Chapter 8 Energy Transport

Background

• Chapter 5 dealt with how the neutron population varies in time

• Chapter 6 and 7 dealt with the s patial distribution of neutrons in the core

– It was determine that a critical reactor could operate at any power level and that the equilibrium would hold

• N o t en ti re l y t rue !

• A t high power levels, temperature changes will create important transients

Objectives

• Find simple expressions to approximate fuel temperature and coolant temperature in a reactor

– S teady-state temperatures

– Transients

• D etermine temperature effect on reactivity

C o r e Power D e n s i t y

P''' = P / V

• P ’’’ is the power density

• P i s the t o t a l c o r e power

• V is the core volume

P o w e r Peaking F a c t o r

F q = P m '' ' ax / P'''

• F q is the power peaking factor

• P ’’’ max i s the m a x i m u m p o w e r d e n s i t y i n the core

P = P ' m ''

ax V

F q

• We can combine the two p revious e q uations and find the above relation

• Cores are designed to operate at a given power P

– Maximizing the ratio P’’’max / Fq will allow for smaller r e a c t o r d e s i g n a t g iven p o w e r P

• Reduces construction cost

– Main job of reactor physicist is to maximize this ratio

• Control r ods

• New designs

• Varying enrichment

• …

• P ’’’ max depends primarily on material properties

Temperature and pressure that can be tolerated by fuel, coolant, structure

• Fq can be lowered by playing with enrichment loading, position of control

rods ,

poisons ,

reflector , …

f r (r)

f q (r ,z) = f r (r)f z (z)

f z (z)

For a uniform bare reactor – F q = F r F z

From chapter 7, we’ve seen the solution for such a reactor

– F r = 2.32

– F z = 1.57

More complicated geometries will have higher peaking due to local variations

Image by MIT OpenCourseWare.

Simple heat transfer on fuel e l ement

• Define q ’ has the linear heat rate (kW/m)

– T hermal power produced per unit length

– P ’’’ = q ’ / A cell element

where A cell is the area of the fuel

q’ = PF q A cell / V = PF q / NH N: Number of fuel pins

H: Height of the core

Steady - s t a t e temperatures

• T e m p e r a t u r e d r o p f r o m f u e l t o coolant i s proportional to linear heat rate

T fe ( r , z ) - T c ( r ,z ) = R f ' e q ' ( r ,z )

– R’ fe is the fuel element thermal resistance, details of which are found in Appendix D

• f unc ti on o f th e th erma l con d uc ti v i t y o f th e f ue l an d

the cladding as well as the heat transfer coefficient

• If we average over the volume

T f - T c = R f P

w h ere

R f = 1 R ' fe

It should be noted that w h e n a v e r a g i n g q ’ using the previous relation q’ = PF q / NH, we

must note that the avera g e core factor is 1.

p eakin g

We know Rf and P, but we need more information to evalute Tf and Tc

Coolant heat balance

Outlet temperature = T 0

W ch c p [ T 0 ( r ) - T i ] =

H/ 2

-H/ 2

q' ( r , z' ) dz'

_ Replacing q' and solving for T 0

Flow rate = W ch

T 0 ( r ) =

1 P Wc p NH

f r ( r )

H/ 2

-H/ 2

f z (z) dz + T i

_ Define total core flow rate

W = NW ch

Inlet temperature = T i

Image by MIT OpenCourseWare.

• We get

T 0 ( r ) =

1 Pf r ( r ) + T i p

• If we average radially

T 0 =

1 P + T i p

• Average coolant temperature

T c =

1 ( T o + T i )

• We can than get an expression for the coolant

tem p erature

1 P + T

2Wc p

• And we can replace it in our previous expression

T f = ( R f + 1

P + T i

Fuel Thermal transients

• If c o o l i n g is t u r n e d o f f , w e c a n approximate that the fuel will heat up adiabatically

M f c f

d T f ( t ) = P ( t ) - 1 [ T f ( t ) - T c (t)]

f

M f is the total fuel mass c f is the fuel specific heat

• B o u n d i n g c a se s

– Steady-state d/dt =0 P = ( T f - T c )/ R f

– N o c o o l i n g ( R f tends to infinity) M f c f d T f ( t ) = P ( t )

