

Moderating power measure of energy transfer

Moderator ratio

Moderator Power Ratio

H O 1.35 71

D O 0.176 5670

Be 0.158 143

C 0.060 192

s

a

s

s

a

Moderator power and ratio

13


Space-dependent slowing down Moderating power

Energy transfer to the moderator per unit path length

lethargy gain by the neutron:

s

sdu dx

Monte Carlo game:

Start particles at isotropic plane source

Follow the particles from interval to interval

In each interval, certain probability to scatter

When scattering, particles can reverse d

irection

When scattering, particles gain lethargy


Space-dependent slowing down

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

Interval

Leth

argy

dis

trib

utio

n

u=4u=6u=8

14


Space-dependent slowing down Fermi-age model

Age-diffusion equation

Continuous slowing down model

Takes the form of time-dependent diffusion eq. without absorpti

2

2

on

,,

where , , (slowing down density) and

is the Fermi age:

1

6 is mean squared distance a source neutron travels until it reaches

s

qq

q u u u

u

r

rr

r r



Duderstadt&Hamilton, Nuclear Reactor Analysis, 1976

Non-absorbing slab

Plane source

15



2 2

2

Recall from diffusion theory assumed to be known

is diffusion length

1

6

is mean squared distance a thermal neutron travels until absorption

L

L r

L

birth

death

2L


Space-dependent slowing down2 2 2

2 2

1 1Fermi age Diffusion length

6 6

Migration area

Migration length is 1/ 6 of the rms distance a neutron travels

between birth as a fission neutron and absorption in thermal range

r L r

M L

M


16


Resonance absorption


Flux depression in a resonance.

Neutron flux depression in resonance

Neutron flux spectrum

in the fuel lump often

calculated by collision

probability method

Resonance integral:

FI E E dE

17


Neutron balance in two regions

/

/

/

/

' '1 ' regioFuel

Moderat

n1 '

' ''

1 '

' 'region '

1 'o

' '1 '

1

r

'

F

M

F

M

FF t F

E Fs F

F FMFE

E Ms M

M MFME

MM t M

E Fs F

F FMFE

E Ms M

M MFME

V E E

E EV P E dE

E

E EV P E dE

E

V E E

E EV P E dE

E

E EV P E dE

E


First-flight escape probabilities

is probability that a neutron originating in the fuel

will make its next collision in the moderator

is probability that a neutron originating in the moderator

will mak

FM

MF

P E

P E

e its next collision in the fuel

Both and are usually approximated by the first-flight

escape probabilities assuming a flat source distribution.

In particular:

FM MF

FM esc

P P

P E P

18


Two-region slowing-down equations Three approximations in the slowing down equations:

Flat source approx for and reciprocity theorem

Narrow resonance approximation in the moderator

No absorption in the moderator (1/

FM MFP P

E

/

Stacey, Nuclear Reactor P

Only one equation;

flux)

' '1 '

1 '

Several approximations possible for ' like NR, NRI

only

M, etc

needed

F

FM

E F Fs F FM tF

t F FMFE

F

E E P E EE E P E dE

P

E E

E

hysics, 2000


First-flight escape probability

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R TF

Pes

c

Plate

Cylinder

Sphere

3 1 1S C P

4 2 4F F Ft t tR R R

Small lump: 1escP

1Large lump:

4F

esc F Ft F t

SP

l V

/ 4Wigner rational approx:

1 / 4

FF F t

esc FF F t

S VP

S V

Case et al, Introduction to the theory of neutron diffusion

19


Elastic scatter cross section

10-2

100

102

104

106

0

0.5

1

1.5

2

2.5

3

3.5

4

Ela

stic

sca

tter

cro

ss s

ectio

n (c

m-1

)

Energy (eV)

water

graphite

Mean free path 2.5 cm


Rod shadowing and Dancoff factors

0 0 0 1 0 0 1 1 2

If a neutron can easily interact with neighbouring fuel lumps,

the first flight escape probability is not a good estimate for

Better:

1 1 1 1 1 1 ...

FM

FM esc

M M F M M F M F M

P

P P

P P P P P P P P P

0esc MP P 0 0 11 1esc M F MP P P P

20


Rod shadowing and Dancoff factors

0 0

0

0 0 0

0

Assuming: and this series converges to:

1

1 1 1 1 1

The Dancoff factor 1 is the probability that a

neutron emitted isotropically from a fuel lump, wi

i iM M F F

MFM esc esc

M F F

M

P P P P

P CP P P

P P C P

C P

(Bell and Glasstone, Nuclear Reactor Theory, 1970)

ll enter

a neighbouring fuel lump without interaction in between.


