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0 1 , •'o 2 a ( £ ) * ( £ ) = { 2 f r EkT—^ +
(4-114)
or canceling out the integrating factor from both sides
2 a * ).
(4-115)
This solution again has a very plausible interpretation when it is recognized that exp( —2 a i?) is just the attenuation that would occur between the source point and the observation point r. Notice that this immediately reduces to our vacuum result for the case in which 2 a = 0 . We can also obtain an exact solution for the situation in which 2 a depends on position. We need only use a slightly more complicated attenuation factor exp( - 2 a t f )
exp[ - a (r, r ^- R 6 ) ] ,
w h e r e a ( r , r ' ) is k n o w n a s t h e optical thickness
(4-116)
o r optical depth of t h e m e d i a a n d is
defined by a ^ O ^ j f ^ ^ S ^ r - ^ ) ,
R = r' —r.
(4-117)
Note that a is essentially a measure of the effective absorption between points r' and r. E X A M P L E : Consider once again our point source, only this time assume that it is imbedded at the origin of an infinitely large medium characterized by a uniform absorption cross section 2 a . Then repeating our earlier analysis using Eq. (4-115) yields
«')-
SQ exp( —2 r) / a » 47jr1
(4-118)
which is similar to our vacuum result, with the exception of an additional attenuation factor e x p ( - 2 a r ) due to the absorption. This very important result can be easily generalized to the situation in which the source is located at an arbitrary point r' by merely shifting the coordinate system origin to find «H r )=
S0exp(-Salr-r'l) r n — ^ — • 47r|r-r'
(4"119)
NEUTRON TRANSPORT
/
133
Field point
FIGURE 4-13.
Coordinates characterizing a disturbed source s(r, E, 12).
We can finally use this result to synthesize the neutron flux resulting from an arbitrary distribution of isotopic sources, S (r), in an infinite absorbing medium as
(4-120)
C. The One-Speed Diffusion Equation We now turn our attention toward the development of an approximate description of neutron transport more amenable to calculation than the neutron transport equation itself. To make life simple, we will first work within the one-speed approximation represented by Eq. (4-96). Let us first note the explicit forms taken by the neutron conservation equation and the corresponding equation for the current density J in the one-speed case: ^ + V - J + Et<J>(r,0 = 2 s <#>(r,0+5(r,/), V
f dO fiS2(p(r,(l,t) + 2 t J(r,£) = /x 0 2 s J(r,t) + Sj(r,t).
(4-121) (4-122)
Here we have noted explicitly the simplifications that occur in the inscattering term when the one-speed approximation is introduced. More specifically,
(4-123) and (4-124) o But (4-125)
134
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
where we have defined the average scattering angle cosine jCt0 as 2tt r
= -47TZ. r V f
+ l
d t Jf 4l7r
rfsro-Q's^s2-no-
(4-126)
As an aside, it should be noted that one can easily calculate /x0 for the case of elastic scattering from stationary nuclei when s-wave scattering is present. For then w e k n o w t h a t in t h e C M s y s t e m , ocm(0c) = oL(0L)d(lL, we find
= os/4tt.
H e n c e if w e use
oCM(0c)dtlc
sm0cd0ccos0Li:CM(0c) s
J
0
= \ f %in 0C cos 0L d0c. 2 Jn
(4-127)
But recall 1 + A cos 0 C
cos 0L = 2
'\Ja -\-2A
.
(4-128)
cos0c+l
If we substitute this into Eq. (4-127) and perform the integration, we find the very simple result Po=37-
(4-129)
Now that we have justified the forms of Eqs. (4-121) and (4-122) let us consider howA A A we might eliminate the annoying appearance of the third unknown, A We will accomplish this by assuming that the angular flux is only weakly dependent on angle. To be more specific, we will expand the angular flux in angle as =
+
+
+
(4-130)
and neglect all terms of higher than linear order in S2. Actually a slightly different notation for the unknown functions
+
+
(4-131) Notice that we have labeled the unknown expansion coefficients as the flux and current. That this notation is perfectly consistent can be seen by noting from Eq. (4-131) that jf
daq>(T,a,t)
= 4>(r9t)±
JT < / f l + ^ J ( r , r ) - J ^ f a = ^(r9t)9
(4-132)
NEUTRON TRANSPORT
/
135
and J
d(ia
=
+ j
y
dM
+ ±\j
rfnn^n+j^r,/)
( T , t )J[
x
(r
9
t ) f dQQSl
Jf
(4-133)
However one can easily demonstrate (see Problem 4-14) that the integral of the product of any two components of £2 gives 4 77
/
3 0
477
i,J —
x,y,z.
(4-134)
i¥=j
Hence f dti S2cp(r,S2,?)=J(r,?). 477
(4-135)
Of course Eqs. (4-132) and (4-135) are identical to our original definitions of the flux and current earlier in Eqs. (4-17) and (4-19). We will now use the approximate form of the angular flux in Eq. (4-131) to evaluate the second term in Eq. (4-122): v- f
da
aa
f AIT
4t7
da
1 ^ T"
(4-136) a
Next note that the integral of the product of any odd number of components of 12 vanishes by symmetry: f d An
i
=
0
if l9m9 or n is o d d .
(4-137)
If we use both Eqs. (4-134) and Eq. (4-137) we can evaluate V- f Att
da aticp = |V
(4-138)
3
Hence by assuming that the angular flux depends only weakly on angle—more specifically, that the angular flux is only linearly anisotropic—we have managed to express the third unknown appearing in Eq. (4-122) in terms of the neutron flux <J>(r, /). We have now achieved our goal of obtaining a closed set of two equations for two unknowns, <Mt>0 and J(r,f):
I ^
+ V • J + Za(r)«Kr, 0 = 5 (r, /),
(4-139)
We have noted here that 2 a (r) = 2 t ( r ) - 2 . ( r ) ,
(4-141)
136
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
a n d d e f i n e d t h e macroscopic
transport
cross
section
2 t r (r) = 2 t (r)-Mo2 s (r).
(4-142)
(We will comment on this definition in a moment.) These two equations are known in nuclear reactor analysis as the Px equations (in the one-speed approximation) since the approximation of linearly anisotropic angular dependence in Eq. (4-131) in one-dimensional plane geometry is equivalent to expanding the angular flux in Legendre polynomials in ju, = cos0 and retaining only the / = 0 and / = 1 terms q>(x9ii3t)^(x9t)^P0(ii) + J(x9t)\Px(ii)9
(4-143)
hence the name P t approximation. Notice that this could be easily generalized to o b t a i n t h e PN
approximation.
In principle we could now use the Pl equations to describe the distribution of neutrons in a nuclear reactor. However it is customary to introduce two more approximations in order to simplify these equations even further. First we will assume that the neutron source term ^(r,!^,*) is isotropic. This implies, of course, that the source term Sj(r,/) vanishes in the equation for the current density. As we mentioned earlier, this approximation is usually of reasonable validity in nuclear reactor studies. As our second approximation, we will assume that we can neglect the time derivative v~l 3 J / 9 / in comparison with the remaining terms in Eq. (4-140). This would imply, for example, that 1
9|J|
that is, that the rate of time variation of the current density is much slower than the collision frequency t>2 r Since t?2t is typically of order 105 s e c - 1 or larger, only an extremely rapid time variation of the current would invalidate this assumption. We will later find that such rapid changes are very rarely encountered in reactor dynamics. Hence we are justified in rewriting Eq. (4-140) as ±V<j>(r, t) + 2 t r (r)J(r, t) = 0.
(4-145)
We can solve Eq. (4-145) for the neutron current density in terms of the neutron flux (4-146) If w e d e f i n e t h e neutron
diffusion
coefficient
D by
D (r) = [32*00]" 1 = [ 3 ( 2 t -
~\
(4-147)
then we can rewrite Eq. (4-146) as
J(r, /) = — D (r)V<£>(r, /).
(4-148)
Hence we have found that in certain situations the neutron current density is proportional to the spatial gradient of the flux. This very important relation arises
NEUTRON TRANSPORT
/
137
quite frequently in other areas of physics where it is known as Fick's law. It is also o c c a s i o n a l l y r e f e r r e d t o a s t h e diffusion
approximation.