• All the power stays in the fuel, fuel temperature increases and can eventuall y lead to meltin g

• More convenient form

d T f ( t ) = 1 P ( t ) - 1 [ T f ( t ) - T c ( t )]

dt M f c f 

IJ i s t h e core t h erm a l t i me constant

Coolant thermal transient

• C o n s e v a t i o n e q u a t i o n in the c o o l a n t

M c c p

d T c ( t ) = 1

f

[ T f ( t ) - T c ( t )] - 2 Wc p [ T c ( t ) - T i ]

Heating term 1 [ T f ( t )- T c ( t )]

f

Cooling term 2 Wc p [ T c ( t )- T i ]

• W e ca n r e writ e h as

d T ( t ) = 1

dt  '

[ T ( t ) - T ( t )] - 1

 ''

[ T c ( t ) - T i ]

where

 ' =

M c c p 

M f c f

and

 '' =

M c c p 

2 Wc p

• C oolant usuall y follows fuel surface transient quite rapidly

– W e c a n ignore the energy storage t e r m of the

coolant equation

T c ( t ) =

1 1 + 2 R f Wc p

[2 R f Wc p T i + T f ( t )]

• C ombining with the fuel transient expression

d T ( t ) =

dt

1 P ( t ) - 1

M f c f ~ 

[ T f

( t ) - T i ]

Chapter 9 Reactivity Feedback

Background

• Temperature increase w ill create feedback mechanisms in the reactor

– Do pp ler broadenin g

– T hermal expansion

– Densit y cha n g es which will induce s p ectral shifts

• T hese changes will impact the reactivity, thus causing transients

Reactivity Coefficients

• D y n a m i c r e a c t i v i t y was d e f i n e d b y

 ( t ) = k ( t ) - 1

k ( t )

• We can relate a change in reactivity to a chan g e in k

dp = dk/k 2 ~

dk/k

= d (ln k )

• The advantage is that we change a multiplication of terms into a sum of terms

F u e l Temperature C oefficient

• D oppler broadening of the resonance capture cross-section of U-238 is the dominant effect in LWR reactors

– Lots of U-238 (red) present

– S imilar effect with Th-232 (green)

Image removed due to copyright restrictions.

Fuel Temperature C oefficient

• E ffect is felt in resonance esca p e p robabilit y (p)

• N o effect on ε ,because …

• M inor effect on η and f

– E specially in the presence of Pu-239

• D oppler effect arises from the temperature d epen d ence o f th e cross-sec ti on on th e re l a ti ve speed between neutron and nucleus

– Resonances are s meared in energy has temperature increases.

– Thermal resonances are more important

F u e l Temperature C o e fficient

• W e c a n a p p r o x i m a t e i t b y

 f =

1 k ~

1 p

k T f

p T f

• Y ou can eva l ua t e i t us i ng f ormu l as f rom Chapter 4 used to determine p (see book)

• O r, you can a l so run s i mu l a t i ons a t different fuel temperatures and compare the e s t i m a t e o f the e i g e n v a l u e

M o d e r a t o r Temperature Coefficient

• We s e e k to evaluate

 m = 1 k

k T m

• T he biggest impact of the moderator temperature comes f r o m a s s o c i a t e d c h a n g e s in density

– A s temperature increases, moderator ( and coolant ) will see a decrease in their density

• Less water molecules, means less moderation, leading to a s p e c t r a l s h i f t

M o d e r a t o r Temperature Coefficient

• D ecrease in slowin g down efficienc y will lead to an increase in resonance absorption

– V alue of p will decrease

• Lower coolant density will also have an impact on the thermal utilization (f)

– V alue of f will increase

• F a s t f ission will increase slightly, but e ff ect is negligible

• The combine effect is usually negative , but in

some reactors with solid moderators (i.e. graphite), the coefficient might be positive over certain temperature ranges .

