Point Kinetic Equations

C H A P T E R

2

Point Kinetic Equations

T h e kinetic equations discussed in t h e previous chapter describe t h e time behavior of t h e n e u t r o n p o p u l a t i o n in too m u c h detail for most of t h e practical uses of reactor d y n a m i c s . I n m a n y applications, we are interested only in t h e d o m i n a n t features of t h e t i m e behavior of t h e n e u t r o n population, such as t h e variation of t h e total n u m b e r of n e u t r o n s or t h e total power generation in t h e m e d i u m as a function of t i m e . Details such as t h e angular d e p e n d e n c e of t h e n e u t r o n s are almost never n e e d e d in reactor d y n a m i c s . I n some applications, even t h e spatial distribution of t h e n e u t r o n p o p u l a t i o n is not of great interest. I t is, therefore, desirable to cast t h e basic kinetic equations into a simpler form which contains only t h e d o m i n a n t aspects of t h e n e u t r o n population of practical interest, b u t leaves out all t h e u n d e s i r e d details. T h e p u r p o s e of this chapter is to obtain a set of equations w h i c h will describe t h e time behavior of t h e total power generated in t h e m e d i u m . T h e s e equations are called t h e " p o i n t reactor kinetic'' (or simply point kinetic) equations. T h e y were first obtained in a systematic m a n n e r by H u r w i t z [1] in 1949, and later by Ussachoff [2] in 1955 a n d by H e n r y [ 3 , 4 ] in 1955 using adjoint fluxes. M o r e recently, Gyftopoulos [5] derived these equations in 1964 with some modifications in H e n r y ' s approach, and Becker [6] reformulated t h e m in 1968 using a variational principle and extending L e w i n ' s derivations [7] of 1960. O u r derivation in this chapter will follow H e n r y ' s derivation. T h e Gyftopoulos modification a n d Becker's formulation will be discussed at t h e e n d of t h e chapter. 48

49

2.1. Mathematical Preliminaries

2.1. Mathematical Preliminaries T h e p u r p o s e of this section is to i n t r o d u c e t h e m a t h e m a t i c a l concepts, terminology, a n d s y m b o l s w h i c h will b e used in t h e derivation of t h e point kinetic equations in t h e following sections. A. Scalar Product T h e * 'scalar'' p r o d u c t of two complex functions 0(r, v) a n d (r, v) is a complex n u m b e r | > defined b y

R

w h e r e 0* is t h e complex conjugate of i/j. W e note t h a t t h e integration in t h e configuration space is e x t e n d e d only over t h e v o l u m e of t h e reactor, whereas t h e velocity integration is e x t e n d e d over t h e entire velocity space. I n some applications, t h e symbol | (/>> will denote a vector whose c o m p o ­ n e n t s are functions of r, v ; i.e., | } = c o l ^ ^ r , v),..., 0Ar(r, v ) ] . T h e definition of t h e scalar p r o d u c t of t w o vectors | > a n d | 0> is (2) T h e s y m b o l < | can b e i n t e r p r e t e d as a row vector whose c o m p o n e n t s are t h e complex conjugate of t h e c o l u m n | */>>. T h e n , (2) implies b o t h a scalar p r o d u c t of two vectors in t h e usual sense, a n d an integration over r a n d v. If a x a n d a 2 are two complex n u m b e r s , t h e following relations follow from (1) a n d (2) (these relations are used in t h e s u b s e q u e n t m a n i p u l a t i o n w i t h o u t explanation):

<«< iA 14>> *i*<^ I ^>

< | OIJ^! +

22

<4> 14>>*

«1<0 I l>

(3a) +

2
(3b)

a

4>2>

(3c)

B. Adjoint Operators I n order to p r e p a r e for our discussion of o r t h o n o r m a l m o d e s , we need to i n t r o d u c e t h e concept of adjoint operator. L e t O be an operator operating on a function (r, v) such t h a t Ocf) is another function of r

2. Point Kinetic Equations

50

a n d v . It may b e an integral or differential operator acting on b o t h variables r a n d v . A n operator 0 + is defined to be t h e " a d j o i n t " of O if

=<0|

(4)

h o l d s for all (r, v ) and ip(r, v ) .

W h e n t h e integration over r in t h e definition of scalar p r o d u c t is e x t e n d e d over a finite region in t h e configuration space (as is the case in reactor applications w h e r e the finite region is the v o l u m e of t h e reactor), one has to include t h e b o u n d a r y conditions in t h e definition of t h e adjoint operator. W e consider t h e functions <£(r, v ) t h a t satisfy t h e regular b o u n d a r y condition (cf Section 1.3), namely (/>(r, v ) = 0 for n • v < 0, w h e r e r is on t h e outer surface of the reactor. W e shall show presently that (4) can b e satisfied for differential operators only if the functions I/J(T, v ) are allowed to satisfy a different b o u n d a r y condition, w h i c h is referred to as t h e + " a d j o i n t " b o u n d a r y condition. W e shall d e n o t e these functions by (r, v ) . T h e following examples will illus­ trate the m e t h o d of finding t h e adjoint of an operator, and t h e associated adjoint b o u n d a r y condition Example 1. Differential operator. T h e adjoint of v * V is — v • V with t h e adjoint b o u n d a r y condition +(r, v ) = 0

for

h • v > 0,

r e S

(5)

T h e proof is as follows:

<<£+ | 0<> /> == J dH J

dh | # + * ( r , v ) v • V<£(r, v ) ]

2 =

dv v j dQ j d*r

*(r, v ) V • v(r, v)]

w h e r e we have used v ' V = V • v 0 , which follows from the fact1 that t h e gradient operator operates only r. U s i n g G r e e n ' s t h e o r e m , we obtain

2

<+ | 0> -

dv v | J dQ j

ds[h- v+*]

3 -

+ By

j dQ j

i r[(vVf)*flj

3 Green's

theorem,

$ d r ( y • A ) == j ds (n • A ) . Let

R

8

A =

and

use

51

2.1. Mathematical Preliminaries

+ w h e r e s is t h e outer surface of t h e reactor. Since satisfies (5), t h e surface integral vanishes because either + or 4> will always be zero on t h e surface in integrating over SI. T h e r e m a i n i n g t e r m can b e recognized as < — v • V+ | >, w h i c h proves t h e assertion. Example

2.

Integral operator. T h e adjoint of J d*v' |V278(v' - > v, r)]

(6a)

j d*v'[vZB(v^v',r)]

(6b)

is

which is obtained by interchanging t h e v and v ' in t h e integrand. N o t e that t h e r e is no restriction on the adjoint functions + on t h e b o u n d a r i e s . T h e proof starts with the definition of t h e scalar p r o d u c t .

3

f

<+ [ Ocf>> = j dh j d v <£+*(r, v ) j d*v' [v'Zs(v

- > v, r) <£(r, v')]

If the order of integration over v a n d v ' is interchanged, t h e r i g h t - h a n d side becomes

3 j dh j d v <£(r, v ) j d*v' [vZs(v - > v', r) ^+*(r, v')] = I t follows from this example as a special case that t h e adjoint of f0(v)jd*v'[Zt{r,v')v']

(7a)

is vZfav)

j d*v'f0(v')

(7b)

+ If t h e adjoint of an operator is identical to itself, namely 0 == O, t h e n it is called "self-adjoint." It is easy to show, for example, that V • DV is self-adjoint for functions (r) that vanish on the outer surface (Problem 1). T h e above definitions can be extended to matrix operators and to their adjoints. A matrix operator O = [ O ^ ] is a square matrix whose elements are operators as defined above. If | (/>> is a c o l u m n vector, t h e n O | > is also a c o l u m n vector, defined by O \ } = | Oy =

c

o

i

[

o

^

(

8

)

52

2. Point Kinetic Equations

with t h e s u m m a t i o n convention on J in each element of | 0>. T h e adjoint of O is a s q u a r e matrix defined by

+ G

-

[0+]

(9)

which is obtained by interchanging t h e c o l u m n s and rows of O a n d taking t h e adjoints of each element. T h e proof is left as an exercise ( P r o b l e m 2). T h e following relations are self-evident:


> = > =

1 o^>

(io)

T h e s e equalities define t h e symbol <+ | O | (/>>, which will be used extensively in t h e following. C. Eigenvalue Problem T h e p r o b l e m of finding t h e nontrivial solutions of a h o m o g e n e o u s equation of t h e form 0\n>

=K\n>

(11)

with certain b o u n d a r y a n d regularity conditions on t h e solutions is called an eigenvalue p r o b l e m . T h e set of complex n u m b e r s Xn for which (11) has nontrivial solutions are called t h e eigenvalues and t h e c o r r e s p o n d i n g solutions |
0 l^» > = /*„|* > +

+

+

w

(12)

with t h e adjoint b o u n+d a r y conditions. T h e eigenfunctions a n d t h e eigenvalues of O and 0 are related to each other. T h e following p r o p e r ­ ties are proven in s t a n d a r d texts [8, 9] on functional analysis in certain function spaces and for certain classes of operators. I n our analysis, these properties will always be satisfied. 1. T h e sets of eigenvalues {An} a n d {/xn} are complex conjugates of each other, i.e., for a given Xn , t h e r e is a+\in such t h a t \xn = A n*. 2. T h e eigenfunctions {| 0n>} and {| <£ n>} form a biorthogonal set, i.e.,

+

(m

I n> =

where Smr)is t h e K r o n e c k e r delta.

§mn

(13)

2.2. Stationary Reactor and the Multiplication Factor

53

+ 3. {| n>} a n d {| n>} are complete, i.e., any vector e x p a n d e d into {| n>} as l<£> =

| y can be

£ a \n> n n=0

(14a)

+

where a

n

=

<<^ | c/>y

(14b)

4. T h e eigenvalues of a self-adjoint operator are real a n d positive, a n d its eigenfunctions form a c o m p l e t e o r t h o n o r m a l set. 5. T h e c o m p o n e n t s of | n> satisfy (closure p r o p e r t y ) 8(r - r') S(v - v ' ) 8„ = £ ^ w( ri' , V) ^ ( r , v )

(15)

( P r o b l e m 3). 2.2. S t a t i o n a r y R e a c t o r a n d t h e M u l t i p l i c a t i o n F a c t o r T h e p u r p o s e of this section is to i n t r o d u c e t h e concept of criticality a n d n e u t r o n cycle, a n d to define t h e multiplication factor in a stationary noncritical reactor. W e start w i t h t h e basic kinetic equations (1.3, E q . 3) a n d (1.3, E q . 4), w h i c h can be w r i t t e n in a c o m p a c t way by defining t h e o p e r a t o r s : L = — £1 - Vv(u) — Z(r, u, t) v(u) + j du' j dQ' [v(u') 27s(r, u' -+u,£lMQ = £

[/o'(«)/47r]

j

Mt = I [/i(«)/47r] J H=L

du'

j

dQ' [v(u') vi(u')(l

t)]

(1)

- 00 2V(r,

J dQ' [ftV("') «(«') &(r,

+ M0

u', t)]

*)]

(2)

(3) (4)

as follows: = diCJ^jdt

Hn+Y

j

XJfit

= Mitt - XJiCi

+ 5

(5a) (5b)

w h e r e we have s u p p r e s s e d t h e a r g u m e n t s , a n d multiplied (5b) b y fi(u)