Before we consider the physical implications of this relationship, let us use it to simplify the P l equations. If we substitute this into Eq. (4-139) we find i
^-V-Z)(r)V
(4-149)
This very important equation is known as the one-speed neutron diffusion equation, and it will play an extremely significant role in our further studies of nuclear reactors. We will discuss its solution in considerable detail in Chapter 5, and we will use it as the basis of a very simple but very useful model of nuclear reactor behavior. Let us now return to consider the diffusion approximation [Eq. (4-148)] in more detail. Notice that it implies that a spatial variation in the neutron flux (or density) will give rise to a current of neutrons flowing from regions of high to low density. Physically this is understandable since the collision rate in high neutron density regions will be higher with the corresponding tendency for neutrons to scatter more frequently away toward lower densities. The rate at which such diffusion occurs
y FIGURE 4-14.
A schematic representation of Fick's law.
138
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
depends on the diffusion coefficient which, in turn, is inversely proportional to the transport cross section 2 t r . It is c o n v e n i e n t t o i n t r o d u c e t h e c o n c e p t of a transport \ x =(2tr) ~
1
= ( 2 t ~ /*02s) ~ 1 •
mean free
path (4-150)
The transport mfp can be regarded as a corrected mfp accounting for anisotropics in the scattering collision process. Since Jl0 is almost always positive—that is, biased in the direction of forward scattering—the transport m f p Atr will always be somewhat larger than the actual mfp, A = (2 t )~ 1 . This essentially accounts for the fact that neutrons experiencing forward scattering tend to be transported somewhat further in a sequence of collisions than those being isotropically (or backward) scattered. Hence since Z) = X t r /3, we see that diffusion is also enhanced in a material with pronounced forward scattering (e.g., hydrogen), although this of course also depends on the magnitude of the macroscopic cross sections. It is important to keep in mind that the diffusion approximation is actually a consequence of four different approximations: (a) the angular flux can be adequately represented by only a linearly anisotropic angular dependence [Eq. (4-131)], (b) the one-speed approximation, (c) isotropic sources, and (d) the neutron current density changes slowly on a time scale compared to the mean collision time [Eq. (4-144)]. Actually only the first of these approximations is really crucial. The remaining approximations can be relaxed provided we are willing to work with the Pl equations rather than the neutron diffusion equations (as one frequently is). It is natural to ask when the angijlar flux is sufficiently weakly dependent on angle so that the diffusion approximation is valid. More detailed studies of the transport equation itself indicate that the assumption of weak angular dependence is violated in the following cases: (a) near boundaries or where material properties change dramatically from point to point over distances comparable to a mean free path, (b) near localized sources, and (c) in strongly absorbing media. In fact strong angular dependence can be associated with neutron fluxes having a strong spatial variation. Usually if one is over several mean free paths from any sources or boundaries in a weakly absorbing medium, the flux is slowly varying in space, and diffusion theory is valid.
D. The Energy-Dependent Diffusion Equation Let us now try to repeat this analysis for the case in which the neutron energy dependence is retained. Again we will approximate (4-151) so that we can evaluate V •f
d&
•M TT
(4-152)
If we now use this in the pair of Eqs. (4-79) and (4-91), we arrive immediately at
NEUTRON TRANSPORT the energy-dependent
Pl
139
equations
1 9
f 00
JQ a t
dE"2s(E'-+E
)<>(r, £ ' , * ) + S (r,E, t),
(4-153)
r00
j_ 9J . 11 v
/
+ |V*
+ 2t(r,£)J(r,J?,0=
/
dE'2-(E'^>E)J(T,E'9t)
+
Sl(r9E9t).
(4-154) Continuing our analogy with the derivation of the one-speed diffusion equation, we will again assume: (a) isotropic source S j = 0 (b) | J | _ 1 9 | J | / 8 f < t ; 2 t ( r , i s ) so that Eq. (4-154) can be rewritten as r cc 2 t ( r , £ " ) J ( r , E , t ) — \ dE'2
(E'^>E)J(r9E'9t)
•'o
i = - ±W(r9E9t).
(4-155)
^
However we can now see a problem that appears to prevent a straightforward generalization of Fick's law to include energy dependence. For we cannot bring J(r,£", t) out of the scattering integral since it depends on the integration variable E'.
Of course, if we were allowed to assume isotropic scattering in the LAB system, then 2 S i ( £ , / - ^ £ ) = 0 and we could find (4-156) But the assumption of isotropic scattering is far too gross for most reactor calculations. We could proceed formally by merely defining an energy-dependent diffusion coefficient r 00
f
j^o 2t(r >£)
d E ' ^ E ' ^ E y ^ E ' j )
| Ji(r,E,t)
(4-157)
which would automatically yield J(r,£,/)=
-D(r,E)V
(4-158)
Of course, this approach is highly artificial because as defined in Eq. (4-157) D(r,E) still depends on J(r,E,t). One common procedure for avoiding this difficulty is to neglect the anisotropic contribution to energy transfer in a scattering collision by setting (4-159) so that /* 0 0
/
J
0
d E ' ^ E ' ^ E y ^ E ' ^ J L ^ i E y ^ E j ) .
(4-160)
140
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Then we find a natural generalization of the diffusion coefficient: D ( r , E ) = | [ Z t ( r , E ) - ji0Zs(r,E)]~'.
(4-161)
Actually none of these derivations of an energy-dependent diffusion equation are particularly satisfying because we have ignored the fact that in neutron transport processes, spatial transport, directional changes, and energy changes are intimately mixed. We can only provide a more satisfactory derivation of Eq. (4-158) after we have discussed comparable approximations characterizing neutron energy transfer in scattering collisions. Such a derivation must await our discussion of neutron slowing down in Chapter 8. For the present, we will simply assume that the generalization of Fick's law to include energy-dependence is given by Eq. (4-158) with an energy dependent diffusion coefficient as defined by Eq. (4-161). If we now substitute this into Eq. (4-153), w e a r r i v e a t t h e energy-dependent
r
= /
diffusion
equation
00
dEf^(Ef->E)<$>(
r9E'9t)
+ S{r9E9t).
(4-162)
This equation plays a very important role in nuclear reactor analysis since it is frequently taken as the starting point for the derivation of the multigroup diffusion equations. These latter equations represent the fundamental tool used in modern nuclear reactor analysis.
E. Diffusion Theory Boundary Conditions Since the neutron diffusion equation has derivatives in both space and time, it is apparent that one must assign suitable boundary and initial conditions to complete the specification of any particular problem. Since the diffusion equation itself is only an approximation to the more exact transport equation, we might suspect that we can use the transport theory boundary conditions as a guide in our development of appropriate diffusion boundary conditions. It will suffice to consider this development within the one-speed approximation. Recall that the transport theory boundary conditions we discussed earlier were: A
A
Initial condition:
(4-163)
Boundary condition:
(4-164)
We can obtain the appropriate initial condition for the diffusion equation by merely integrating the transport condition over angle to obtain: Initial condition: <J>(r, 0) = 4>0(r)
(4-165)
The boundary conditions are a bit harder to come by. Actually we will require several types of boundary condition, depending on the particular physical problem of interest. W e will group these boundary conditions into one of several classes:
NEUTRON TRANSPORT
/
141
1. GENERAL MATHEMATICAL CONDITIONS ON T H E FLUX
Although strictly speaking they are not boundary conditions, we should first mention those mathematical properties that the function (r, t) must exhibit in order to represent a physically realizable neutron flux. For example, <J>(r, t) must be a real function. Furthermore since both the neutron speed v and density N cannot be negative, we must require that (M) be greater than or equal to zero. In most cases we can also require that
Consider next an interface between two regions of differing cross sections. N o w clearly the correct transport boundary condition is that
f o r a11
®,
(4-166)
where cpj is the angular flux in region 1, while
d 6
FIGURE 4-15.
An interface boundary.
(4-167)
142
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
and f dM
= f
= J 2 (r s ,r).
(4-168)
Hence the interface diffusion theory boundary conditions are simply those corresponding to continuity of flux and current density across the interface:
- Z)1Vl(rs,/)= - D2V $ 2 (r s ,f).
(4-169)
We will occasionally find it mathematically expedient to imagine an infinitestimally thin source of neutrons S at an interface boundary. Then the interface boundary conditions are modified to read:
• J2(rs> 0 -
' J i( r s> 0 = S (0>
(4-170)
where e s is the unit normal to the surface. 3. VACUUM BOUNDARIES Recall that our transport theory boundary condition was merely a mathematical statement that there could be no incoming neutrons at a free or vacuum boundary
for 0 * ^ S < 0 , all r c on S.