Dancoff factors for a double heterogeneous fuel design

is the probability that the neutron will

enter a fuel kernel in another pebble without

collision in between

InterC

is the probability that a neutron leaving

a fuel kernel will enter another kernel in the

same pebble without collision with graphite

IntraC

21


Total Dancoff factor

*1*

1

*

11

1

is probability per unit path that a neutron

will collide with a moderator nuclide or will

enter a fuel kernel.

Note that if the fuel pebble conta

esc IIfk fk fk FZesc

IO II OI

P R TC C P R C C

T T T

Bende , Nucl Sci Eng, 113:147-162 (1999)

ins

no moderator zone then:

1, =1 and FZ fk fkIO OI

et al

T T C C C

Intra Inter


!=======================================! Program dancoff - calculate Dancoff factor for pebble bed HTR! J.L. Kloosterman, Delft University of Technology! Reactor Institute Delft, Mekelweg 15, Delft! Email: [email protected]! Website: www.JanLeenKloosterman.nl! Phone: +31 15 278 1191!=======================================

program dancoffparameter (pi=3.14159265)real intra,inter,infdan

!-----read the radius of the fuel grain (cm) (usually 0.025 cm)read(5,*) rg1

!-----read the number of fuel grains per pebbleread(5,*) grn

!-----read the radius of fuel zone of pebble (usually 2.5 cm)read(5,*) rp1

!-----read the radius of graphite moderator shell (usually 3 cm)read(5,*) rp2

!-----sigma of graphite is 0.4097 1/cmsigma = 0.4097rg2 = rp1 / (grn**(1./3.))

!-----calculate dancoff factordan1 = intra(rg1,rg2,rp1,rp2,sigma)dan2 = inter(rg1,rg2,rp1,rp2,sigma)cinf = grain(rg1,rg2,sigma)write(6,1100) dan1,dan2,cinf,dan1+dan2

!=========================================1100 format(' Intra Dancoff factor ',f8.4,/, &

' Inter Dancoff factor ',f8.4,/, &' Infinite med Dancoff ',f8.4,/, &' Total Dancoff factor ', f8.4)

!=========================================end

real function grain(r1,r2,sigma)!-----calculates the inf medium fuel grain Dancoff factor!-----Eq. A.12 in thesis Evert Bende

call trans(r1,r2,sigma,tio,toi,too)grain = tio*toi/(1.0-too)returnend

real function intra(rg1,rg2,rp1,rp2,sigma)!-----calculates the intra-pebble Dancoff factor!-----Eq. A.24 in thesis Evert Bende

call trans(rg1,rg2,sigma,tio,toi,too)star = -alog(too)/chord(rg2)intra = grain(rg1,rg2,sigma)*(1.0-pf(rp1*star))returnend

real function inter(rg1,rg2,rp1,rp2,sigma)!-----calculates the inter-pebble Dancoff factor!-----Eq. A.26 and A.28 in thesis Evert Bende

call trans(rg1,rg2,sigma,tio,toi,too)star = -alog(too)/chord(rg2)prob = pf(rp1*star)

!-----Eq. A.26 in thesis Evert Bendetii = 1.0-(4./3.)*rp1*star*probfac = (1.0-tii)/(1.0-tii*tio*toi)inter = grain(rg1,rg2,sigma)*grain(rp1,rp2,sigma)*prob*facreturnend

subroutine trans(r1,r2,sigma,tio,toi,too)!-----calculates transmission probabilities in a sphere!-----Eqs. A.3, A.5, A.6, and A.7 in thesis Bende

nint = 100r12 = r1**2r22 = r2**2dy = r1/real(nint)tio = 0.0do i=1,nint

y = (i-0.5)*dyy2 = y**2u = sqrt(r22-y2)-sqrt(r12-y2)tio = tio + y*exp(-sigma*u)*dy

enddotio = 2.0*tiotoi = tio/r22tio = tio/r12a = sigma*sqrt(r22-r12)too = (1.-(1.+2.*a)*exp(-2.*a)) / (2.*r22*(sigma**2))returnend

real function chord(r)!-----calculates mean chord length of a sphere!-----under Eq. A11 thesis Bende

chord = 4.0*r/3.0returnend

real function pf(rlam)!-----calculates the first flight escape probability for a sphere!-----Eq. A.9 in thesis Evert Bende

twor = 2.*rlampf = .75*(1.-(1.-(1.+twor)*exp(-twor))/(twor*rlam))/rlamreturnend

Download from www.janleenkloosterman.nl

(click on reports)

22


Dancoff factor results pebble-bed

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of grains

Cto

tal

250 m

100 m


Dancoff factor results prismatic

3

2Sphere CylR R

Kruijf&Kloosterman, Annals Nucl Ener, 30:549-553, 2003

4Average chord length for a convex body always

Plate Cylinder Sphere

halfwidth radius radius

42 2

3

c s

c s

V

S

a R R

a R R

23


Dancoff factor results prismatic

Talamo, Annals Nucl Ener, 34:68-82, 2007


Doppler temperature effect

24


Due to the vibration of the nucleus, the effective resonance broadens.