2. Point Kinetic Equations

54

for convenience. H e r e Q(r, t) has b e e n redefined to absorb 1 \ATT. T h e physical m e a n i n g of these operators, which motivates their i n t r o d u c t i o n , can b e d e d u c e d from their definitions: L describes t h e t r a n s p o r t , a b s o r p ­ tion, a n d scattering of n e u t r o n s in a n o n m u l t i p l y i n g m e d i u m ; M0 d e t e r m i n e s t h e rate of p r o d u c t i o n of p r o m p t n e u t r o n s w h e n it operates on t h e angular density; hence, it can b e called t h e p r o m p t - p r o d u c t i o n operator. Similarly, Mi can b e called t h e p r e c u r s o r - p r o d u c t i o n operator, which d e t e r m i n e s t h e rate of p r o d u c t i o n of t h e delayed n e u t r o n p r e ­ cursors of t h e ith. type w h e n it operates on n(r, u, Sly t). Finally, H describes n e u t r o n s in a multiplying m e d i u m in t h e absence of delayed n e u t r o n s ; it is called t h e B o l t z m a n n operator. W e note t h a t t h e operators H a n d Mi d e p e n d on t h e composition of t h e m e d i u m , and describe t h e m e d i u m completely. T h e y are, in general, time d e p e n d e n t , insofar as t h e cross sections are functions of t i m e . W e shall a s s u m e for t h e present that t h e properties of t h e m e d i u m do not d e p e n d on t h e n e u t r o n population, i.e., feedback is not present. I n this case, t h e kinetic equations (5) are linear. T h e effect of feedback will be considered at t h e e n d of this chapter. I n the absence of feedback, t h e t i m e d e p e n d e n c e of H a n d Mi is explicit, and is due to t h e changes i n t r o d u c e d externally in t h e composition and configuration of the m e d i u m . I n this section, we shall focus our attention on a reactor with t i m e i n d e p e n d e n t cross sections (stationary reactor), in t h e absence of external n e u t r o n sources, i.e., S^r, v , t) = 0. T h e angular n e u t r o n density in a stationary reactor is still a function of time, a n d either increases or decreases. A source-free, multiplying m e d i u m with t i m e - i n d e p e n d e n t cross sections is " c r i t i c a l " if it can s u p p o r t a stationary n e u t r o n p o p u l a ­ tion N0(ry uy SI) which is not zero everywhere. It is supercritical or s u b critical if w(r, Uy Sly t) is increasing or decreasing respectively in t i m e . T h e time behavior involved in this definition is the asymptotic behavior at long times following initial disturbances. T h e n e u t r o n population is a function of t i m e even in a critical reactor d u r i n g t h e transients because criticality is a p r o p e r t y of t h e m e d i u m , b u t not of t h e n e u t r o n population. T h e steady angular n e u t r o n density Af0(r, w, SI) in a critical reactor satisfies t h e following e q u a t i o n : JE N (Vy

00

Uy SI) =

0

(6a)

which is obtained from (5) by setting t h e t i m e derivatives to zero and eliminating Ci0 (r). H e r e , J$?0 is defined by JR^H+YMI^L

+ M

(6b)

2.2. Stationary Reactor and the Multiplication Factor

55

w h e r e M is t h e modified multiplication operator: (7)

j in which f (u)

is defined by 6

/'(u) = ( i -/»0/o'(«)+ £ # / « ( « )

(8)

i=l

W e note t h a t M differs from M0 in t h e energy distribution of the fission n e u t r o n s , i.e., f0(u) in M0 is replaced by f(u) in M. T h e latter is the weighted average of t h e p r o m p t and delayed n e u t r o n energy distributions. T h e energy d e p e n d e n c e of t h e second t e r m in (8), t h e yield-averaged delayed n e u t r o n s p e c t r u m , is s h o w n in ( F i g u r e 1.1.3). T h e steady-state equation is a h o m o g e n e o u s equation which is to be solved with t h e regular b o u n d a r y conditions requiring N0(r, u, SI) to b e c o n t i n u o u s a n d positive everywhere in t h e reactor v o l u m e (cf Section 1.3). T h e condition w h i c h m u s t b e satisfied by Jf?0 (i.e., by the material and geometric properties of the m e d i u m ) such that J^0N0 = 0 will have a nontrivial solution is referred to as t h e criticality condition. W e n o w i n t r o d u c e t h e concept of effective multiplication factor by considering a noncritical stationary reactor. Since t h e reactor is not critical, (L + M) iV 0(r, u, Si) = 0 has no other solution b u t t h e trivial j suppose t h a t we modify t h e m e a n n u m b e r one AT0(r, u, SI) == 0. L e t us of n e u t r o n s per fission v for each isotope by multiplying t h e m by ( l / & e )f, fkeeping all t h e other nuclear a n d geometric properties t h e same. T h i s modification is equivalent to multiplying t h e p r o d u c t i o n operator M by (1/^eff)- Adjusting t h e value of t h e positive n u m b e r k e Jt we can t always associate a critical fictitious system with a given noncritical reactor. T h e p r o p e r value for kett is d e t e r m i n e d by requiring t h a t [L + (l/£eff)M]7V 0(r, «,n)

=0

(9)

have a nontrivial solution. It is clear t h a t keff m u s t be unity if t h e actual reactor is already critical. I n order to explain t h e physical m e a n i n g of keU , which has been i n t r o ­ duced as a formal mathematical device, we mentally break, following Ussachoff [2], t h e c o n t i n u o u s chain process into n e u t r o n cycles. W e begin a cycle by i n t r o d u c i n g n e u t r o n s into t h e reactor with t h e space and velocity distributions a p p r o p r i a t e to t h e steady state. T h e s e n e u t r o n s disappear eventually t h r o u g h absorption and leakage, m a r k i n g t h e end

56

2. Point Kinetic Equations

of a cycle. S o m e of t h e absorbed n e u t r o n s cause fission, a n d p r o d u c e n e u t r o n s which originate t h e next cycle. W e shall show that kett is t h e ratio of t h e n u m b e r of fission n e u t r o n s emitted in a given cycle to t h e n u m b e r of fission n e u t r o n s emitted in t h e preceding cycle. T h e r e f o r e , kett can b e identified as t h e effective multi­

plication

factor.

L e t us denote t h e n u m b e r of fission n e u t r o n s emitted in t h e critical reactor per second per u n i t v o l u m e at r and v by Q(r, u, SI), Q(r,v)^MN (r,u,Sl)

(10)

0

L e t us inject instantaneously Q(r, v ) / & eff n e u t r o n s into t h e actual noncritical reactor at t = 0. A s s u m e t h a t t h e r e are no n e u t r o n s in this reactor prior to t = 0. T h e angular density q(r, u, SI, i) of t h e n e u t r o n s t h a t originate directly from Q(r, v)jkeU will satisfy t h e following t i m e d e p e n d e n t equation: dqjdt=Lq

(11)

with t h e initial condition t h a t q(r, u, SI, 0) = Q(r, u, Sl)lkett . W e note that (11) describes only t h e removal of n e u t r o n s by leakage and a b s o r p ­ tion, as implied by the definition of a n e u t r o n cycle. By integrating (11) from 0 to oo a n d taking into account t h e initial condition a n d the fact that q(r, u, SI, t) = 0 as t —> oo, we obtain [Q(r,

II, n ) / * e „ ] +

f

Lq(r,

u, ft, t) dt

=

0

(12a)

u, SI, t) dt

=

0

(12b)

or [Q(r,

u, n ) / * e f f ] + L

Jf "

q(r,

o S u b s t i t u t i n g Q(r, u, SI) from (10) in (12b), and c o m p a r i n g t h e resulting equation to (9), we find t h a t t h e time-integrated angular density $o u, SI, i) dt is equal to iV 0(r, u, SI). [It may seem at first sight t h a t t h e r e is a m i s m a t c h of dimensions because of t h e time integration on #(r, u, SI, t). However, this is not t h e case, because ^(r, u, SI, t) has t h e same dimension as £)(r, u, SI), w h i c h has t h e dimension of a source.] H e n c e , we can also write (10) as Q(r, u,Sl)

= f Mq(r, u, SI, t) dt Jo

(13)

w h i c h indicates t h a t Q(r, u, SI) is t h e total n u m b e r of fission n e u t r o n s in t h e actual reactor p r o d u c e d directly by t h e £)(r, u, Sl)lkett n e u t r o n s in t h e interval (0, oo), which were i n t r o d u c e d instantaneously at t = 0.

57

2.3. Adjoint Angular Density and Neutron Importance

T h u s , t h e ratio of t h e n u m b e r of n e u t r o n s in t h e s u b s e q u e n t cycle, i.e., Q(ry uy Si) to t h a t of n e u t r o n s in t h e p r e c e d i n g cycle, i.e., Q(r> uy Si)LKEFT , is indeed equal to KETT . W e w o u l d like to e m p h a s i z e t h a t t h e source n e u t r o n s , Q(r, u> Si)/KETT , are i n t r o d u c e d into t h e actual reactor with t h e same distributions in velocity a n d position as they have in t h e stationary reactor. H e n c e , t h e effective multiplication factor c o m p u t e d by (9) is called t h e static m u l t i ­ plication factor [10].

2.3. A d j o i n t A n g u l a r Density a n d N e u t r o n I m p o r t a n c e I n this section, w e shall i n t r o d u c e t h e concept of adjoint angular density [2, 11, 12] in a source-free critical reactor, w h i c h plays an i m p o r t a n t role in reactor theory. W e have already defined t h e adjoint of an operator in Section 2 . 1 . U s i n g this definition a n d t h e examples in t h a t section, we find t h a t t h e adjoint of t h e operator Jf?0 is J4T0+ = SI • Vv(u) — U(ry u) v(u) + j du' J dQ' {Z^r, u^u'ySl+ Z U\u')l^}

A") 2V(r, u)} v(u)

SI') (1)

3

I n (1), t h e sign of t h e s t r e a m i n g operator has b e e n changed, a n d (uy Si) a n d (u'y SI') in t h e i n t e g r a n d s have b e e n i n t e r c h a n g e d . W e define t h e " a d j o i n t " angular density [11, 12] N0+(ry uy Si) as t h e nontrivial solution of ^N^{ryUySl)

=0

(2)

with t h e adjoint b o u n d a r y condition iV0+(r, Uy Si) = 0

for

n • SI > 0,

reS.