Coolant Void Reactivity Coefficient

• In LWRs and BWRs, this coefficient is alwa y s ne g ative

– C oolant and moderator are the same, thus losing the coolant also implies loosing all the moderation

• In CANDU and RBMK , this coefficient is p ositive

– Loosing the coolant as very little impact on the moderation

• C auses slight increase in fast fission

• C auses sli g ht increase in resonance esca p e p robabilit y

– Before Chernobyl, void reactivity coefficient of RBMK was 4.7 beta, after re-design it was lowered to 0.7 beta

– C ANDU have a very small positive reactivity coefficient that can be controlled easily

Fast Reactor Coefficients

• Leakage plays a more important role in fast reactor transients

– D ecreasing density will make the spectrum harder

• Larger value of η , thus increase in k

– Migration length would also increase

• More leakage , th u s decreasing k

– Overall effect is usually positive

• Doppler effect is smaller in m agnitude

– Thermal resonances are more affected

I s o t h e r m a l T emperature Coefficient

• In man y reactors, the entire core is brou g ht ver y slowly from room temperature to the operating inlet coolant temperature

– R e a c t o r a t l o w p o w e r

– E xternal heat source

– D ecay heat

• R easona b l e approx i ma ti on i s t o assume th a t th e core behaves isothermally

T f = T c = T i

• We can thus define the isothermal temperature

coefficient

d  fb 1 k 1 k

 = = +

 T =  f   c

T dT k T f k T c

Temperature D e f e c t

• This coefficient a l l o w s u s t o e s t i m a t e the amount of reactivity needed to maintain c r i t i c a l i t y a t h i g h t e m p e r a t u r e ( h o t z e r o power)

• T h i s r e a c t i v i t y i s o b t a i n e d b y i n t e g r a t i n g the isothermal temperature coefficient from r o o m t e m p e r a t u r e t o hot t e m p e r a t u r e

D T = T i  T ( T ) dT

T r

P o w e r c o e f f i c i e n t

• A far m o r e u s e f u l c o e f f i c i e n t , i t t a k e s i n t o account impact of temperature changes when r e a c t o r i s o p e r a t i n g a t f u l l power

d  fb

1 k

dT f 1 k

dT c

 P = dP

=

k T f

dP + k

T c dP

• If we assume that power changes are slow compared to the time required for heat removal, we can use the steady-state temperature profiles from Chapter 8 and derive them with respect to Power

T c =

2 1 P + T i

d T c = 1

Wc p

dP 2 Wc p

( Wc p ) d P 2 Wc p

P o w e r c o e f f i c i e n t

• T h e p o w e r coefficient i s t h u s e x p r e s s e d i n

terms of both the fuel coefficient and the moderator coefficient

 P = ( R f + 1 ) 1 k + 1 1 k

 P = R f  f + (2 Wc p ) -1 (  f +  c )

• T h u s , a s p o w e r i s i n c r e a s e d , p o s i t i v e reactivity is required to overcome negative c o e f f i c i e n t s a n d m a i n t a i n c r i t i c a l i t y

P o w e r D e f e c t

• A s p o w e r i n c r e a s e s t o its o p e r a t i n g level, additional negative reactivity is introduced b y an increase in tem p erature

• We can evaluate the power defect by the followin g

D p = T f ( p ) 

T i f

( T f

) dT f +

T c ( p )  e ( T c ) dT c

T i

where T f (P) and T c (P) are the fuel and coolant temperatures at power P

Typical values

• T emperature Defect

• P ower Defect

• Good exercise: Lewis 9 . 4

Excess Reactivity

• Defined as the value of rho if all control poisons and rods were removed from the core

– Large excess reactivity are avoided because they need lots of poison to compensate at BOC (beginning of cycle) and require extra care

– Creates dangerous scenarios (e . g . high worth control rods become a problem if ejected)

– Strict limits are thus p laced on excess reactivit y and on the reactivity limits of control devices

• Large amount of small control rods

coefficients are nice

• T emperature feedback causes excess reactivity t o d e c r e a s e

– N eed to pull out control rods

from a stability and • If you shutdown, temperature

safety point of v i e w , large negative values can c r e a t e excess reactivity problems

• P l o t d e p i c t s

– C old shutdown (a)

– C old critical (b)

– H ot zero power critical

– Full power (d)

(c)

decreases and excess reactivity is

increased

– N eed to insert control rods as you reduce power

T emperature defect

b

a

Power defect

c

d

BOL

T ime

EOL

Image by MIT OpenCourseWare.