(4-171)
a
FIGURE 4-16.
A vacuum boundary.
NEUTRON TRANSPORT
/
143
Once again diffusion theory will only be able to approximate this boundary condition. Notice in particular that the boundary condition is given only over half of the range of solid angle (corresponding to incoming neutrons). Hence suppose we again seek to satisfy the transport boundary condition in an "integral" sense by demanding f
J
dOet>a
f
J
2it-
• j(r s , &,t) = J_ (r s , /) = 0.
(4-172)
2TT~
where we have recognized that this integral condition is equivalent to demanding that the inwardly directed partial current J _ vanish on the boundary. Unfortunately diffusion theory is capable of only approximating even this integral condition, since it cannot yield the exact form for J ± . Indeed if we use the Px approximation [(4-131)] for the angular flux, we find that the partial current densities J + are approximated in diffusion theory by J + (r, t) = f
(4-173)
Hence our diffusion theory approximation to the transport boundary condition Eq. (4-171) is just / _ ( r s , / ) = ^ ( r s , / ) + | e s - ^ ( r s , r ) = 0.
(4-174)
For convenience consider this boundary condition applied to a one-dimensional geometry with the boundary at * = x 1
D
d
=0
or 1
dx
Yd •
<4"175)
Notice that this relation implies that if we "extrapolated" the flux linearly beyond the boundary, it would vanish at a point x s = x s + 2 D = x&+I\tr
(4-176)
For this reason, one frequently replaces the vacuum boundary condition / _ ( * , ) = o,
(4-177)
(4-178)
by the slightly simpler condition
where xs is referred to as the "extrapolated" boundary. More advanced transport theory calculations of the extrapolated boundary indicate that one should choose £ s = * s + z0,
(4-179)
144
/
T H E O N E - S P E E D D I F F U S I O N M O D E L O F A NUCLEAR REACTOR JJ*%> J+(XS)
^ d i f f u s i o n theory
\
Linear extrapolation
J •H F I G U R E 4-17. The behavior of approximate and exact representations of the neutron ftux near a vacuum boundary.
where the "extrapolation length" z 0 for plane geometries is given by z 0 = 0.7104Atr.
(4-180)
More complicated extrapolation length formulas can be derived for curved boundaries, 14 although they are rarely necessary since Eq. (4-180) suffices unless the radius of curvature of the boundary is comparable to a mean free path. It should be remembered that the true flux does not vanish even outside the boundary. The diffusion theory flux is a poor representation of the true flux near the boundary (as we saw earlier, diffusion theory is not valid near a boundary). The boundary conditions we have derived are intended to yield the proper flux only in the interior of the reactor, that is, several mean free paths away from the reactor boundary. These boundary conditions complete our description of neutron transport within the diffusion approximation. The neutron diffusion equation will play a very fundamental role in our development of nuclear reactor analysis methods. We will begin our study of nuclear reactor behavior using one-speed diffusion theory, since while this description has only limited quantitative validity, it does allow us to illustrate rather easily the principal concepts of nuclear reactor theory, as well as to develop the mathematical techniques used in more sophisticated models. With this background, we will then develop and apply the principal tool of the nuclear reactor designer, multigroup diffusion theory.
NEUTRON TRANSPORT
/
145
REFERENCES 1. W. G. Vincenti and C. H. Kruger, Jr., Introduction to Physical Gas Dynamics, Wiley, New York (1965); K. Huang, Statistical Mechanics, Wiley, New York (1963). 2. P. F. Zweifel, Reactor Physics, McGraw-Hill, New York (1973). 3. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press (1958). 4. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, Princeton, N. J. (1970). 5. K. M. Case, F. de Hoffmann, and G. Placzek, Introduction to the Theory of Neutron Diffusion, Vol. I, Los Alamos Scientific Laboratory Report (1953). 6. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, Mass., (1967); B. Davison, Neutron Transport Theory, Oxford University Press, (1958). 7. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Wiley, New York (1966). 8. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods, Wiley, New York (1969). 9. W. M. Stacey, Jr., Modal Approximations: Theory and an Application to Reactor Physics, M.I.T. Press, Cambridge (1967). 10. M. Becker, The Principles and Applications of Variational Methods, M.I.T. Press, Cambridge (1964). 11. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, Princeton, N. J., (1970), pp. 214-249. 12. L. I. Schiff, Quantum Mechanics, 3rd Edition, McGraw-Hill, New York (1968). 13. A. Messiah, Quantum Mechanics, Vol. I., North-Holland, Amsterdam (1958). 14. Reactor Physics Constants, USAEC Document ANL-5800, 2nd Edition (1963). 15. K. D. Lathrop, Reactor Technol. 15, 107 (1972).
PROBLEMS 4-1
4-2
4-3
We have defined the angular neutron density /i(r, in terms of the neutron energy E and the direction of motion fi, but one could as well define an angular density that depends instead on the neutron velocity v, n(r,\,t). Calculate the relationship between these two dependent variables. Two thermal neutron beams are injected from opposite directions into a thin sample of 2 3 5 U. At a given point in the sample, the beam intensities are 1012 neutrons/cm 2 sec from the left and 2X10 1 2 neutrons/cm 2 • sec from the right. Compute: (a) the neutron flux and current density at this point and (b) the fission reaction rate density at this point. Suppose that the angular neutron density is given by
An
n( r , f i ) = ^ ( l - c o s 0 ) ,
4-4
where 9 is the angle between S2 and the z-axis. If A is the area perpendicular to the z-axis, then what is the number of neutrons passing through the area A per second: (a) per unit solid angle at an angle of 45° with the z-axis, (b) from the negative z to the positive z direction, (c) net, and (d) total? In a spherical thermal reactor of radius R, it is found that the angular neutron flux can be roughly described by *(T,E9G)-
4^exp(-
Compute the total number of neutrons in the reactor.
.