(3)

I n order to attach a physical m e a n i n g to t h e adjoint angular density, we consider t h e following p r o b l e m [2]. S u p p o s e a n e u t r o n is injected into a critical reactor at t = 0 at t h e space point r ' with a velocity v ' , a n d a s s u m e t h a t t h e r e are n o n e u t r o n s in t h e reactor prior t o t = 0. W e w a n t to d e t e r m i n e t h e t i m e - d e p e n d e n t angular density n(r, w, Sly t) as a function of r a n d v for all s u b s e q u e n t times, a n d in particular as t —> oo. F o r t h e t i m e being, we shall ignore t h e delayed n e u t r o n s for t h e sake of simplicity. T h e n , n(r, uy Slf t) satisfies dnjdt = Hn

(4)

58

2. Point Kinetic Equations

with t h e initial condition «(r, w , 0 ) = 8(r - r') S(u - u') 8(Sl - &)

(5)

E q u a t i o n (4) is obtained from t h e kinetic equations (2.2, E q s . 5a a n d 5b) by omitting the delayed n e u t r o n s and t h e source t e r m . I n order to find t h e solution of (4), s u p p o s e t h a t it is possible to find t h e eigenfunctions of t h e operator H (Boltzmann operator) by solving t h e following equation: Hn =

°>nn

(6)

with t h e regular b o u n d a r y conditions. W e assume t h a t t h e eigenvalues of con can be arranged in increasing order of the m a g n i t u d e of their real parts in case they are complex. T h e eigenvalues and t h e eigenfunctions of t h e Boltzmann operator are not k n o w n in general, except for some very special cases, e.g., t h e one-speed model with isotropic scattering and plane s y m m e t r y [13]. I n general, t h e eigenvalue s p e c t r u m contains a discrete set and a c o n t i n u u m , as is t h e case in space- a n d / o r energyd e p e n d e n t kinetic p r o b l e m s , and, therefore, t h e index n takes on discrete and continuous values. Since t h e p u r p o s e of t h e present discussion is only to u n d e r s t a n d t h e physical m e a n i n g of t h e adjoint angular density, we shall not get involved in questions of existence, completeness, etc., and instead proceed formally. However, the assumptions i n t r o d u c e d in this formal presentation can b e justified in m a n y practical particular cases, e.g., t h e one-speed diffusion model to b e discussed subsequently. Since the Boltzmann operator is not self-adjoint, we have to consider the adjoint eigenvalue p r o b l e m also, i.e., =

(7)

+ so that {n} and {n} will form a complete b i o r t h o n o r m a l set. T h e n , we can expand the t i m e - d e p e n d e n t angular density n(r, u, £1, t) in t h e functions n(r, u, &) as 00

n(r, «, a, t) = £ «„(r\ «', Si', t) -£K (r, u, Si)

(8)

+ w h e r e t h e expansion coefficients are of course given by an = <n | »>. Substituting (8) into (4) and using (6), we obtain an as

l

an(r\

u', Sl\ t) = an(v\

u' ,£l' , 0) e

(9)

59

2.3. Adjoint Angular Density and Neutron Importance

r

f

T h e initial values an(r , u', Sl , 0) m u s t be d e t e r m i n e d by t h e initial condition on n{r, u, SI, t): 8(r -

r') S(S1 - SI') 8(u -

«') = L "N(r\ *, Sl\ 0) cf> (r, u, SI)

n

+

u s scalar p r o d u c t s , we M u l t i p l y i n g b o t h sides by „ (r, u, Si) aTn d h forming get o n( r ' , « ' , a', 0) = n+(r', « ' , Q')n{v, u, Si, t)

= J

>

(10)

^ + ( r \ «', £2') ^ ( r , u, Si)

F r o m t h e physical consideration that in a critical reactor t h e density cannot increase indefinitely, we assert t h a t t h e a>n have negative real parts for n ^ 0, a n d a>0 = 0. T h e last equality follows from t h e fact t h a t t h e reactor is critical, a n d hence H(/>0 = 0 has a u n i q u e nontrivial solution. I t is also clear t h a t t h e eigenfunction 0c o r r e s p o n d i n g to w0 = 0 is t h e steady-state angular density iV 0(r, u, Si). T h u s , t h e coefficients of all t h e higher m o d e s in (10) decay exponentially in time, a n d t h e asymptotic angular density is obtained as (r\ noo

u\ SI'; r, «, SI) = N0+(r',

(11)

SI') 7V0(r, u, SI)

w h e r e we have s h o w n t h e d e p e n d e n c e of on r', u', SI' explicitly. W e n o w i n t r o d4u c e t h e concept of * ' i m p o r t a n c e . ' ' T h e ' i m p o r t a n c e ' ' of a n e u t r o n injected into a critical reactor at r' with a lethargy u in t h e direction of SI' is t h e total n u m b e r of fissions per second in t h e entire reactor at a long t i m e following t h e injection of t h e n e u t r o n at t = 0. T h e i m p o r t a n c e function is readily obtained from (11) by multiplying b o t h sides by ^ f( r , u) v(u) a n d integrating over r a n d v:

7(r', II', SI') = 7V0+(r',

n')27 | f

(12)

JV0>

+ e conclude from this result t h a t t h e adjoint angular density W i V 0( r ' , u', SI') is proportional to t h e i m p o r t a n c e of n e u t r o n s at r' m o v i n g with a lethargy u' in t h e direction of SI' in sustaining t h e chain reaction in t h e reactor. T h e proportionality constant in (12) is obtained with t h e normalization of A^0+(r, w, SI) and A^0(r, u> SI) as = 1. I n general, E q . (12) can be written as 7(r', II', SI') = Ar0+(r', u\ Sl'){vEz \ iV0>/<7V0+ | iV0>

+

(13)

which is clearly i n d e p e n d e n t of t h e choice of normalization for i V 0 a n d

N.

0

60

2. Point Kinetic Equations

+

+

Since t h e eigenvalue p r o b l e m H N0+ = 0 yields i V 0 with an i n d e ­ t e r m i n a t e factor, we can make only the following statement u n a m b i g ­ uously: T h e adjoint angular density is a m e a s u r e of t h e "relative i m p o r t a n c e " of n e u t r o n s at two different positions with t w o different velocities in t h e reactor. T h e foregoing analysis can easily be extended to include t h e delayed n e u t r o n s . I n this case, t h e t i m e - d e p e n d e n t angular density satisfies the full set of kinetic equations dn/dt = Hn + YJ HfiCi)

(14a)

1=1

(14b) which are to b e solved with t h e initial conditions n(r, u, £ly 0) = 8(r - r') S(ft - fl') 8(u - u') a n d Q(r, 0) = 0 for all i = 1, 2,..., 6. E q u a t i o n s (14) can be expressed in a compact way by defining a matrix operator J f by ' H M1 M2

Ax -K 0

M6

0

A2 " • 0

A6 0 (15)

0

•• •

-A,

a n d a c o l u m n vector ~n(r, uySly t)

Ci(r,

as

(16)

_C 6(r, 0/,(u).

j r 1 = d | w ) > / a

(17)

W e assume again t h a t it is possible to find the eigenvectors a n d the eigenvalues of (cf Section 2.1) X+ | + = » „ | jr |*f > l = o n* | ^ + >

(18a) (18b)

w h e r e the adjoint matrix operator JT+ is given explicitly by

Jf+ =

H+ Ax

Mj+ -A,

M 6+' 0

(19)

2.3. Adjoint Angular Density and Neutron Importance

61

(See P r o b l e m 4 for an illustrative example.) T h e eigenvectors | n> are referred to as period m o d e s [14] or, m o r e frequently, as co-modes [15-18]. W e e x p a n d | */*(£)> in t e r m s of {| ^> n)} as follows:

I = £ an(r', «', Sl',0) I e""* where

, flw(r',« ,n',O)=<^w+|0(O)>

(20)

(21)

Since | 0(O)> = col[8(r - r') 8(11 - 11') S(S1 - ft'), 0,..., 0]

(22)

we find

+ w h e r e Nn(ry Hence,

*w (r', II', Q', 0) = Nn+(r',

u\ SI')

(23)

+

uy SI) is t h e first c o m p o n e n t of t h e eigenvector | n>. | 0(O> = £ iVw +(r', u', SI') | <£n> ^

(24)

71=0

or, taking t h e first c o m p o n e n t of (24), OO + n(r, u, Q, 0 = Z A^ (r', «',«') Nn{t, u, Si) e™*'

(25)

n=0

w h e r e Nn(ry uy Si) is t h e first c o m p o n e n t of t h e vector. I t is a s s u m e d t h a t t h e o j n all have negative real p a r t s w i t h increasing o r d e r in m a g n i t u d e . T h e criticality of t h e reactor implies t h a t co0 = 0, with t h e c o r r e s p o n d i n g eigenvector satisfying | 0 > = 0. T h e first c o m p o n e n t of | c/)O YY i.e., N0(ry uy SI) satisfies, as+ can be readily seen from (14), 3f0NQ (ry uy SI) = 0. Similarly, ^ + / V 0 ( r , uy SI) = 0. H e n c e , t h e a s y m p ­ totic distribution is u\ SI'; r, ii, SI) = N0+(r\

u'ySI') N0(r, u, SI)

w h i c h is identical to (11). T h e definition of t h e i m p o r t a n c e is therefore + as in t h e case with no delayed n e u t r o n s , p r o v i d e d N (r, u SI) t h e same Q y a n d i V 0( r , uy Si) are i n t e r p r e t e d as t h e steady-state angular density a n d its adjoint w i t h t h e delayed n e u t r o n s .

62

2. Point Kinetic Equations

2.4. Reduction of t h e K i n e t i c Equations W e are n o w in a position to transform t h e basic kinetic equations into a m o r e tractable form, from w h i c h t h e point reactor kinetic equations can be d e d u c e d by making certain approximations. S u c h a precise derivation of t h e point reactor kinetic equation is n e e d e d for various reasons; for example, we m u s t take into account t h e delayed n e u t r o n s p r o d u c e d by fast fission, a n d consider t h e fact t h a t t h e delayed n e u t r o n s do not have t h e same energy distribution as t h e p r o m p t n e u t r o n s . F u r t h e r m o r e , t h e definition of other quantities, such as t h e generation time, external source, etc., m u s t be sufficiently precise to allow t h e m to be c o m p u t e d for a given reactor composition and geometry. I t is therefore essential to establish t h e point reactor kinetic equations so that t h e u n d e r l y i n g a s s u m p t i o n s restricting t h e validity of these e q u a ­ tions and t h e precise physical m e a n i n g of t h e quantities appearing in t h e m will be clear. F o r this p u r p o s e , we partition t h e angular n e u t r o n density [2, 3] n(r, uy SI, t) into a shape function (r, uy £1, t) a n d a t i m e function P(t) such that TL(Vy Uy Sly T) =

P'(f)

y Uy Sly T)

(l)

T h i s separation of n(ry uy SI, t) into t h e p r o d u c t of two new u n k n o w n functions is not u n i q u e . However, later, we shall impose further n o r m a l ­ ization restrictions to m a k e t h e choice of P(t) a n d cf)(r, u, SI, i) u n i q u e . W e substitute (1) in t h e basic kinetic equations (2.2, E q . 5): PWIDT) + (DPLDT) = PH(T)CF> + £ XJiCt + S d{UC% )\dt = PMtM

- xjtCt

(2a) (2b)

W e have written t h e t i m e d e p e n d e n c e of t h e operators H(t) and Mt(t) explicitly, to r e m i n d ourselves t h a t t h e reactor p a r a m e t e r s are explicit functions of time. W e neglect feedback effects for p r e s e n t purposes, and assume that t h e geometry of t h e reactor is u n c h a n g e d . W e imagine a critical, source-free ' 'reference" reactor of t h e same geometry, a n d with similar nuclear properties. I n fact, if t h e actual reactor is critical in t h e initial or final state, t h e n t h e reference reactor can b e chosen as t h e actual reactor in t h a t particular state. Being critical, t h e reference reactor is completely described by t h e stationary B o l t z m a n n + operator J f 0 . W e assume t h a t t h e steady-state angular density A^0(r, u, SI) a n d its adjoint N0(ry u, SI) are k n o w n . M u l t i p l y i n g (2a) and (2b) by

2.4. Reduction of the Kinetic Equations

63

+ N0(r, u, Si) a n d , integrating t h e resulting e q u a t i o n s over r, u, a n d Si, w e obtain, in t h e scalar p r o d u c t notation, + 14>>{dPldt) + P(dldt)(N,+
| <£> = P(N0+

| H(t) | >

+ i*i
d
ffiMdt =

+

(3a)