Shutdown Margin

• A minimum shutdown margin is imposed by the NRC

– R eac ti v it y requ i re d t o s h u td own th e reac t or no ma tt er i n w hi c h con diti on (cold critical is the one with the most excess reactivity)

– T he stuck rod criteria is usually applied

– N ormally 1-5% of excess reactivity

• Going from curve a to b removes the excess margins to get to cold critical

• As the core is heated, the excess r eactivity curve goes from b to c, with the d ifference being the t emperature defect

– S low temperature increase to reduce mechanical stresses on pipes and pressure vessel

• As the reactor goes up in power, we approach curve d

– R ema i n i ng excess reac ti v it y i s w h a t a ll ows th e core t o opera t e f or a given cycle

– T ypical LWR cycle 1-2 years

• C ore designers try to predict excess reactivity curves

– Schedule outages

– Prepare reloading

• C ores are usually reloaded in 3-4 batches, thus in a PWR you re p lace about 60 assemblies at each c y cle

• T ypical assemblies will thus stay in the core for 3 cycles or 4.5 years

• F uel is then sent to spent fuel pools for at least 5 years

• P oo l con fi gura ti on i s i mpor t an t t o avo id cr iti ca lit y acc id en t s

• W hen pool is full, oldest spent fuel elements are put in dry casks

• If they fall short on reactivity, they can reduce power to reduce temperature and increase excess reactivity

• If they under predict the excess reactivity, it indicates that they loaded more fresh fuel bundles than they needed

– R eactor is still shutdown on schedule due to mobilization of workforce

– $$$

• Outages usually last 3 - 4 w eeks

Reactor Transients

• If rapid changes of power occur, steady - state temperatures cannot be used

– Rod e j ection

– Loss of coolant

– Loss of flow

• We can develop a simple reactor d y namics model based on the kinetics relation, and the temperature transient models

• P ower

d P ( t ) = [  ( t ) -  ] P ( t ) +  

C ~ ( t )

dt 

i i i

• Precursors

d C ~ ( t ) =  i P ( t ) -  C ~ ( t ) i = 1, 2, 3, 4, 5, 6

dt i  i i

• Fuel temperature

d T f ( t ) = 1

P ( t ) - 1

[ T f ( t ) - T i ]

dt M f c f ~ 

• Coolant temperature T c ( t ) = T i + 2 R f

1

Wc p

T f ( t )

Feedback effects

• T h e r e a c t i v i t y w i l l a l s o h a v e t o i n c l u d e the temperature feedback effects

 ( t ) =  i ( t ) - |  f | [ T f ( t ) - T f (0)] - |  c | [ T c ( t ) - T c (0)]

• If the reactor is initially critical at power P 0 we can evaluate the temperatures and precursor concentrations using the steady- state relations

Demo – Step insertion

• Beta = 0 . 0065

• F u ll power = 3000 MWth

– T inlet = 300 C

– T fuel = 1142 C

Step of 0 . 2$ -

Full Power

• T fuel at 10 seconds = 1173 C

Image by MIT OpenCourseWare.

• Prompt jump b r i n g s p o w e r t o 3 7 0 0 M W t h

– S tabilizes to 3100 MWth with feedback

Step of 0 . 2$ - Low P o w e r ( 1 M W t h )

Image by MIT OpenCourseWare.

• F uel temperature eventually reaches 333 C

• P ower eventually stabilizes to 120 MWth

Step of 1$ at F u l l p o w e r

• P ower spikes to 27500 MWth

• Stabilizes t o 3 5 6 7 M W t h

Image by MIT OpenCourseWare.

• F uel temperature reaches 1293 C

R a m p i n s e r t i o n 1 $ / s w i t h F eedback

• F uel temperature increase is greater at high power

Image by MIT OpenCourseWare.

• At low power, negative reactivity feedback is too slow, thus reactor reaches prompt critical, until temperature increases

• B o t h s i t ua t i on converge t o t h e same power even t ua l l y

Shutdown ( - 5 * B e t a )

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22.05 Neutron Science and Reactor Physics

Fall 20 09

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