146
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
4-5
Demonstrate that in an isotropic flux, the partial current density in any direction is given by /+=<#>/4. 4-6 One of the principal assumptions made in the derivation of the neutron transport equation is that there are no external forces acting on the neutrons, that is, that the neutrons stream freely between collisions. Suppose we were to relax this assumption and consider such a force acting on the neutrons (say, a constant gravitational force in the z-direction). Then we might expect additional terms to appear that involve this force. Derive this more general transport equation. 4-7 Explain briefly whether or not the transport equation as we have derived it adequately describes the spatial and angular distribution of neutrons: (a) in a flux of the order of 100/cm 2 -sec, (b) scattering in a single crystal, (c) passing through a thin pure absorber, and (d) originating within a very intense nuclear explosion in which the explosion debris are moving outward with very high velocities. 4-8 Develop the particular form of the transport equation in spherical and cylindrical geometries. To simplify this calculation, utilize the one-speed form of the transport equation (4-100) in which time dependence has been ignored. 4-9 A n interesting model of neutron transport involves transport in a one-dimensional rod. That is, the neutrons can move only to the left [say, described by an angular density n_(x9E9t)] or to the right [n+{x,E,t)\. One need only consider forward scattering events described by 2 S + ( £ " - » £ ) or backward scattering events described by 2~(.E"—»£)• Perform the following: (a) derive the transport equations for n+ and n_, (b) make the one-speed approximation in this set of equations, that is, 'S
4-10
and (c) describe the boundary and initial conditions necessary to complete the specification of the problem if the rod is characterized by a length L. Consider the following differential equation:
^ +
4-11
'
x2f(x)
=
2x(4-x),
where / ( x ) is defined on the interval 0 < JC < 4. Discretize this equation by first breaking up the independent variable range into four segments of equal length. Next use a finite-difference approximation to the d2f/dx2 term to rewrite the differential equation as a set of algebraic equations for the discretized unknown f i x ^ — f . For convenience, assume boundary conditions such t h a t / ( 0 ) = 0 = / ( 4 ) . Solve this set of equations for the ft and then plot this solution against x using straight-line interpolation. Consider the steady-state one-speed transport equation assuming isotropic scattering and sources in a one-dimensional plane geometry S(x) 2 Expand the solution to this equation in the first two Legendre polynomials < p ( x , ju,) s
+
Px(
/i),
Substitute this expansion into the transport equation, multiply by Pq(/i) and P x { ju) respectively, and integrate over /A to obtain a set of equations for the unknown expansion coefficients
NEUTRON TRANSPORT 4-12 4-13
4-14
/
147
Use Simpson's rule to write a numerical quadrature formula for the angular integral f 1 \diup(x,fi) for N equal mesh intervals. Develop the multigroup form of the transport equation as follows: First break the energy range 0 < £"< 10 MeV into G intervals or groups. N o w integrate each of the terms in the transport equation, Eq. (4-43), over the energies in a given group, say EG < E < EG_ (Remember that the group indexing runs backwards such that 0— EG < EG_ j < • • • EG < EG_! < • • • EX< E0= 10 MeV.) Now by defining the group fluxes as the integral of the flux over each group, and the cross sections characterizing each group as in Eq. (4-69), determine the set of G equations representing the transport equation. By writing out the components of the direction unit vector in polar coordinates, demonstrate explicitly that
f 477 f
d & f y
=0
and
f
dtitifij
=
3
Jaw
1
J
•
i¥=j
Demonstrate that f 4ljrd(iQlxSl™Slnz = 0 if /, m, or n is odd. Verify Eq. (4-138) using the identity in Problem 4-15. Explicitly demonstrate by integration that ju0 = 2 / 3 A for elastic scattering from stationary nuclei when such scattering is assumed to be isotropic in the CM system. 4-18 Consider an isotropic point source emitting SQ monoenergetic neutrons per second in an infinite medium. Assume that the medium is characterized by an absorption cross section 2 a s but only by negligible scattering. Determine the rate at which neutron absorptions occur per unit volume at any point in the medium. 4-19 Compute the neutron flux resulting from a plane source emitting neutrons isotropically at the origin of an infinite absorbing medium. Hint: Just represent the plane source as a superposition of point sources. 4-20 In a laser-induced thermonuclear fusion reaction, a tiny pellet is imploded to super high densities such that it ignites in a thermonuclear burn. In such a reaction some 1017 14 MeV neutrons will be emitted essentially instantaneously (within 10" 1 1 sec). Compute the neutron flux at a distance of 1 m from the reaction as a function of time, assuming that the chamber in which the reaction occurs is evacuated. 4-23 Compute the thermal neutron diffusion coefficients characterizing water, graphite, and natural uranium. Then compute the extrapolation length z 0 characterizing these materials. 4-24 Assuming that the diffusion approximation [Eq. (4-131)] is valid, compute the partial current densities in the z direction defined by Eq. (4-22). Use the one-speed approximation. 4-25 Consider the time-dependent one-speed Px equations assuming isotropic sources and plane symmetry. Eliminate the current density J(xJ) to obtain one equation for the neutron flux <#>(•*,/). Compare this equation with the one-speed neutron diffusion equation and indicate what differences you might expect in solutions to the two equations. 4-26 Try to construct solutions to the one-speed transport equation in an infinite sourceless medium:
4-15 4-16 4-17
0
r
+1
~Y J ^
dfl'
by seeking solutions of the form
where v and x( m)
are
to be
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
4-27 Consider the time-independent one-speed transport equation under the assumption of isotropic sources and scattering +
4tt J
f d&
hit
.
By regarding the right-hand side of this equation as an effective source, use the result [Eq. (4-120)] to derive an integral equation for the neutron flux <>(r).
5 The One-Speed Diffusion Theory Model
In this chapter we will develop the one-speed diffusion model of neutron transport. This model plays an extremely important role in reactor theory since it is sufficiently simple to allow detailed calculations and also sufficiently realistic to allow us to study many of the more important concepts arising in nuclear reactor analysis. Of course any model characterizing all of the neutrons in a reactor by a single speed (or energy) and treating their transport from point to point as a diffusion process cannot be expected to yield accurate quantitative estimates. Nevertheless if the cross sections appearing in this theory are properly chosen, one can use the one-speed diffusion model to make preliminary design estimates. Moreover many of the mathematical techniques we will use to solve and analyze this model are in fact identical to those applied to the more sophisticated models (e.g., multigroup diffusion theory) used in modern nuclear reactor design. The rigorous mathematical derivation of the one-speed diffusion model from the neutron transport equation has been given in Chapter 4. In this chapter we will give a more heuristic physical derivation of the one-speed diffusion equation and then apply this equation to study nuclear reactor behavior. The solution of the diffusion equation draws upon many familiar topics from a field that has become known as mathematical physics, including methods for solving boundary value problems involving both ordinary and partial differential equations. 1-3 For most problems of practical interest in nuclear reactor studies, one must employ methods from numerical analysis as well to allow the solution of the diffusion equation on a high-speed computer. We would expect that many of these topics (e.g., separation of variables methods in the solution of partial differential equations or Gaussian elimination for solving systems of algebraic equations) are already quite familiar to the advanced undergraduate. However we will continue our effort to make this 149
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
presentation as self-contained as possible by briefly reviewing such methods when they arise (although numerous references will also be provided). Throughout this development we would again caution the reader to avoid letting the smokescreen generated by these various mathematical techniques obscure the fundamental physical concepts governing the behavior of the neutron population in the reactor.
I. THE ONE-SPEED DIFFUSION EQUATION A. Derivation of the Diffusion Equation We will suppress the neutron energy dependence by assuming that all of the neutrons can be characterized by a single kinetic energy. Such a one-speed (or one-group) approximation greatly simplifies the mathematical study of nuclear reactor behavior. Of course such an approximation is also highly suspect, particularly in light of the fact that neutron energies typically encountered in a reactor span a range from 10" 3 to 107 eV, and neutron cross sections depend sensitively on energy over most of this range. We will later be able to show, surprisingly enough, that if one regards the one-speed approximation as an average description and chooses the appropriate average cross sections, then in fact the one-speed model can actually be used to obtain a quantitative description of a nuclear reactor. As yet a further justification for our exhaustive study of the one-speed approximation, we would remark that most energy-dependent theories (e.g., multigroup diffusion theory) are solved by performing a sequence of one-speed calculations for each successive energy group. Hence the methods we develop for analyzing our one-speed model will later be extended directly to more sophisticated descriptions. We will characterize the neutron distribution in the reactor by the neutron density N(r,t) which gives the number of neutrons per unit volume at a position r at time t. Actually we will find it more convenient to work with the neutron flux, 4>(r, t) — vN (r, t), since then we can compute the rate at which various types of neutron-nuclear reactions occur per unit volume by merely multiplying the flux by the corresponding macroscopic cross section. For example, the rate at which fission reactions occur per cm 3 at a point r would be given by 2f(r)<J>(r,/)We will derive an equation for the neutron flux by merely writing down a mathematical statement of the fact that the time rate of change of the number of neutrons in a given volume must be simply the difference between the rate at which neutrons are produced in the volume and the rate at which they are lost from the volume due to absorption or leakage.
THE ONE-SPEED DIFFUSION THEORY MODEL
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151
To be more precise, consider an arbitrary volume V of surface area S located anywhere within the reactor. We will examine this "control" volume carefully to determine how the neutron population within it changes. Evidently the total number of neutrons in V at a time t can be obtained by simply integrating over the volume f d3rN{x,t)
= f d3r — J y
J
V
(5-1)
V
Hence the time rate of change of the number of neutrons in V must be just
= Production in F— absorption in F — net leakage from F.
(5-2)
We can easily write down mathematical expressions for the gain and loss terms. If we define a neutron source density S(r,t), then Production in F = fd3rS(r,t).
(5-3)
Jy
Since the absorption rate density at any point in F is just 2 a (r) <J>(r>0> it i s obvious that the total rate of neutron loss due to absorption in F is just A b s o r p t i o n in V=
f d3r^a(r)<j>(rtt).
Jy
(5-4)
The term describing neutron leakage out of or into F is a bit more difficult. If J(r, /) is the neutron current density, then the net rate at which neutrons pass out through a small surface element d S at position rs is J ( r s , t ) - d S . Hence the total net leakage through the surface of F is just Net leakage from V= f dS-J(r,/).
(5-5)
Js
Now we could combine all of these terms back into Eq. (5-2) as they stand, but first it is convenient to convert Eq. (5-5) into a volume integral similar to the other terms. The common way to convert such surface integrals into volume integrals is to use Gauss's theorem to write f JC
rfS-J(r,/)= f d3rV * J(r,/). J1/
(5-6)
If we now substitute each of these mathematical expressions into Eq. (5-2), we find
I
,
d3r
r
i
3
- ^ - - S + v 31
a
S^+V-J
= 0.