P - A,<7V0+ | /

(3b)

W e m u s t n o w i m p o s e a " n o r m a l i z a t i o n " condition o n t h e s h a p e function to e n s u r e u n i q u e n e s s , w h i c h w e choose as (other choices will b e discussed later in Section 2.5). {djdt)(N+

| <£> = (d/dt) j

dh J du jdSl

[AT0+(r, u, SI)
T h e physical i n t e r p r e t a t i o n of this condition is as follows: N0+(r, u, SI) is p r o p o r t i o n a l t o t h e i m p o r t a n c e of n e u t r o n s (cf 2 . 3 , E q . 12). H e n c e , > is t h e total i m p o r t a n c e of n e u t r o n s in t h e reference reactor with a distribution function {r, u, SI, t). According to (4), t h e shape function m u s t b e so chosen t h a t t h e total i m p o r t a n c e in t h e reference reactor will r e m a i n constant in t i m e even t h o u g h (r?u, SI, t) itself m a y vary slowly in t i m e locally. A n alternative i n t e r p r e t a t i o n of t h e n o r m a l ­ ization condition (4) can b e obtained b y e x p a n d i n g (r, u, SI, t) in t h e eigenfunctions of , (r, u, SI, t) = >N +

0

0

£ } Nn

(5)

N o t e t h a t all t h e expansion coefficients are functions of t i m e . T h e condition (4) requires t h e coefficient of t h e f u n d a m e n t a l m o d e [i.e., t h e r u of o n N (r, u, Si) in t h e function space s p a n n e d b y projection 0 ^ n ( > > ®)] to b e i n d e p e n d e n t of t i m e . I t is also clear from this inter­ pretation that t h e normalization condition (4) does n o t imply t h a t t h e s h a p e function b e i n d e p e n d e n t of t i m e locally. + T h i s discussion enables u s to clarify t h e physical m e a n i n g of t h e t i m e function. M u l t i p l y i n g b o t h sides of (1) b y N0, w e find t h a t

+

+

^(0 = < ^ o N > / < ^ o l ^ >

(6)

w h i c h states t h a t P(t) is t h e ratio of t h e total i m p o r t a n c e of n e u t r o n s with a distribution n(r, u, SI, t) to t h e i m p o r t a n c e of those n e u t r o n s t h a t have a distribution
64

2. Point Kinetic Equations

time, and can be scaled as unity. T h e n , P(t) becomes t h e instantaneous value of t h e total i m p o r t a n c e of t h e n e u t r o n population in t h e actual + reactor which is necessary to sustain a chain reaction in t h e reference reactor. Since < i V 0 | n) is t h e coefficient of t h e fundamental m o d e in t h e expansion of n(r, u, £1, t), P(t) is simply t h e t i m e d e p e n d e n c e of t h e fundamental m o d e . I t is emphasized that P(i) is not t h e total n u m b e r of n e u t r o n s in t h e reactor v o l u m e at t h e t i m e t. Also notice t h a t P(t) is i n d+ e p e n d e n t of t h e normalization of t h e adjoint angular density Af 0(r, u, &) as indicated by (6). W e n o w r e t u r n to E q s . (3a) a n d (3b), a n d observe t h a t t h e n o r m a l ­ ization condition (4) removes t h e second t e r m on t h e left of (3a). T o i n t r o d u c e t h e concept of p e r t u r b a t i o n , i.e., t h e deviations of t h e reactor p a r a m e t e r s of t h e actual reactor from those of t h e reference reactor, we define a p e r t u r b a t i o n operator 8J^(t) as 8jT(t) == H(t) + t

M,(0 - Jt0 = L(t) + M(t) - Jf0

(7)

M o r e explicitly, 8Jt?(t) == -v(u)

8Z(r, uy t) + j du' j dJCl' 8{Zs(r, u'->u,&-

j

t)

j

+ Z [f (")l^]

A*') Z* (r, u\ t)} v{u')

(8)

j

w h e r e S2^(r, u, t) measures t h e variations of t h e cross sections about their reference values, i.e.,

Si^(r,

(r, u9t) = Ei{ry u, t) — Zj0

u)

w h e r e t h e subscript j denotes a, f, or s. S u b s t i t u t+ i n g i H(t) from (7) into (3a), a n d observing that < N 0+ | Jf0 \ <£> = < ^ 0 V I <£> = 0 for any function with t h e regular b o u n d a r y conditions (this is a crucial point in t h e derivation), we obtain t h e desired form of t h e kinetic e q u a t i o n s : dPjdt = Mt)

- j5)/Z] P(t) + Z A A W

+ S(t)

(9a)

4=1

dCt/dt

= (ft/Z) P(t) - Xfilt)

(9b)

with t h e following definitions: Reactivity:

P( ( ) - ( l / W | 8 ^ ) | ^ >

(10)

65

2.4. Reduction of the Kinetic Equations

Effective delayed n e u t r o n fraction: ft EE (l/F)>

?=z &

(11)

(12)

Effective concentration of delayed n e u t r o n p r e c u r s o r s : (13) Effective source: S(T) = (IIFIKN0+ | S)

(14)

M e a n p r o m p t generation t i m e : /^(l/F)

(15)

Normalization factor:

and

F^(N0+\M(t)\}

= £ | J Jfl'

{[/'(W)/47T]

(16)

v'(w') ^ ( r , w', 0 v(u')} ( 1 7 )

T h e physical m e a n i n g of t h e symbols a n d t h e reason for n a m i n g t h e m as indicated will b e discussed presently. First, however, t h e following general r e m a r k s a b o u t t h e equations (9) are in order, (a) T h e s e equations are exact a n d completely equivalent to t h e basic kinetic equations in a different form. T h e quantities appearing in this equation still contain t h e u n k n o w n s h a p e function (r, w, t), w h i c h can only be solved t h r o u g h t h e original space- a n d e n e r g y - d e p e n d e n t kinetic equation. T h e advantage of this new form lies in t h e fact that it lends itself easily to various physical approximations w h i c h are used to investigate t h e t i m e behavior of P(t), this q u a n t i t y being one of t h e most i m p o r t a n t aspects of t h e n e u t r o n population in reactor dynamics. T h e s e a p p r o x i m a t i o n s will be discussed in t h e following section, (b) T h e choice of t h e n o r m a l ­ ization factor F is arbitrary in t h e sense t h a t t h e kinetic equations (9) are entirely i n d e p e n d e n t of it. However, t h e m a g n i t u d e s of p, , and / d e p e n d on F . T h e particular definition used in (17) is chosen so that t h e reactivity can approximately be i n t e r p r e t e d as ( & eff— l ) / & eff in t e r m s of t h e effective multiplication constant defined in (2.2, E q . 9), as will be

66

2. Point Kinetic Equations

s h o w n later, (c) T h e arbitrariness in t h e choice of F implies t h a t t h e quantities p, /, cannot b e defined as physical quantities in an absolute sense, a n d t h a t only t h e ratios {pjl) a n d (fijl) can b e defined u n a m b i g ­ uously. T h i s conclusion has t h e i m p o r t a n t consequence t h a t only t h e ratios of these quantities can b e m e a s u r e d experimentally for a given reactor. EFFECTIVE DELAYED N E U T R O N

FRACTION

W e can express t h e effective delayed n e u t r o n fraction m o r e explicitly b y substituting Mi in (11) from (2.2, E q . 3) a n d using t h e normalization in (17): ft = I (ft Jr d*r j | du j dQ [JV0+(r, u, «)/<(«)] X j du' | dQ' [v\u') v(u') 2V(r, u\ t)(r, u\ £2', *)]

X J

J

|V(z/') a(w') Z?(r, w', t) (r, a', ft', *)] j)

(18)

T h e distinction b e t w e e n and t h e actual delayed n e u t r o n fraction from t h e / t h isotope, is a p p a r e n t in (18). I t is t h e ratio of t h e i m p o r t a n c e of all t h e delayed n e u t r o n s of t h e zth g r o u p emitted per second in t h e entire reactor t o t h e i m p o r t a n c e of all fission n e u t r o n s , delayed or p r o m p t , emitted per second in t h e entire reactor. Since t h e energy s p e c t r u m of t h e delayed n e u t r o n s fi(u)> is m u c h lower (a few h u n d r e d keV) t h a n t h a t of p r o m p t n e u t r o n s (a few M e V ) , their i m p o r t a n c e in a s t h e r m a l reactor is greater (smaller leakage probability) than the impor­ tance of p r o m p t n e u t r o n s . T h e r e f o r e , * greater t h a n unity, a n d t h e difference m a y b e 20 or 3 0 % [3]. Explicit formulas for calculating in t h e age-diffusion a p p r o x i m a t i o n will be given later. EFFECTIVE SOURCE

T h e definition of t h e effective source follows from (14) as S(t) = y

(19)

which indicates t h a t S(t) is proportional to t h e i m p o r t a n c e of all t h e external n e u t r o n s i n t r o d u c e d per second in t h e entire reactor. If t h e

67

2.4. Reduction of the Kinetic Equations

distribution of t h e source n e u t r o n s in position a n d velocity is t h e s a m e as (r, uy Sly t), t h e n S(t) = S0(t). W e can also i n t e r p r e t S(t) as t h e coefficient of t h e f u n d a m e n t a l m o d e in t h e expansion of 5 ( r , uy SI, t) in t e r m s of t h e eigenfunctions of t h e B o l t z m a n n operator ^ . M E A N P R O M P T GENERATION

TIME

T h e definition of / is given by (15) as

+

/ = <7V0+ I <£>/<^o I M | } = <7V0+ I n)KN0+

\M\n)

(20)

H e n c e , it is t h e ratio of t h e total i m p o r t a n c e of all n e u t r o n s in t h e reactor at t i m e t to t h e i m p o r t a n c e of all fission n e u t r o n s p r o d u c e d per second in t h e entire reactor. I n other w o r d s , t h e total a m o u n t of i m p o r t a n c e created by t h e fission n e u t r o n s in t h e m e a n p r o m p t generation t i m e / is equal to t h e i n s t a n t a n e o u s value of t h e i m p o r t a n c e of n e u t r o n s p r e s e n t at t i m e t. REACTIVITY

T h e concept a n d t h e definition of reactivity in a reactor whose nuclear properties are changing continuously in t i m e r e q u i r e closer attention. Various definitions [19, 20] of reactivity are obtained by a p p r o p r i a t e choice of t h e s h a p e function in P(0

=

W I » I W W I M | ^ >

Method 1. T h e crudest, a n d t h e simplest, a p p r o x i m a t i o n is to a s s u m e t h e s h a p e function to be p r o p o r t i o n a l to t h e steady-state distri­ b u t i o n iV 0(r, Uy SI) in t h e critical reference reactor. If we d e n o t e t h e proportionality constant by (1/P 0)> this a p p r o x i m a t i o n implies n(Ty Uy Sly t) ^ [P(t)iPQ ] N0(ry Uy SI)

(21a)

T h e normalization condition (4) is automatically satisfied. W i t h i n t h e limitation of t h e p e r t u r b a t i o n a p p r o x i m a t i o n , we m a y i n t e r p r e t P(t)fP0 as P(t)IP0 ^

<«2* I » > / < ^ f 0I 0

^o>

(21b)

w h e r e Ef0 (r, u) is t h e fission cross section in t h e reference reactor. Since P(t) is approximately proportional to t h e i n s t a n t a n e o u s p o w e r (vEt0 | n)y it is often referred to as t h e reactor power (this point will be discussed further in 2.5). T h e kinetic p a r a m e t e r s p, fit , a n d / are

68

2. Point Kinetic Equations

i n d e p e n d e n t of t h e proportionality constant effective source (cf E q . 19) d e p e n d s on P 0 as

1 / P 0.