The area remains virtually constant.



2

E

E

E E

barn

100

50

5

Scatter

Re

Broaden

ed resonance 50

5 barn

sonance 100 ba

bar

r

n

n

Capture probability in resonance is

100

1 1 105

Capture probability in broadened resonance is

2 2 50

1

2

1

1 5 15

cap

tot

cap

tot

E E

E

E

E E

E E

E E

E

E


25


Doppler coefficient: 0refT T

d

dT

( )T

TrefT

ref ( 0)



Thermal scattering kernel

Lamarsh, Introduction to Nuclear Reactor Theory, 1965

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

The

rmal

sca

tter

ing

kern

el P

(E

E')

E'/E

100*kT

10*kT

kT

Hydrogen

26


Thermal scattering kernel

Massimo, Physics of High Temperature Reactors, 1976

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1T

herm

al s

catt

erin

g ke

rnel

P(E

E

')

E'/E

40*kT

10*kT

kT

Carbon


Thermal neutron spectrum

0

Thermal region characterized by upscattering of neutrons

Under the assumptions of no absorption and no sources:

' ' '

Result is a Maxwellian neutron number density:

mE

s sE E E E E dE

3/ 2

1/ 2

03/ 2

2exp

and corresponding neutron flux density:

2 2 exp

o

M

n EM E E

kTkT

n EE E

m kTkT

27



10-3

10-2

10-1

100

0

0.5

1

1.5

2

2.5

3

The

rmal

Max

wel

l spe

ctru

m

M(E

)

Energy (eV)

293 K

kT=0.025 eV

5·kT

03/ 2

0

1/ 2

03/ 2

2exp

Average neutron energy:

3

2

2 2exp

Most probable energy:

0

2Corresponding velocity: 2200 m/s for T=293 K

M

MT

n EM E E

kTkT

E E M E dE kT

n EE E

m kTkT

EE kT

E

kTv

m



10-3

10-2

10-1

100

0

0.5

1

1.5

2

2.5

3

The

rmal

Max

wel

l spe

ctru

m

M(E

)

Energy (eV)

293 K

600 K

900 K

1200 K

28




Moderator temperature effect

10-2

10-1

100

0

0.5

1

1.5

The

rmal

Max

wel

l spe

ctru

m

M(E

)

Energy (eV)10

-210

-110

00

2

Rep

rodu

ctio

n fa

ctor

1500 K900 K

Pu-239

U-235Pu-241

f

a

29





pcm/K

1100 1100 1100 1.32553

Fuel 1400 1100 1100 1.31285 -2.43

Shell 1100 1400 1100 1.32019 -1.02

Pebble 1400 1400 1100 1.30006 -4.93

Reflector 1100 1100 1400 1.33318 1.44

Fuel Shell ReflecT T T k

30


Double cell-weighting procedure


Single cell-weighting procedure

31


Cell weighting procedure

0 500 1000 1500 2000 25000.8

1

1.2

1.4

1.6

1.8

2

C over U relation

k k

un i fo rm fuel 20%

20%

10%

k

doub le heterogeneous 10%

k

as a function of C/U for 10% and 20% enriched fuel


Core design

Top reflector

Pebble bed

Inner reflector

Bottom reflector

Outlet pipe

Barrel support

Pressure vessel

Outer reflector

Core barrel

Defuel shute

Unloading syst.

32


Neutron flux density

10-2

100

102

104

106

0

0.2

0.4

Neu

tron

spe

ctru

m r

efle

ctor

Energy (eV)

10-2

100

102

104

106

0

5

10

15

Neu

tron

spe

ctru

m f

uel z

one


Concluding remarks• Neutrons slowing down in HTRs need much more collisions than in LWRs; distance traveled is longer.

• Due to large epithermal neutron flux and homogeneously distributed fuel, resonance absorption is important => use less uranium with high enrichment and high specific power.

• Resonance shielding calculations need special double-heterogeneous Dancoff factors.

• Physically, HTR cores are very large, but neutronically they are not.

• Moderator temperature reactivity effect mainly due to shift of Maxwell spectrum and non-1/v absorbers.

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