(5-7)
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
But recall that we chose our volume V to be arbitrary. That is, Eq. (5-7) must hold for any volume V that we would care to examine. However the only way this can occur is if the integrand itself were to vanish. Hence we find we must require 1 7
(
5
-
8
)
Of course this equation is still quite exact (aside from the one-speed approximation), but it is also quite formal since it contains two unknowns,
(5-9)
where the constant of proportionality Z>(r) is known as the diffusion coefficient, while Eq. (5-9) is referred to as Fick's law. Of course to postulate such a relationship between current density and flux implies nothing about its range of validity. Indeed we do not even know what the diffusion coefficient D is. This situation is very common in macroscopic descriptions of physics, and relationships such as Eq. (5-9) are usually referred to as "transport laws" (not to be confused with the transport equation—we are talking the language of the physicist now) while the proportionality coefficients are known as "transport coefficients." Examples are numerous a n d include Fourier's law of thermal conduction (thermal conductivity), Stokes' law of viscosity (shear viscosity), Ohm's law (electrical resistivity), to name only a few. In all cases, one is forced to go to a microscopic description in order to evaluate the transport coefficient and examine the range of validity of the transport law. However this is of course exactly what we did by deriving the diffusion approximation [Eq. (5-9)] from the neutron transport equation in Chapter 4. There we found that D = (32 t r ) ~ 1 = [3(2, — /X0^s) ] ~ \
(5-10)
where is the average cosine of the scattering angle in a neutron scattering collision. Furthermore, we found that Eq. (5-9) was valid provided it was used to describe the neutron flux several m f p away f r o m the boundaries or isolated sources, the medium was only weakly absorbing, and provided the neutron current was changing slowly on a time scale comparable to the mean time between neutron-nuclei collisions. It is important to keep these limitations in mind as we apply this approximation in nuclear reactor analysis. Henceforth we will accept the diffusion approximation [Eq. (5-9)] as providing a valid expression for the neutron current density in terms of the neutron flux. If we substitute Eq. (5-9) into Eq. (5-8), we arrive at the one-speed neutron diffusion
THE ONE-SPEED DIFFUSION THEORY MODEL
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153
equation
+
(5-11)
This equation will form the basis of much of our further development of nuclear reactor theory.
B. Initial and Boundary Conditions We must augment this equation with suitable initial and boundary conditions. Although these conditions have been developed in a more rigorous fashion from neutron transport theory in Chapter 4, we will remotivate them here using plausible physical arguments. The appropriate initial condition involves specifying the neutron flux for all positions r at the initial time, say / = 0: Initial condition: <J>(r,0) = 0(r),
all r.
(5-12)
The boundary conditions are a bit more complicated and depend on the type of physical system we are studying. The principal types of boundary conditions we will utilize include the following: 1. V A C U U M B O U N D A R Y
At the outside boundary of a reactor, one would like to construct a boundary condition corresponding to the fact that no neutrons can enter the reactor through this surface from outside. Implicit in this fact is the assumption that the reactor is surrounded by an infinitely large vacuous region. Of course no reactor is surrounded by a vacuum, but rather by air, concrete, and a host of other materials. It is frequently convenient to assume that the reflection of neutrons back into the reactor from such materials is negligible so that nonreentrant boundary conditions apply. There is only one problem; diffusion theory is incapable of exactly representing a nonreentrant boundary condition. The closest one can come would be to demand that the inwardly directed partial current
•M's) =
+ ^es-V
(5-13)
vanish on the boundary (clearly an approximation, since this expression for J _ already is approximate). Actually we really shouldn't worry much about a consistent free surface boundary condition within the diffusion approximation, for we have already indicated that diffusion theory is not valid near the boundary anyway. It can only be expected to hold several mfp inside the boundary. Hence what we should really look for is a "fudged-up" boundary condition which, although it may have little physical relevance at the boundary, does in fact yield the correct neutron flux deep within the reactor where diffusion theory is valid. More detailed transport theory studies indicate that the proper boundary
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
condition to choose is one in which the diffusion theory flux (r, t) vanishes at a distance z o = 0.7104A tr
(5-14)
outside of the actual boundary of the reactor. This extrapolated boundary is usually denoted symbolically with a tilde. For instance, if the physical surface is at rs, then the flux will be taken to vanish on the extrapolated boundary (rs, t) = 0. A side comment here is appropriate. For most reactor materials, Atr is quite small, usually being of the order of several centimeters or less. When it is recognized that most reactor cores are quite large (several meters in diameter), then it is understandable that one frequently ignores the extrapolation length z0 and simply assumes the flux vanishes on the true boundary. Furthermore few realistic reactor geometries are surrounded by a free surface. Rather they are surrounded by coolant flow channels or plenums, structural materials, thermal shields or such. Hence while the concept of a free surface is useful pedagogically for painting a picture of an idealized reactor geometry surrounded by a vacuum of infinite extent, it is rarely employed in modern nuclear reactor analysis. 2. INTERFACES (MATERIAL DISCONTINUITIES) The structure of a nuclear reactor core is extremely complex, containing regions of fuel, structural material, coolant, control elements, and so on. Hence while one rarely encounters situations in which the material cross sections 2(r) depend continuously on position, one frequently is faced with what might be termed "sectionally uniform" cross sections that change abruptly across an interface separating regions of differing material composition. Our usual procedure in treating such discontinuities in material properties will be to solve the diffusion equation in each region separately and then attempt to match these solutions at the interface using appropriate boundary conditions. Once again the diffusion equation is not strictly valid within several mfp of the interface. However in this case one can argue that conservation of neutron transport across the interface demands continuity of both the normal component of the neutron current density J(r,/) and the neutron flux 0(r,/). This condition is occasionally modified to account for the physically fictitious but mathematically expedient convenience of including an infinitesimally thin neutron absorber or source at the interface. Then while the neutron flux is still continuous across the interface, the normal component of the current experiences a jump:
e s -[J(r s + )-J(r s ")]=5,
(5-15)
where e s is the interface surface normal, while S would represent a source term (if positive) or an absorption term (if negative). (See Figure 4-15.) 3. OTHER TYPES OF BOUNDARY CONDITIONS It is frequently convenient to impose other types of boundary conditions upon the neutron flux. For example, we know the flux must be nonnegative, real, and finite. Actually in our mathematical modeling of neutron sources we will
THE ONE-SPEED DIFFUSION THEORY MODEL
/
155
occasionally encounter a situation in which the neutron flux becomes singular at a localized source (e.g., a point source). However since such sources are mathematical idealizations, this singular behavior doesn't bother us, and in general we will demand that the flux be finite away from such sources. This condition is particularly useful in geometries in which certain dimensions are infinite. We will also occasionally be able to use symmetry properties to discard physically irrelevant solutions of the diffusion equation. For example, in onedimensional slab geometries, we can choose the coordinate origin at the centerline of the slab, and then use symmetry to eliminate solutions with odd parity [i.e., Other types of boundary conditions are encountered in practice. A very common problem in reactor calculations involves the determination of the flux in a small subregion of the reactor fuel lattice, a so-called unit cell repeated throughout the lattice. For example, such a cell might contain a single fuel rod surrounded by coolant (a fuel cell) or several fuel assemblies along with a control element (a control cell). Since these unit cells are repeated in a regular fashion throughout the core lattice, one can argue that there should be no net transfer of neutrons between cells, that is, that the neutron current density J(r) vanish on the cell boundaries. This is an example of a boundary condition on the current. In such cell calculations it is also frequently necessary to obtain a diffusion theory solution in the vicinity of a strong absorber (e.g., a fuel rod or control element). The appropriate boundary condition at the interface between the diffusing medium and the absorber is handled much like that characterizing a free surface. That is, one uses a transport-corrected boundary condition on the flux or current to yield the proper diffusion theory solution at a distance of several mfp from the interface. We will
Equivalent fuel cell
o
o
d
1 o
o
o
d
o
o
o
O 1 o o
Moderator
Ci iat n I A
•
Pa
I
Fuel e l e m e n t
C o n t r o l cell
Fuel cell
FIGURE 5-2.
Typical unit fuel and control cells.