However,

the

+

S(t) ^ [I
(21c)

so t h a t it has t h e right d i m e n s i o n in t h e kinetic equations. T h e m e a n p r o m p t generation t i m e / and t h e effective delayed n e u t r o n fractions b e c o m e i n d e p e n d e n t of t i m e in t h e case of t h e constant-shape-function a p p r o x i m a t i o n if t h e p e r t u r b a t i o n does not affect t h e multiplication operators M and Mi . However, it is consistent with t h e first-order p e r t u r b a t i o n a p p r o x i m a t i o n to ignore t h e changes in St\r, u, t) in calculating ^ , /, a n d F, even t h o u g h they do change. A l t h o u g h it is t h e crudest, t h e first-order p e r t u r b a t i o n approximation is t h e only a p p r o x i m a t i o n t e c h n i q u e which allows further analytical t r e a t m e n t of t h e point kinetic equations. Method 2. T h e second m e t h o d of a p p r o x i m a t i o n consists in choosing t h e shape function as proportional to t h a t solution of [L(t) + ( 1 / M M(t)]NkJr,

u, Sl,t) = 0

(22)

t h a t is everywhere positive within t h e v o l u m e of t h e reactor ( " a d i a b a t i c " approximation). T h i s equation describes a critical reactor t h a t possesses identical nuclear properties to those of t h e actual reactor at t h e instant t, and is m a d e critical by a fictitious value of keft if t h e instantaneous configuration is not already stationary (cf 2.2, E q . 9). Clearly, b o t h kett and Nkett (r, u, Sly t) will d e p e n d on t i m e parametrically because t h e stationary e q u a t i o n (22) will be different at different times. [ T h e function A ^ e (fr ,f u, SI, t) should not be confused with t h e t i m e - d e p e n d e n t angular density n(r, u, SI, t).] Since t h e p a r a m e t e r s of t h e p e r t u r b e d reactor are k n o w n as a function of time, we can d e t e r m i n e t h e t i m e d e p e n d e n c e of keft a n d Nken (r, u, SI, t) by solving (22) for each t. W e can n o w c o m p u t e t h e reactivity p(t) by replacing t h e shape function <£(r, u, Sly t) in (21) by Nk^(r, u, SI, t), recalling t h a t

hje =

L{t) -f M(t)

-

je0

(cf E q s . 7 and 8), a n d using M(t) Nkeff = —keU L(t) p(t) = (kett -

I) I kett

(23) Nkett . T h e result is (24)

indicating t h a t t h e original identification of t h e symbol p(i) as reactivity is indeed consistent with t h e conventional definition of reactivity in t e r m s of t h e effective multiplication factor. T h e reactivity defined with

69

2.4. Reduction of the Kinetic Equations

respect to t h e stationary distribution Nk (r, u, SI, i) is referred to as t h e " s t a t i c " reactivity [14, 19, 20]. A w o r d of caution is needed at this point. As soon as we choose t h e s h a p e function as Nkeft (r, u, SI, t), which d e p e n d s on t i m e parametrically, + we have no guarantee in general t h a t t h e normalization condition (4) is satisfied, because t h e r e is no reason for e ftof b e i n d e p e n d e n t of time, unless t h e nuclear properties of t h e p e r t u r b e d reactor are t i m e i n d e p e n d e n t . T h i s question does not arise in t h e p e r t u r b a t i o n analysis because N0(r, u, SI) is b y its definition i n d e p e n d e n t of t i m e . However, if t h e variations in the shape function Nkeff (r, u, SI, t) are slow, we can still use t h e point kinetic equations w i t h o u t additional t e r m s . T h e possibility of allowing a t i m e - d e p e n d e n t normalization will b e discussed in Section 2.5. A-MODES

T h e solutions of t h e eigenvalue p r o b l e m defined by L ^ = -(1/A)M^

(25)

are referred to as reactivity m o d e s or simply A-modes. T h e equation is L+fa+ = -(1/A*) T h e o r t h o n o r m a l i t y relation for the functions

+

<^ |M|^> = V

+

adjoint (26)

ha n d A is (27)

T h e r e is one eigenfunction, designated by <£A q, that is positive every­ w h e r e t h r o u g h o u t t h e reactor v o l u m e . T h e corresponding eigenvalue A 0 is real and positive. T h e stationary distribution Nk (r, u, SI, t) a n d the effective multiplication factor ketf in (22) correspond to *0(r, u, SI) and A 0 . It therefore follows that t h e second m e t h o d of a p p r o x i m a t i o n is equivalent to replacing t h e shape function in p(t) by the lowest-reactivity m o d e at each instant of time. T h e t i m e - d e p e n d e n t angular density is approximately given by n(r, u, SI, t) = P(t) Nk (ry u, Sly t), w h e r e P(t) is obtained from t h e solution of the point kinetic equations (9) with t h e t i m e - d e p e n d e n t p(t), ^(t) and l(t). T h e latter two quantities are also d e t e r m i n e d by using Nk (r, u, Sly t) as t h e shape function in their definitions (18) and (20). Both t h e solution of (22) for Nk (r, u, Sly t) and (9) for P(t) in general require m a c h i n e calculations. Method 3. A t h i r d m e t h o d of a p p r o x i m a t i o n is achieved by choosing the shape function as t h e fundamental a>-mode defined by (2.3, E q s . 18).

2. Point Kinetic Equations

70

By eliminating t h e delayed n e u t r o n precursor densities in these equations, we obtain t h e following alternative definition of t h e oj-modes: \H + £ Mt] Nn(ry uy£1) = con [l + £

1 ATw (r, */, £1)

(28)

T h e o r t h o n o r m a l i t y relation for t h e functions Nn(ry uy £1) a n d their adjoint follows from (2.3, E q . 19) as

Z n^

4 V I AO +

I f ' 1 ^

8 = «*

2 9 ( )

w h i c h+ can also be o b t a i n e d directly from (28) by multiplying b o t h sides b y Nn(rf u, £1), integrating over r, uy a n d £1, and making use of t h e adjoint equation. T h e fundamental co-mode is t h e o n e t h a t corresponds to t h e alge­ braically largest eigenvalue co0 , a n d will b e d e n o t e d by iV w(r, uy £1). I t is t h e asymptotic distribution in a stationary reactor following an initial p e r t u r b a t i o n (cf 2.3, E q . 25). T h e asymptotic t i m e d e p e n d e n c e is described by exp[oj 0*]. T h e inverse of oo is called t h e " a s y m p t o t i c " (or stable) ^reactor period. T h e fundamental co-mode c o r r e s p o n d i n g to t h e instantaneous con­ figuration at t i m e t of t h e actual reactor u n d e r consideration will be d e n o t e d by N^r, uy £ly t). I n contrast to t h e definition of t h e funda­ mental reactivity m o d e Nk (r, uy £1, t)y we do not, in t h e present case, adjust t h e n u m b e r of n e u t r o n s per fission to make t h e instantaneous configuration critical. T h e r e f o r e , i V ^ r , uy £ly i) exp[oj 0(£ + f)] would be t h e asymptotic behavior (r —* oo) of t h e n e u t r o n population, h a d t h e core properties of t h e actual reactor r e m a i n e d u n c h a n g e d for r ^ 0. Clearly, this asymptotic behavior w o u l d b e different at different t. T h e t h i r d m e t h o d of a p p r o x i m a t i o n t h e n consists in replacing t h e shape function ^>(r, uy £ly t) in t h e definition of , /, and p(t) by iV W(r,o uy £ly t). T h e reactivity is obtained from [21] as

3 p(t)=*> \l+t

Q

where

i +,

1

( ° )

+ ft(0

=
Kt) = KN0+

| M | iVMo >

| M | NWo y

(31) (32)

71

2.4. Reduction of the Kinetic Equations

E q u a t i o n (30) is referred to conventionally as t h e i n h o u r equation, which relates t h e reactivity to t h e inverse reactor period OJ0. T h e t i m e behavior of t h e n e u t r o n p o p u l a t i o n is a p p r o x i m a t e d by P(t) Nw (r, uy SI, t) in t h e t h i r d m e t h o d , w h e r e P(f) is to b e d e t e r m i n e d from t h e point kinetic equations with t h e t i m e - d e p e n d e n t p a r a m e t e r s / and defined above. T h e reactivity defined b y (30), using t h e persisting n e u t r o n distribu­ tion A ^ o( r , Uy SI, t) is referred to as t h e d y n a m i c reactivity [21]. DISCUSSION

+

By operating on (28) by
(f). — [' £&).Trbrl +

<*>

where (,,//)„ = (&/*)» -

< N 0+ \ H + Y

MT | Nn}KN0+

/ < i V I Nn}

| Nn}

(34a) (34b)

I t s solution yields t h e t i m e constants cjn associated with all t h e higher m o d e s Nn as well as with t h e fundamental . It was pointed out by H e n r y [17] t h a t t h e six other roots cx)n ^ OO NJ Jof t h e i n h o u r e q u a t i o n for particular values of (pjl)n a n d (pjl)n are not in general eigenfunctions of (28). T h e reason is t h a t t h e eigenfunctions Nn are different functions of r , Uy and SI for different eigenvalues w n . T h e r e f o r e , c o n C O i N, G which c o r r e s p o n d to t h e same eigenfunction, cannot b e eigenvalues. However, some of t h e eigenfunctions fall into clusters of seven such t h a t t h e eigenfunctions Nnj , / = 0, 1,..., 6, in t h e nth cluster are approximately t h e same function of r , u, and SI even t h o u g h t h e corre­ s p o n d i n g eigenfunctions a>nj are different. Since t h e p a r a m e t e r s (p/l)nj a n d (Pill)nj will be approximately t h e same for a given cluster, t h e seven roots of t h e i n h o u r equation c o m p u t e d with any one of these Nnj will yield approximately t h e seven eigenvalues
2. Point Kinetic Equations

72

illustrates these properties of t h e co-modes in t h e one-speed diffusion approximation with one g r o u p of delayed n e u t r o n s . I t was also d e m o n s t r a t e d by H e n r y [17], considering t h e case of t h e t w o - g r o u p P - l a p p r o x i m a t i o n with isotropic scattering applied to a b a r e slab, t h a t t h e eigenfunctions r e p r e s e n t i n g higher angular a n d energy m o d e s cannot be identified with a cluster (Problem 4). T h e n e u t r o n density c o m p o n e n t of these eigenvectors are strongly d e p e n d e n t on t h e eigenvalues. I t is c o n c l u d e d t h a t [17] t h e i n h o u r equation is useful only if Nn for J = 0, 1,..., 6 are all approximately t h e same functions of r , u, and SI. F o r t h e fundamental, this is usually t h e case [14], a n d h e n c e t h e r e m a i n i n g six roots of (30) can b e identified as t h e delayed eigenvalues. ( F o r further discussion of t h e co-modes see H e n r y [17] and Gozani [19].) W e conclude this section with t h e following remark. T h e reactivity [strictly speaking, p(t)/l] b e c o m e s a linear functional of t h e p e r t u r b a t i o n s w h e n it is c o m p u t e d in t h e c o n s t a n t - s h a p e approximation. Only in this case are t h e various reactivity changes resulting from p e r t u r b a t i o n s in different nuclear properties a n d locations additive. I n all other a p p r o x i ­ mations, t h e s h a p e function itself varies w i t h t h e p e r t u r b a t i o n a n d t h u s t h e reactivity fails to be additive.