RR\A
Fuel assembly
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
give explicit examples of such boundary conditions when we consider cell calculations in Chapter 10. We should reemphasize that the application of neutron diffusion theory in reactor analysis is successful in large part because the diffusion equation and its boundary conditions have been modified using more exact transport theory corrections. For example, Z> = [3(2 t — jw 0 2 s )] -1 contains a correction to account for anisotropic scattering. Furthermore the boundary conditions on the flux or current at a free surface or adjacent to a highly absorbing region contain transport corrections to yield the proper neutron flux deeper within the diffusing region. Such transport corrections frequently yield diffusion theory estimates that are far more accurate than one would normally expect, especially when we recall the rather strong approximations required to derive the neutron diffusion equation.
C. A Summary of the One-Speed Diffusion Model To summarize then, the model we will initially use to describe the neutron population in a nuclear reactor consists of the neutron diffusion equation: ±^-V-D(Tysr* + \(rWT,t)-S(r,t)
(5-16)
along with suitable initial conditions:
all r
(5-17)
and boundary conditions: (a) Free surface:
(5-18)
Here the diffusion coefficient Z>=A t r /3 = [3(2 t — /IqEJ] - 1 while the extrapolation length characterizing a free surface boundary condition is z 0 = 0.7104 Atr. The neutron diffusion equation [Eq. (5-16)] may be classified as an example of a linear partial differential equation of the parabolic type. 4 This type of equation has been thoroughly studied by mathematicians and physicists alike for years, since it also describes processes such as heat condition, gas diffusion, and even a wave function (notice, if we stick an in front of the time derivative, we have essentially just the Schrodinger equation familiar from quantum mechanics). As we proceed to apply this equation to nuclear reactor analysis, we will review several of the more popular schemes available for solving such equations. In many cases we will deal with situations for which the medium in which the neutrons are diffusing is uniform or homogeneous such that D and 2 a do not depend on position. Then the one-speed diffusion equation simplifies to +
=
(5-19)
The explicit form taken by this equation will depend on the specific coordinate system in which we choose to express the spatial variable r. For convenience, we have included the explicit forms taken by the Laplacian operator, V2, in the more common coordinate systems in Appendix B.5
THE ONE-SPEED DIFFUSION THEORY MODEL
/
157
W e will frequently consider situations in which the flux is not a function of time. Then Eq. (5-19) becomes — D V2
(5-20)
This equation is known as the Helmholtz equation and is also a very familiar beast in mathematical physics. It is useful to divide by — D to rewrite Eq. (5-20) as s
1
w h e r e w e h a v e d e f i n e d t h e neutron
diffusion
length
(r)
(5-21)
L
L = Vi>/2a .
(5-22)
W e will later find that L is essentially a measure of how far the neutrons will diffuse from a source before they are absorbed. W e now turn our attention to the application of this model to some important problems in nuclear reactor theory. W e will first study neutron diffusion in "nonmultiplying" media—that is, media containing no fissile material. Then we will turn to the study of the neutron flux in fissile material and begin our investigation of nuclear reactor core physics.
II. NEUTRON DIFFUSION IN NONMULTIPLYING MEDIA W e will first apply Eq. (5-20) to study the diffusion of neutrons from a steady-state source in a nonmultiplying medium. All of the mathematical techniques we will use are standard methods which arise in the solution of ordinary or partial differential equation boundary value problems and are discussed in any text on mathematical physics or applied mathematics. 1 " 3
A. Elementary Solutions of the Diffusion Equation 1. P L A N E S O U R C E IN A N I N F I N I T E M E D I U M
Perhaps the simplest problem in neutron diffusion theory is that of an infinitely wide plane source located at the origin of an infinite, homogeneous medium. The source is assumed to be emitting neutrons isotropically at a rate of S0 neutrons/cm 2 *sec. Since both the source plane and the medium are of infinite extent, the neutron flux <£>(r)^
l
S(x)
S0
where we have mathematically modeled the source by a Dirac S-functiom Hence we just have an inhomogeneous ordinary differential equation to solve with a
158
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
slightly weird source. Note that if we restrict Xt^O, the source term disappears from Eq. (5-23): d2
dx
1
LL
Our approach will be to solve this homogeneous equation for x ^ O , and then use a boundary condition at x = 0 to "fix u p " these solutions. We could obtain this boundary condition directly by integrating Eq. (5-23) from x = 0— e to x = 0 + € across the source plane and then taking the limit as e ^ O to find Dd<$>
d
dx
dx
= Jx(Q+)-Jx(
0~) = So.
(5-25)
[See Problem 5-2]. However we might also merely note that this is just a special case of the more general interface boundary condition [Eq. (5-15)]. If we use the symmetry of the geometry to assert that Jx(0+)= -Jx(0~) = J(0), then our boundary condition at the source plane becomes just lim
(5-26)
This source boundary condition makes sense physically, since it merely says that the net neutron current at the origin on either side must be just half of the total source strength. We are not through with boundary conditions yet. Since we have a second-order derivative, d2/dx1, we need another boundary condition. We will use the boundary condition of finite flux as x ^ c o . Hence the mathematical problem to be solved is
dxL
Lr
with boundary conditions:
(a) v 7 (b)
lim
d
=
S0
dx 2 lim <|>(x)
(5_27)
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/
159
We will then use symmetry to infer the solution for x < 0 . To solve this equation, we note the general solution 4>(x) = A exp( - j-) + B exp ( j-).
(5-28)
Applying the boundary conditions, we find
or A =
S0L 2D
Hence our solution is
,>0,
(5-29)
and by symmetry we can infer
svx ^e
x ) =
X p
(l)'
*<0.
(5-30)
Hence the neutron flux falls off exponentially as one moves away from the source plane with a characteristic decay length of L. As one might expect, the larger L (i.e., the smaller 2 a ), the less the neutron flux is attenuated as we move into the medium. Notice also that the magnitude of the flux is proportional to the source. That is, doubling the source strength will double the neutron flux
-4— In
2tL+ 1 2tL- 1
l + 3L 2 2 s 2 aJ Ei 0 ' • 1+ 3L 2 t 2 Ji0
(5-31)
Fortunately if 2 a < 2 t (as it must be for diffusion theory to be valid), one can
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T H E ONE-SPEED DIFFUSION M O D E L OF A NUCLEAR REACTOR
\ \
Transport theory \
Diffusion
theory
FIGURE 5-4. A comparison between the diffusion theory and transport theory solutions of the plane source problem.
expand Eq. (5-31) to find
(5-32)
L —
V 3 2 X For example, in graphite 2 t = . 3 8 5 c m " 1 while 2 a = . 0 0 0 3 2 . Hence 2 a / 2 t = 8.3x 10~ 4 , which implies that the correction to L given by higher terms in this expansion (that is, by transport theory) is only about 0.03% in this material. 2. POINT SOURCE IN AN INFINITE MEDIUM As a variation on this theme, let us repeat this calculation for the case of an isotropic point source emitting S0 neutrons/sec at the origin of an infinite medium. Since the source is isotropic, there can be no dependence of the neutron flux on angle. Hence the diffusion equation in spherical coordinates reduces to 1
d
2d$
1 ,/ x
r >0.
n
(5-33)
We will use our previous problem as a guide, and seek solutions such that the boundary conditions are (a)
l i m Airr2J
(/*) =
r—> 0
(b) v
'
S0,
lim <2>(r)
r—>00
One can readily verify that the fundamental solutions to Eq. (5-33) are of the form r _ 1 e x p ( ± r/ L); hence we are led to seek exp(-r/L) ^>(r) =
A r
exp(r/L) +
B r
(5-34)
Applying the boundary conditions, we find that (b) implies that we choose 5—0,
THE ONE-SPEED DIFFUSION THEORY MODEL
while (a) implies that A = S0/4itD.
/
161
Hence the solution is S0exp(-r/L)
A n interesting application of this result is to calculate the mean-square distance to absorption in a nonmultiplying medium. Note the number of neutrons absorbed between r and r + dr is just S0cxp(-r/L) 4tirD
\
)(4^Vr)(2a)
(5-36)
and thus the probability that the neutron is absorbed in dr is just P ( r ) d r - j j e x p ( - j - ) d r .
(5-37)
We can then calculate /• 00 =
v
^o
drr^p (r) = 6L2.
(5-38)
Hence the neutron-diffusion length L has the interesting physical interpretation as being 1 / V6 of the root mean square (rms) distance to absorption L2=i
(5-39)
That is, L measures the distance to which the neutron will diffuse (on the average) away from the source before it is absorbed. It should be stressed that we have calculated the rms distance from the source to the point of absorption, not the total path length traveled by the neutron. This path length will be very much longer since the neutron suffers a great many scattering collisions before it is finally absorbed. For example, in graphite the thermal diffusion length is 59 cm. Hence the rms distance to absorption from a point source is « / 2 » 1 / / 2 = V6 L = 144 cm. If we recall that the m f p characterizing thermal neutrons in graphite is 2.5 cm and also recall that the average number of scattering collisions suffered by the neutron before absorption is 1500, then it is apparent that the average path length or track length traveled by the neutron is about 3700 cm, considerably larger than V 6 L . One can actually use Eq. (5-39) to define the neutron diffusion length in situations in which diffusion theory would not apply (e.g., strongly anisotropic scattering or large absorption). Then one would first determine the flux (r) resulting from an isotropic point source using transport theory (or whatever description is relevant) and then calculate L by using f d3rr2
2
= W > = - r 6
J d r <j>(r)
,
(5-40)
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
where the integral is taken over all space. The cylindrical geometry problem of a line source at the origin of an infinite medium can be worked out in a very similar way. However to do so here would "deprive" the reader of an opportunity to try his own hand at such diffusion theory problems. Hence we have left the line source as an exercise in the problem set at the end of the chapter. We will instead turn our attention to problems in finite geometries. 3. FINITE SLAB GEOMETRIES Let us now modify our isotropic plane source by assuming that it is imbedded at the center of a slab of nonmultiplying material of width a surrounded on both sides by a vacuum (see Figure 5-5). We will set up this problem in a manner very similar to that for infinite plane geometry, except that we will add vacuum boundary conditions on either end of the slab. d2
1
z
L'
dx
with boundary conditions: d<\>
SQ
(a)
lim - D — = — , dx 2
(b)
*(±§)-o.
Here we have replaced the boundary condition at infinity by the vacuum boundary condition—in this case, using an extrapolated boundary a/2 = a/2 + z0. If we again seek a general solution of the form of Eq. (5-28), then applying the boundary condition (b) implies
x
= -all
FIGURE 5-5.
X =
0
x
= a/2
A plane source at the origin of a finite slab.
x
THE ONE-SPEED DIFFUSION THEORY MODEL
/
163
Then boundary condition (a) implies
Our final solution is therefore - ( a - x )
exp SL
K
M
^
l+exp(-f)
)
\(S-2\x\)i
sinh SL 2D
2 L
(5-42) cosh
(fi)
This solution is sketched in Figure 5-5. It looks somewhat similar to the infinite medium result [Eq. (5-29)], except for dropping off more rapidly near the boundaries due to neutron leakage. We should caution the reader once again that the solution is not valid within several mfp of the vacuum boundary (just as it is not valid near the source plane at the origin). A variant on the above problem involves replacing the vacuum by a material of different cbmposition than the slab itself (as sketched in Figure 5-6). The general procedure for attacking such multiregion problems is to seek solutions of the diffusion equation characterizing each region, and then match these solutions using interface boundary conditions. Once again we can use geometrical symmetry to allow us to restrict our attention to the range 0 < x < oo. In region ( l ) we will seek a solution of d2
0<x<
a
(5-43)
dx
where L 1 = y / ) 1 / E a i region (5)
is the diffusion length characterizing region (T). Similarly in d2
FIGURE 5-6.
A multiregion or reflected slab.
— <X<00.
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Now we will need several boundary conditions. We can use our earlier conditions [Eq. (5-27)] (a) (b)
So lim • / , ( * ) = — 2(.x)
as
In addition we will use the interface conditions
(C) •.(f)-fc(f) (d)
A(f) =^(f)
or
-D,
dd><j
dfa dx
2
Using our earlier work as a guide, we can seek general solutions
-j- + .Bjsinh i
in region (7) i
+
in region ( 2 )
and apply the boundary conditions to find (after a bit of algebra) B2 = 0,
SL 2D,
A2 =
slxL2
B{=-
SLX 2D x
9
DXL2 cosh ^ - 1 - j + D2L{ sinh ^
j (5-44)
^ e o s h ^ j + ^ ^ s i n h ^ )
CXP (4) iCosh +DiL2Sinli DL 2
(2t)
(2z:)
We have sketched the form of this solution in Figure 5-6. Several features of this solution are of some interest. Note that while the neutron flux is continuous across the interface, the derivative of the flux is not. This later discontinuity is, of course, a consequence of the fact that the diffusion coefficients in the two regions differ; hence to obtain continuity of current J, we must allow a j u m p in d§/dx across the interface. We have compared the solution for this problem with that obtained earlier for the slab surrounded by a vacuum. It should be noted that the flux in the central region falls off somewhat more slowly when the vacuum is replaced by a diffusing material. This can be readily understood by noting that the material surrounding the slab will tend to scatter neutrons back into the slab that would have otherwise
THE ONE-SPEED DIFFUSION THEORY MODEL
/
165
been lost to the vacuum. Such materials used to reduce neutron leakage are known as reflectors. Any material with a large scattering cross section and low absorption cross section would make a suitable neutron reflector. For example, the water channels surrounding LWR cores act as reflectors. In the HTGR, graphite blocks are added to the top and bottom of the reactor core to serve as neutron reflectors. A concept very closely related to that of neutron reflectors is the reflection coefficient or albedo, defined as the ratio between the current out of the reflecting region to the current into the reflecting region: (5-45) To make this concept more precise, suppose we want to attach a reflecting slab of thickness a to a reactor core (or perhaps a medium with a neutron source in it such as the slab geometry we have just considered). If we are given a current density Jm = J+ entering the reflecting slab surface, which we locate at x = 0 (see Figure 5-7) for convenience, then we can solve the diffusion equation characterizing the reflecting region: (5-46)
subject to boundary conditions / + ( 0 ) = 7 i n , <j>(a) = 0. We can then solve for the flux (x) in the reflecting slab and use this solution to compute the albedo a as (see Problem 5-11):
"(¥Mi) " •+(¥Mf)'
(5-47)
It is of interest to plot the albedo for the slab reflector versus slab thickness as shown in Figure 5-8. For thin reflectors, very few of the neutrons are reflected and hence the albedo is small. As the reflector becomes very thick, the albedo
x = 0
x = a
x
FIGURE 5-7.
The albedo problem.
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
1.0
1 - 2D/L 1 + 2D/L
a
a
FIGURE 5-8. The albedo for a finite slab plotted versus slab thickness.
approaches an asymptotic limit dependent only upon the material properties D and L:
(5-48)
For example, the albedos characterizing infinitely thick reflectors of graphite, H 2 0 , and D 2 0 are 0.93, 0.82, and 0.97 respectively. The albedo can be used to replace the detailed solution in the reflecting region by an equivalent boundary condition at the edge of the core using Eq. (5-45). That is, if we use our earlier definitions of the partial current densities J ± , then the effective boundary condition on the flux in the region x < 0 is just
(5-49)
Used in this manner, the albedo becomes a very useful device for obtaining boundary conditions for reactor core calculations. One can continue this game of solving the diffusion equation in various onedimensional geometries indefinitely. As we mentioned earlier, it is simply an exercise in ordinary differential equations. A variety of two- and three-dimensional problems can also be studied. However since these latter problems involve partial differential equations, we will defer their treatment until after we have introduced the separation of variables approach for solving such equations in Section 5-III-C. If we really want to get masochistic, we can remove some of the symmetry in our earlier problems so that the original partial differential equation (5-20) would have to be solved directly—for example, a point source set off-center in a sphere. However there is very little in the way of new physics to be learned from such exercises. Hence we will bypass further examples in favor of moving directly to more general problems. In particular, we will study how our previous solutions for plane and point sources can be used to determine the neutron flux resulting from an arbitrary distribution of neutron sources.
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167
4. GENERAL DIFFUSION PROBLEMS Recall that the neutron flux resulting from an isotropic point source of strength S0 located at the origin of an infinite medium was found to be o e x p ( — T") w
T
< 5 - 50 >
•
Suppose this source was located at the point r' instead. Then the flux could be found by a simple coordinate translation as exp < 5 - 51 >
4ttI) |r— r
Next suppose we have several point sources at positions rj, each of strength S r Then we can use the fact that the diffusion equation is linear to invoke the principle of superposition and write
2
Distributed source
—T1T\ AttD
FIGURE 5-9.
r — r-7\—
•
Superposition of several point sources.