2.5. A l t e r n a t i v e D e r i v a t i o n s of t h e P o i n t K i n e t i c Equations T h e r e d u c t i o n of t h e kinetic equation into point kinetics as described in t h e previous section contains two crucial steps: (a) multiplication of t h e + kinetic equation (2.2, E q s . 5) by t h e adjoint angular density i V 0 ( r , u, SI) a p p r o p r i a t e to a critical source-free reference reactor, a n d integration over r , u, a n d SI; and (b) separation of t h e angular density as a p r o d u c t of a t i m e a n d a s h a p e function, n(r, w, SI, t) = P(t) (r> u, SI, t) such that satisfies t h e following normalization condition: (d/dtKNo+lfr

=0

(1)

T h e first step enables one to express t h e reactivity (cf 2.4, E q . 10) in t e r m s of SJf, which characterizes t h e deviations of t h e nuclear properties of t h e reactor from those of t h e reference reactor. T h e normalization condition (1) leads to t h e interpretation of P(t) as t h e t i m e variation of t h e coefficient of t h e f u n d a m e n t a l m o d e in t h e expansion of w(r, w, SI, t) into t h e set {Nn(r, u, SI)}. A l t h o u g h P(t) satisfies t h e point kinetic equations, it is not directly interpretable as t h e total instantaneous power in t h e reactor or t h e o u t p u t of a detector characterized by a cross

2.5. Alternative Derivations

73

section 2? D(r, u), which are m o r e significant from t h e safety and experi­ mental points of view. T h e total p o w e r is given by P(t) =
(2)

w h e r e 2? f(r, uy t) is t h e instantaneous fission cross section. T h e detector o u t p u t is obtained as ID (t) = (vZD | ny (3) W e can investigate these cases in general b y i n t r o d u c i n g a " w e i g h t " function w(r, u, t) and choosing t h e t i m e function P(t) proportional to (tv | ri). Since n = P, we m u s t impose t h e following normalization condition on > = 0 (4) so t h a t P(i) ~ (w | /z>, i.e.,

P(0 = <«;|»>/<«; | ^>

(5)

Following t h e steps described in t h e previous section, we can show t h a t P(t) satisfies t h e point kinetic equations (cf 2.4., E q s . 9) with t h e following identification of t h e p a r a m e t e r s w h e n w is constant: [Pli)^(w\H

+ YJMi\yi(w\y

(6)

i

(pill)^^\Mi\^l
(7)

T h e a p p r o x i m a t i o n i n t r o d u c e d in t h e previous section to replace is replaced by t h e fundamental A- or co-modes (in fact, it is not satisfied even in t h e case of t h e p e r t u r b a ­ tion a p p r o x i m a t i o n if t h e weighting function w is allowed to b e a function of time). T h i s difficulty led Gyftopoulos [5] to derive point kinetic equations t h a t do not require (4). If w e relax (4), let wx{t) = (d/dt) \og(w | }

(8)

a n d assume t h a t w is constant in t i m e , we obtain t h e modified kinetic equations d u e to Gyftopoulos: dP/dt = [(p - fall] P + £ A A i=l

+ S - Pwx{t)

dCt/dt = (ft/0 P(t) - XiCi - tojff) Ct

(9a) (9b)

74

2. Point Kinetic Equations

T h i s set contains t h e additional t e r m w^t), w h i c h was assumed to b e zero previously. I t is interesting to note t h a t (9) can b e r e d u c e d to t h e s t a n d a r d form [5, 6] if w e let

w

Pi= \

i(t)l

(10a)

+ ^i(0

(10b)

p = K

T h u s , t h e conventional form of t h e point kinetic equations is obtained, provided different p r e c u r s o r decay constants are u s e d in t h e power a n d p r e c u r s o r equations. T h e additional t e r m wx(t) in (9) vanishes exactly if t h e shape function is i n d e p e n d e n t of t i m e . T h e effect of t h e shape changes has b e e n investigated b y Gyftopoulos [5]. H e c o n c l u d e d t h a t it can be ignored only if t h e shape function changes by a small a m o u n t over a long period of t i m e . If t h e transients u n d e r consideration involve fast s h a p e changes, one has either to change t h e definition of reactivity as indicated in (10), or attack t h e p r o b l e m as a s p a c e - e n e r g y - t i m e - d e p e n d e n t p r o b l e m in t h e first place. F u r t h e r refinements in t h e derivation of t h e point kinetic equations have b e e n p r e s e n t e d by Becker [6] using variational principles. I n his derivations, he allows not only t h e flux shape, b u t also t h e weighting function to vary w i t h t i m e . T h e additional t e r m s disappear if appropriate normalization conditions are i m p o s e d on t h e shape function. If these conditions are found to be inconvenient to work with, t h e n m o r e general forms of t h e kinetic e q u a t i o n s such as those d u e to Gyftopoulos discussed above can be used.

2.6. P o i n t R e a c t o r K i n e t i c Equations w i t h Feedback I n this section, we shall derive t h e point reactor kinetic equations with feedback starting from t h e general description p r e s e n t e d above. T h e motivation a n d t h e m e t h o d of a p p r o a c h are t h e same as those presented in Section 2.4 in obtaining t h e point kinetic equations in t h e absence of feedback. W e begin o u r analysis b y observing t h a t t h e cross sections are n o w functionals of t h e angular density /z(r, u, £1, t), i.e., with j = a, f, or s, Zfa

w, t) = Zfa

u, t; [»])

w h e r e t h e bracket denotes t h e functional d e p e n d e n c e on n. Consequently,

2.6. Kinetic Equations with Feedback

75

t h e operators H a n d Mi a p p e a r i n g in (2.2, E q . 5) are also f u n c t i o n a l of n. H e n c e , o u r starting equations in operator form are Bnldt = H[n] if + f

A,/,Q + S,

(la)

i=l

%fiCi)ldt

= Mt[n] n - XJtCt,

i = 1, 2,..., 6

(lb)

W e n o w consider a stationary reference reactor s u p p o r t i n g a n e u t r o n distribution characterized by N0(rf u, Si). Since a reactor is never truly stationary w h e n t h e b u r n u p a n d b u i l d u p of t h e various nuclear species are included (cf 1.3, E q s . 6-10), we m u s t either a s s u m e t h a t t h e reference reactor is operated at zero power level, a n d hence free from all t h e feed­ back effects, or ignore t h e l o n g - t e r m changes in t h e nuclear species d u e to b u r n u p a n d b u i l d u p by irradiation. I n t h e first case, t h e reference reactor is critical in t h e absence of feedback effects, a n d represents a cold, clean reactor free from fission p r o d u c t s . I n t h e second case, t h e reference reactor is critical in t h e presence of all t h e feedback effects except for those arising from t h e depletion of t h e fuel a n d c o n t i n u o u s b1 u3i l d5u p of stable isotopes; t h e effects of t h e b u r n a b l e poisons, such as X e , are still included. I t is m o r e realistic to visualize t h e reference reactor as in t h e second case, because t h e n t h e reference distribution iV 0(r, uy Si) can b e chosen as t h e steady-state distribution in t h e actual reactor at t h e operating power level before t h e p e r t u r b a t i o n s are i n t r o ­ d u c e d . Since this distribution includes t h e initial feedback effects, a p e r t u r b a t i o n analysis based on it is better justified t h a n choosing iV 0(r, Uy SI) as t h e steady-state distribution in a reactor critical in t h e absence of feedback. T h e steady-state distribution N0(ry u, SI) can be obtained in principle b y solving t h e set of coupled nonlinear integrodifferential e q u a t i o n s derived in t h e previous section. I n operator form, we can formulate this p r o b l e m as No = 0 (2) where we recall (cf 2.2, E q . 6b)

je0[No] = H[N0] +

t

Mt[N0]

(3a)

or, m o r e explicitly, JQN0]

= - v • V - v(u) E(Ty Uy [N0]) + j du' j dQ' v(u') X U ( r , v! —

Sly [N0]) + X [f*(u)l4n] v\u') 2¥(r, u'; [N0])\ (3b)

2. Point Kinetic Equations

76

W e note that t h e operator ^0[N0] has t h e same structure as the steadystate Boltzmann operator defined in Section 2.3. T h e presence of feedback modifies only t h e energy and space dependences of t h e cross + expression of ^ [ i V ] b u t does not affect its form. H e n c e , sections in the 0 + obtained by the same p r o c e d u r e as described its adjoint, ^ [ i V 0] can be i n Section 2.3. U s i n g J^0[N0], we define t h e adjoint angular density as t h e solution of ^oWoi

= 0

(4)

+ with adjoint b o u n d a r y conditions. I n t h e following analysis, iV 0(r, u, Si) and i V 0 ( r , u, Si) will be assumed to be k n o w n functions of r, u, and SI. I n order to obtain t h e appropriate point kinetic equations with feed­ back, we introduce t h e shape a n d +time functions as n(r, u, SI, t) = P(t)(f)(r, u, SI, t), multiply (1) by i V 0 , +integrate over r, u, and SI, and use t h e normalization condition d(N0 \ $y\dt = 0. W e obtain t h e following kinetic p a r a m e t e r s : yKN0+\}

(5a)


(5b)

/<^o l<^> /<#„+I +

S

+

(5c) (5d)

0

where

(6) W e note t h e difference between (5) a n d t h e corresponding equations (10), (11), (13), and (14) of Section 2.4 in t h e absence of feedback. I n (5), the p e r t u r b a t i o n operator 8Jt?[ri\ d e p e n d s on P(t) as well as on t h e shape function
*rw

s I K (0 4

g +

-GFC]

+ MM § f

(7)

2.6. Kinetic Equations with Feedback

where

77

x 8N° \t)

= A?j(r, T0 , t; [N0]) - NJr)

(8)

3AT/[«] = A ^ r , TQ; [n]) - iV, 0(r) ST[n] = T(r; [«])

-

(9)

r o(r)

(10)

H e r e , Ni0 (r) a n d T 0( r ) are t h e e q u i l i b r i u m concentration of t h e ith nucleus a n d t h e local t e m p e r a t u r e at r , respectively, a n d is defined in (3b). ( N o t e t h a t t h e partial derivatives r e m o v e t h e s t r e a m i n g t e r m in this definition.) S u b s t i t u t i n g (7) into (5a), we break u p t h e reactivity into t h r e e p a r t s : ?\i = where

(WO + (WO + (W)

(ii)

x

SPEX/l -

(1/
Spc/l = ( l / < i V I <^»<^o I Z SNXBJTJdNa)

W

(1/W I <£»<^V I ST(3Jt0ldT0)

-

I }

(12)

i

| <£>

| <£>

(13) (14)

T h e t e r m s in (11) represent, respectively, t h e external reactivity changes, reactivity feedback d u e to changes in atomic concentrations, a n d reactivity feedback d u e to t e m p e r a t u r e variations. T h e m e a n i n g of t h e scalar p r o d u c t in t h e presence of t e m p e r a t u r e feedback requires clarification, because t e m p e r a t u r e changes affect t h e size of t h e reactor. A l t h o u g h one can use p e r t u r b a t i o n t h e o r y in­ volving b o u n d a r y variations as discussed b y M o r s e a n d F e s h b a c h [22] to take into account t h e size changes, we prefer to ignore t h e m in this book because their effect on reactivity in large power reactors is negli­ gible. I n small, b a r e cores such as in t h e S N A P reactor, w h e r e t h e size changes constitute an appreciable c o n t r i b u t i o n to feedback reactivity, one can calculate t h e reactivity m o r e directly ( P r o b l e m 5) t h a n by using t h e p e r t u r b a t i o n a p p r o x i m a t i o n , as a result of t h e simplicity of t h e system. I n order to c o m p l e t e t h e derivation of t h e p o i n t kinetic equations with feedback, we m u s t also consider (ft//) defined in (5b). W e observe t h a t it is a functional of n(r, u, SI, i) t h r o u g h MM

= X (&W,(r, T, t; [«]) 3

X | du' j dQ' {**(«') f ( « ' ) [ / i ( " ) / 4 ^ ]
(15)

78

2. Point Kinetic Equations

It is consistent with the p e r t u r b a t i o n approximation to replace (15) by its equilibrium value = £ (&W,(r, T0;

[N0])

0

X j du' j dQ' {[/,(W)/4TT] v\u')

v(u')

a \u'

t

r )})

0

(16)

T h e shape function (r, SI, t) appearing in (11)—(14) m a y be chosen in one of t h e ways discussed in Section 2.4. H e r e , we use t h e first-order p e r t u r b a t i o n approximation, which implies (cf 2.4, E q . 21) n(r, u, SI, t) * [P(t)IP0] N0(r, uy SI)

(17)

w h e r e N0(ry u> SI) is t h e angular n e u t r o n density at equilibrium. T o conclude this section, we s u m m a r i z e t h e above results in a form which will be used later. T h e point kinetic equations in t h e presence of feedback can be written within t h e framework of first-order p e r t u r b a t i o n theory as [P] - ft//] P(t) + f P(t) = [{SpextW + SPS

i=l

Ci(t)

= (ft//) P(t) - A tC,(0,

A , Q 0 + S(t)

i = 1, 2,..., 6

(18a) (18b)

with t h e following identification of t h e p a r a m e t e r s : Pill

+

=

(1/ / < A T 0+ | N0) m

(19a)

| A T 0>

(19b)

(WIWo l/A>

(19c)

i = /
(19d)

=

+

(1/<2V0+ | N0}KN0+

| £ 8N° (dJ?>0ldNj0 ) 3

| N0>

(20)

sPt [pyi

= (S P[P]/Z) + (3 P[P]/Z) c r

8Pe [F]ll

= (1/ 0 0 0 o i0 0

(22)

8pT[P]/l

= (i/ 0 0 0 0 0

(23)

c

(21)

A comparison of (18) with (2.4, E q s . 9) representing t h e point kinetics in t h e absence of feedback reveals t h a t all t h e feedback effects in (18) are accounted for by t h e feedback functional Sp f[P]. T h e c o m p u t a t i o n of this functional for various reactor types will be discussed in C h a p t e r 5.

2.7. Kinetic Parameters in the Diffusion Approximation

79

I t is to b e n o t e d t h a t t h e reactivity p can b e expressed as t h e s u p e r ­ position of t h e external a n d feedback reactivities only in t h e first-order p e r t u r b a t i o n theory. I n general, an external change in t h e atomic concentration will affect not only t h e external reactivity, b u t also t h e feedback reactivity as a result of t h e changes in t h e shape function.

2.7. C a l c u l a t i o n of K i n e t i c P a r a m e t e r s in t h e Diffusion A p p r o x i m a t i o n T h e definitions of t h e kinetic p a r a m e t e r s given by E q s . (19)—(23) of t h e p r e c e d i+ n g section involve t h e angular density iV 0(r, u, Si) a n d its adjoint i V 0 ( r , u, Si). I n t h e diffusion a p p r+orx i m a t i o n , one calculates + ( > u) [or t h e c o r r e s p o n d i n g only t h e scalar flux 0(r, u) a n d its adjoint 0 scalar n e u t r o n densities iV 0(r, u) a n d i V 0 ( r , u)] as discussed in Sec­ tion 1.4. I t is therefore desirable to express t h e definition of t h e kinetic p a r a m e t e r s in t e r m s of t h e scalar fluxes only within t h e framework of + t h e diffusion a p p r o x i m a t i o n . F o r this p u r p o s e , we e x p a n d iV 0(r, u, Si) a n d N0(ry u, SI) in spherical h a r m o n i c s , a n d treat t h e anisotropic t e r m s as being small. T h e first two t e r m s in these expansions are ( P r o b l e m 7) (la)

vN0(ry uySI) = (l/47r)[^ 0(r, u) + 3J 0(r, u) • Si] -\vN0+{r,

u, SI) = (l/4*r)0 o+(r, u) + 3J 0+(r, u) • Si] + +

-

(lb)

w h e r e J 0( r , u) and J 0 ( r > u) are t h e c u r r e n t vector a n d its adjoint. S u b s t i t u t i n g these into t h e definition of (&//), S(t) a n d C^t) a n d ignoring t h e t e r m s containing t h e p r o d u c t s of c u r r e n t s , we obtain (2a) I N0>KN0+ | MJNo] | AT > (P /(2b)

(l/W S(t)

0

0

0

Clt) I where

0

(2c)

( l / < i V 0+ | i V o » < i+V o l / A > W

I ^ o > / < ^ o I M[NJ I AT > 0

(2d)

(3a) M[N0]

dQ M[N0]

=

£ / ' ( « )

f

du' [v\u') v{u') Zi0(r, «')]

JJ J [N +(r, Ju) N„(r, «)] f du f dh 0 0 R ft R

(3b) (4)

80

2. Point Kinetic Equations

T h e last equation defines t h e scalar p r o d u c t for functions of u a n d r. T h i s definition is implied in (2). T h e calculation of t h e reactivity in t h e diffusion a p p r o x i m a t i o n using (2.6, E q s . 2 0 - 2 3 ) is not as straightforward as above. If we write explicitly in these equations using (2.6, E q . 3b), we e n c o u n t e r t h e following integral:

f J°° du' j dQ j dQ' [Us(u ->u,Sl'

• SI) v(u') N0+(u, SI) NQ (u'y SI')]

- j dQ [v(u) S(u) N0+(u, SI) N0(u, SI)]

(5)

w h e r e we o m i t t e d t h e variable r in t h e a r g u m e n t s for t h e present discussion. S u b s t i t u t i n g (1) into (5) a n d keeping t h e t e r m s containing t h e p r o d u c t s of c u r r e n t s , we obtain (Problem 8)

f (1/4TT)[J^ du' {Zs(u — u) N0+(u) 0 o(ii')} ~ Z(u) N0+(u) cf>0(u) -[3lv(u)]Ztr (u)nu)-J(u)]

(6)

w h e r e Stv is t h e t r a n s p o r t cross section, defined by 2tT — 2 — pLUB , /Z being t h e average cosine of t h e angle of deflection in t h e laboratory system in a scattering event. T o c o m p r e s s t h e notation, we define T 0 EEE -27 0(r, u) +

du' [ ^ ( r , « ' - « ) + £ / > (u) v\u') Z^(r, «')]

(7)

a n d obtain t h e following expression for t h e reactivity p = 8 p e t x+ in t h e diffusion a p p r o x i m a t i o n :

8pt

+

P = (1/
| 0} -
(38Utr Jv)

| J >] 0

T h e scalar p r o d u c t s in this equation are defined by (2d). T h i s expression can be cast into a m o r e conventional form by using J0+ = D0 V0+ J 0 = -D0 Vcf>0 A> = l/3^tr 0

2

(9a) (9b) (9c)

a n d noting that 3S27 tr = — S Z ) 0/ D 0. T h e result is

+

P = (1/
| N0»[
[ 8r0 | 0O > - ]

(10)

(8)

2.7. Kinetic P a r a m e t e r s in the Diffusion A p p r o x i m a t i o n

81

T h e various c o n t r i b u t i o n s to reactivity discussed in t h e previous section can be obtained from (3) or (10) by considering t h e partial derivatives of RO a n d UTTO (or Z) 0). T h e energy integrals appearing in t h e definition of t h e kinetic para­ m e t e r s , E q s . (2) a n d (8), can be r e d u c e d to s u m m a t i o n s by a d o p t i n g t h e m u l t i g r o u p formalism i n t r o d u c e d in Section 1.4. W e shall illustrate t h e application of these formulas by considering simple reactor m o d e l s . Example 1. O n e - g r o u p diffusion m o d e l . I n t h e o n e - g r o u p model, c/)(r) satisfies V • D V ^ + (VZ -

0

0

U

EH )

diffusion

M*) = 0

(11)

in t h e absence of external sources. T h e cross sections are allowed to b e p o s i t i o n - d e p e n d e n t . T h e operators MJ[N0], M[N0], and RO defined in E q s . (3) and (7) r e d u c e to MIN0]

=

M[N0]

= WZT,

ftW 27to

(12a) (12b)

T 0 = - 2 7 ao + VST<>

(12c)

F r o m (2), we find / = [J^ DH*(R)]LV DH [VSTJHFTR)]

(13)

w h e n t h e fission cross section is i n d e ­ w h i c h reduces to / = (\jvvSt^ p e n d e n t of position. I n t h e one-speed model, ft = ft . T h e reactivity follows from (8) as

2 P=

jj*^ DH [3S27tr o| Jo | - S ( £ ao - V2T0)^]\/V

JR DH {ShM)

(14)

or from (10) as p=

" \S

d3r [ S Z > K

» ^ + ^ - "Wo ]j/v j DH (E ^) 1V

|2

2

R

T

(15)

[See P r o b l e m s 6, 9, a n d 10 for some simple applications of (14) a n d (15).] Example 2. Modified o n e - g r o u p diffusion m o d e l . A m o r e realistic model, w h i c h takes into account t h e slowing d o w n process, is t h e agediffusion m o d e l (cf Section 1.5C). T h e t h e r m a l flux satisfies t h e following

82

2. Point Kinetic Equations

equations [23] in this model if we assume instantaneous slowing d o w n of t h e fission n e u t r o n s to t h e r m a l energies: (l/v) 8(r, t)/dt = V • Z)(r, t) V ^ r , t) - ^ ( r , t)(r, t) + p(l - j8) ^ [ e x p ( - 5 g% ) ] Zt(r, t)(r, t)

+ £ Atexpt-W]

Qr, 0

(16)

1=1

aC,(r,

- vftii(r, 0 flr, 0 - A,C<(r, 0

(17)

w h e r e B g is the geometric buckling, T Pa n d ri are t h e ages of t h e p r o m p t and delayed n e u t r o n s , and p is t h e resonance escape probability, which is t h e same for b o t h the delayed and p r o m p t n e u t r o n s because t h e resonances occur at lower energies t h a n t h e delayed n e u t r o n energy s p e c t r u m . It is assumed that there is only one species of fissionable material. T h e first and second t e r m s in t h e r i g h t - h a n d side of (16) are t h e leakage and absorption rates of t h e thermal n e u t r o n s . T h e last two t e r m s account for t h e p r o d u c t i o n rate of t h e t h e r m a l n e u t r o n s due to t h e fission process and to t h e decay of delayed n e u t r o n precursors. T h e
2 M0 = v(l - ft) Utp exp(-5g T P)z; 2

(18a)

M, = ftv£fp e x p ( - £ g r >

(18b)

L = - ( 2 7 a - V • DV)v M = v27,p[(l -

2

2

£ ) [ e x p ( - £ T ) ] + I ft e x p ( - £

G P

i=l

GT , ) >

(18c) (18d)

if we absorb t h e f a c t o r e x p ( — B ^ r ^ ) in Q(r, t). W e assume that the u n p e r t u r b e d reactor is a bare, h o m o g e n e o u s and Z1Q are i n d e p e n d e n t of position, and reactor, i.e., D0, that the p e r t u r b a t i o n does not affect the slowing-down properties of t h e m e d i u m , i.e., r p , ri , and p are u n c h a n g e d . W e also assume that t h e p e r t u r b a t i o n is i n t r o d u c e d uniformly in t h e reactor, i.e., 8D, 8Za ,

83

2.7. Kinetic P a r a m e t e r s in the Diffusion A p p r o x i m a t i o n

and 8Zf are i n d e p e n d e n t of position. T h e s e approximations are not essential for t h e calculation of ft , /, a n d p, b u t they simplify t h e analysis, and allow us to express these quantities in a m o r e familiar form. T h e effective delayed n e u t r o n fraction ft follows from (2a) and (2d) as

2

2

PSi = {exp[-fig (rf - r p)]}/j(l - [ P) + £ e x p t - S g -^T p) ] | '

(19)

1=1

which reduces to

ft/ft = [(1 - j8) e x p { - ^ ( r p -

(20)

)T +d

if we assume t h a t t h e ages of t h e delayed n e u t r o n g r o u p s are t h e same, namely, ri ^ r d for all i. T h e m e a n generation time / follows from (2d) by substituting M from (18d):

(1//) =

vupJSto

1 { (1

t

- i8)[ex P(-5 g%)] +

2 Pi exp(-5 gr,)|

i=l

(21)

'

where ZfQ is t h e u n p e r t u r b e d fission cross section. W e can cast (21) into a m o r e conventional form by using t h e fact t h a t t h e u n p e r t u r b e d reactor is critical, namely

2

Z)0V^0- 27^0 = -vpZh

f(l -

ft[exp(-rp£g)]

+ £ ft e x p ( - r , ^ ) l cf>0

^

(22)

T h u s , (21) becomes Z = /0/(l +L*Bg*)

12

(23)

w h e r e l0 = XjZ^v a n d L = {DJZ^) ! . T h e reactivity follows from (10) as in (15). T h e presence of delayed n e u t r o n s and the inclusion of t h e fast nonleakage probabilities are accounted for in this expression t h r o u g h t h e fission cross section in t h e d e n o m i n a t o r a n d t h e criticality condition. = T o emphasize this point, we express t h e effective multiplication factor ^eff 1/(1 ~~ p)- W e can verify that

2

*eff = <<£o I M | <£0>/<<£0 | Za - Z)V | 0}v = K where

1(1

-

= vpZt\Z^

ft[exP(-£g%)]

+

Z Pi e x p ( - 5 gS - ) j / ( l + L*Bg*)

, i.e., t h e infinite m e d i u m multiplication factor.

(24)

2. Point Kinetic Equations

84 P R O B L E M S

1.

(a) S h o w that V • Z)(r) V is self-adjoint for functions ^>(r) t h a t 2 2 vanish o n t h e outer surface of t h2e reactor. 2 (b) S h o w t h a t t h e adjoint of ( D V + a • V+Bn) is ( Z ) V - a • V+Bn) for t h e same class of functions, (cf C h a p t e r 1, P r o b l e m 14.) +

2.

S h o w that, if O = [O^] is a matrix operator, its adjoint is 0 = [ O ^ ] . H i n t : Consider O | > = COL[Oin cf>n] a n d 0 + 1 +> = c o l [ O J i0 n+ ] with s u m m a t i o n convention o n N. Verify ( s u m m a t i o n o n / a n d N) | O0> =

| O n>

=



=

3.

in +

+

< 0 ^ n

| ^ >

=

F i n d t h e expansion of t h e c o l u m n vector | / > whose elements are all zero except for i^(r, v ) = S(r — r ' ) S(v — v ' ) a n d prove 00

8(r - r ' ) 8(v - v') 8H= £ ni (r', v')<)>nj {r, v) n=0

4.

w h e r e nj are t h e c o m p o n e n t s of t h e eigenvector | . U s i n g t h e o n e - g r o u p diffusion equation w i t h o n e g r o u p of delayed n e u t r o n s , s h o w t h a t t h e o>-modes for an infinite, bare, slab reactor of thickness d are

1^) = ^

1

.(v27 + j8)/(A +

f

] ^ n s /i n [ ( W+ l)(7r/a)x]

co ).

W/

w h e r e coO T are t h e t w o roots of A + (&eff),

Hint: Use

1 +

w,

+ 1)T

2 2

D(d /dx )

+ (1 - j8) v27 - 2 7 f v27 8

fi

B A

n = 0, 1, 2,...

Problems

85

a n d solve 3f

I nj> =

<*>nj I nj}

with t h e b o u n d a r y condition Nnj (0) = Nnj (d) = 0. Observe t h a t Nnj (x) = ^4nj sin[(w + l)(7rld)x] has t h e s a m e spatial distribution for / = 1, 2 a n d a fixed n. Verify t h e orthogonality

5.

(a) F i n d t h e reactivity change in an initially critical, h o m o g e n e o u s , spherical, b a r e reactor u s i n g t h e o n e - g r o u p diffusion a p p r o x i m a t i o n w h e n t h e reactor t e m p e r a t u r e is raised uniformly b y AT. A s s u m e t h a t t h e t e m p e r a t u r e change affects only t h e density of t h e m e d i u m as N = iV 0(l - ocT). (b) Calculate t h e reactivity change in t h e same reactor using (2.6, E q . 23) a n d c o m p a r e t h e results.

6.

A t h i n , a b s o r b i n g r o d is inserted into a bare, uniform, cubic reactor of edge Hy along t h e z axis (see F i g u r e P 6 ) . U s i n g first-order

1

1 AX,S

FIGURE P 6 .

p e r t u r b a t i o n t h e o r y in t h e absence of delayed n e u t r o n s , s h o w t h a t t h e reactivity change d u e to t h e rod is

P{z)

= p{H)[(zjH)

-

(1/2TT) sin(27r*/#)]

2. Point Kinetic Equations

86 7.

T h e spherical h a r m o n i c expansion of t h e angular flux [18] is (£L)=0Y00 (£L) +

£

m=0,±l

YLM (N)IM +

-

w h e r e t h e expansion coefficients (&) dQ. L e t Qx , i 3 y , £2gbe t h e Cartesian c o m p o n e n t s of t h e u n i t vector £1. (a) S h o w that t h e " s p h e r i c a l " c o m p o n e n t s of £1, defined as Q0 = Qz,

fi±

= T ( l / V 2 ) ( Q X ± i » y)

are given by = (3/4TT)V2 F lw,

, ± l

m

=

0

(b) U s i n g t h e spherical c o m p o n e n t s of two vectors A and B, show that their scalar p r o d u c t is

y A B = £AnBm*9

in = 0 , ± 1

M

(c) S h o w t h a t

y E i » ^ m = (3/4^)j-n M

noting 8.

that

the

spherical

components

of

the

current

J

are

H i n t : First, verify j dQ Zs(u —> u, £1' • £i) £1 = £L'2JB1 (u' -+ u) where

27(w' — > u) si

= ITT \ J -l

dfx [fjL£s(u'

-> w, /x)]

and t h e n use

du' U(M') ^ i(w'

J

S

«)] ^ ]{u) U (u) /x s

0

U (u) jl = J s

9.

J dX? 27( — > w', jit) p, w

s

(a) Calculate t h e reactivity in an initially critical, bare, h o m o g e n e o u s , spherical reactor of radius R using t h e o n e - g r o u p diffusion model d u e to a concentric spherical void of radius r.

References

87

(b) Calculate t h e reactivity using t h e first-order p e r t u r b a t i o n theory assuming r <^R, a n d c o m p a r e it to t h e result of (a). H i n t : U s e (2.7, E q . 14) in (b). 10. F i n d t h e reactivity change in a perfectly reflected h o m o g e n e o u s , slab reactor of thickness a w h e n t h e absorption cross section is changed uniformly in t h e slab b y a n a m o u n t S27a . R E F E R E N C E S 1. J. Hurwitz, Jr., Nucleonics 5, 62 (1949). 2. L. N . Ussachoff, Proc. Int. Conf. Peaceful Uses At. Energy, Geneva, 1955, P/656. Columbia Univ. Press, New York, 1955. 3. A. F. Henry, Computation of parameters appearing in the reactor kinetics equation. WAPD-142. December 1955, Westinghouse Atomic Power Division, Pittsburgh, Pennsylvania. 4. A. F. Henry, Application of reactor kinetics to the analysis of experiments. Nucl. Sci. Eng. 3, 52 (1958). 5. E. P. Gyftopoulos, in " T h e Technology of Nuclear Reactor Safety" (T. J. Thompson and J. G. Beckerly, eds.), Vol. 1, pp. 175-204. M . I . T . Press, Cambridge, Massachu­ setts, 1964. 6. M . Becker, A generalized formulation of point nuclear reactor kinetic equations. Nucl. Sci. Eng. 31, 458 (1968). 7. J. Lewins, / . Nucl. Energy Part A 12, 108 (1960). 8. R. Courant and D . Hilbert, "Methods of Mathematical Physics," Vol. I. Wiley (Interscience), New York, 1953. 9. B. Friedman, "Principles and Techniques of Applied Mathematics." Wiley, New York, 1956. 10. E. E. Gross and J. H. Marable, Nucl. Sci. Eng. 7, 281 (1960). 11. A. Weinberg, Amer. J. Phys. 20, 401 (1952). 12. R. Ehrlich AND H. Hurwitz, Nucleonics 12, 23 (1954). 13. K. M . Case and P. F. Zweifel, "Linear Transport Theory." Addison-Wesley, Reading, Massachusetts, 1967. 14. E. R. Cohen, Proc. U. N. Int. Conf. Peaceful Uses At. Energy, 2nd, Geneva, 1958, A/conf., P/629. United Nations, New York, 1958. 15. A. F. Henry, Trans. Amer. Nucl. Soc. 6, 212 (1963). 16. A. F. Henry, Trans. Amer. Nucl. Soc. 9, 235 (1965). 17. A. F. Henry, Nucl. Sci. Eng. 20, 338 (1964). 18. R. V. Meghreblian and D . K. Holmes, "Reactor Analysis." McGraw-Hill, New York, 1960. 19. T . Gozani, T h e concept of reactivity and its application to kinetic measurements. Nukleonik 5, 55 (1963). 20. N . Corngold, Trans. Amer. Nucl. Soc. 7, 211 (1964). 21. E. E. Gross and J. H . Marable, Nucl. Sci. Eng. 7, 281 (1960). 22. P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," p. 1038. McGrawHill, New York, 1953. 23. S. Glasstone and M . C. Edlund, "Nuclear Reactor Theory." Van Nostrand, Princeton, New Jersey, 1952.