(5"52)
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Finally suppose we have an arbitrary distribution of sources characterized by a source density S(r). Then the flux resulting from this distributed source is just
f I d V —
r').
47tZ>|r~r |
J
(5-53)
This is frequently rewritten as
(5-54)
where
(5
4
diffusion
kernel for an infinite
medium.
[The expression
"55)
kernel8
is a mathematical term used to denote a function of several variables (including the variables of integration) in an integral of the form f dx'K(x,x')f(x').
(5-56)
Here K(x,x') would be the known kernel, while f ( x ) might either be known [as the source density S(r)] or unknown as the flux
A
/• 00
/ dW \
/V
/S
A
dE'2,s(E'^E,W-^to)
(5-57)
IT A.
A
where 2 S ( £ " — i s known as the scattering kernel.] As a second example of such kernels, consider the flux resulting from a plane source at the origin <#,(*) = f § e x p ( - M ) .
(5-58)
If this had been located at x\ then the flux at a position * would be * W - § e x p ( - ! i ^ l ) . Hence in general, for S^>S (x')dx',
(5-59)
we find
= f
L
00
J — 00
dx'Gpl(x,x')S(x'),
(5-60)
THE ONE-SPEED DIFFUSION THEORY MODEL w h e r e w e h a v e i d e n t i f i e d t h e plane
source diffusion
kernel for an infinite
Gpl(x,x')=2^exp(-i^i).
/
169
medium
(5-61)
Once again we will allow the reader the privilege of deriving the line source diffusion kernel. Actually both the plane source and line source kernels could have been superimposed from the point source diffusion kernel. Notice that these kernels all depend only on the difference in spatial coordinates, that is, r — r' or x-~ x'. Such displacement kernels arise because we have assumed an infinite, homogeneous medium. For such infinite geometries we can use these kernels to compute the neutron flux arising from any source distribution. Unfortunately most geometries of interest are not uniform and are certainly not infinite. Hence we must generalize this discussion to determine the diffusion kernels characterizing other geometries and boundary conditions. Before attempting this generalization it is useful to step back a moment and try to obtain a general mathematical perspective of just what problem it is that we really wish to solve. Actually all we are doing is trying to solve an inhomogeneous differential equation of the form M<j>(x) = S(x), w h e r e M is a differential
operator
(5-62)
s u c h as
(We will leave it as understood that one must also apply suitable boundary conditions.) There are a variety of techniques available to solve such inhomogeneous problems. The approach we have been using thus far is known as the Green's function method: 9 (a) GREEN'S FUNCTION
METHODS
In this technique we first construct the solution to M<$>g(x) = 8 ( x - x f ) .
(5-64)
Then if we call
(5-65)
[Proof: M
S(x-x')
EXAMPLE:
Consider Id2
1 \
s
(x)
(5-66)
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
for an infinite medium, — oo < x < + oo. Now define G by d2G
1
S(x-x')
However we have just solved this for G (x,x')
= Gpl(x,xf)
=
exp( - ^
^
)•
(5-69)
Hence the plane source kernel is just the Green's function for this operator, and we find oo /
(xf).
dx' GpX(x,x')S
(5-70)
- OO It should not take much contemplation to convince the reader that this is just the scheme we have been using to construct infinite medium solutions by first determining the appropriate diffusion kernel or Green's function. The construction of the Green's function characterizing a finite geometry is not much more difficult, although one can no longer simply solve for the flux resulting from a source conveniently located at the origin, and then perform a coordinate translation to arrive at the Green's function. This latter complexity should be easily understood when it is recognized that shifting the source "off-center" in a finite geometry will destroy the symmetry of the problem. Hence one no longer finds a simple displacement kernel form for the Green's function. We will provide an alternative scheme for constructing these kernels later in this section. (b) VARIATION OF CONSTANTS
Perhaps a more familiar scheme for the solution of inhomogeneous differential equations is variation of constants,10 in which one first determines the linearly independent solutions of the homogeneous equation M ^ m { x ) = 0, a n d a particular
solution
(5-71)
of t h e i n h o m o g e n e o u s e q u a t i o n M
(5-72)
Since none of these solutions are required to satisfy the boundary conditions pertaining to the specific problem of interest, they are usually rather straightforward to find. Then one seeks the general solution to the problem as <j>(x) = A x
+ <j>part(x)
(5-73)
and applies the boundary conditions to determine the unknown coefficients of the homogeneous solutions, Ax and A2. EXAMPLE:
Consider a uniform source S(x)=
S0 in an infinite medium. Then
THE ONE-SPEED DIFFUSION THEORY MODEL
/
171
with boundary conditions such that <^>(x)
.
(5.75)
Then applying the boundary conditions implies that both A and B must be zero, and hence the solution is
Usually for the method of variation of constants to be of use in more general problems, one must be able to guess part(x) "by inspection." Hence for most problems, the Green's function technique or the method we will describe next is more convenient. (c) EIGENFUNCTION
EXPANSION
METHODS
One of the most powerful methods available for solving boundary value problems is to seek the solution as an expansion in the set of normal modes or eigenfunctions characterizing the geometry of interest. Rather than beginning with a general description of this very important scheme, we will introduce it by considering a specific example. EXAMPLE: We will attempt to determine the neutron flux resulting from an arbitrary distributed source in a finite slab of width a. That is, we wish to solve d2
i
S(x)
a
a
subject to the vacuum boundary conditions:
(a)
(b)
*(-f)-o.
Since we have taken the source S (x) to be arbitrary, we cannot assume symmetry to restrict our attention to the range 0 < x < 5 / 2 . Our approach to solving this problem may at first seem a bit irrelevant. We begin by considering a homogeneous problem very similar to Eq. (5-77): = with boundary conditions:
(5-78)
dx
Here B 2 is just an arbitrary parameter—at least for the moment. Let us now solve this associated homogeneous problem by noting the general solution $ (x) = A j cos Bx + A2 sin Bx.
(5-79)
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Our boundary conditions require ^(±f) =Acos(f)±^2sin(f).
(5-80)
Adding and subtracting these equations, we find that we must simultaneously require Ax
c o s ^ ^ ^ O ,
and A 2 sm in(^)=0.
(5-81)
How do we achieve this? Certainly we cannot set Ax and A2 equal to zero since then we would have the "trivial" solution ^ ( x ) = 0. Instead we must choose the parameter B such that these conditions are satisfied. Of course there are many values of B for which this will occur. For example, if we choose A2 = 0, then any B= Bn=-^,
n = 1,3,5,...
(5-82)
will give rise to a solution xpn (x) = An cos
,
n = 1,3,5,...
(5-83)
which obviously satisfies both the differential equation [Eq. (5-78)] and the boundary conditions. Alternatively, we could have chosen Ax= 0, in which case B = Bn = i f ,
« = 2,4,6,...
(5-84)
yields solutions t n ( x ) = An sin
,
/i = 2,4,6,...
(5-85)
Hence our homogeneous problem can be solved only for certain values of the parameter B. One refers to the values of B2 for which nontrivial solutions exist to the homogeneous problem as eigenvalues: Eigenvalues:
B2 = ^ j
,
n— 1,2,...
(5-86)
The corresponding solutions are referred to as the eigenfunctions of the problem:
An c o s ( ^ ) , Eigenfunctions:
\pn(x)
= .
4
;
s i. n/
n= 1,3,5,...
fl
; mrx \ ( y y
n =
~
,
A 2,4,6,...
(5-87)
v
'
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/
173
The reader may have already encountered eigenvalue problems in a somewhat different form: (5-88) where \pn is the eigenfunction corresponding to the eigenvalue Xn. However by comparing this form with Eq. (5-78), one can easily identify H—d2/dx2, B2, and ^ - ^ ( x ) . Notice that in the example Eq. (5-78) we actually find two types of eigenfunctions: the cosine functions corresponding to odd n and symmetric about the origin, and the sine functions corresponding to even n and antisymmetric about the origin. Had we restricted ourselves to symmetric sources S(x)= S( — x), we could have eliminated the antisymmetric solutions [Eq. (5-85)] from further consideration. We have sketched the first few eigenfunctions for the slab geometry in Figure 5-10. In acoustics these eigenfunctions would be identified as the normal modes or natural harmonics of the system, and this terminology is frequently carried over to reactor analysis. Notice that the An are still undetermined and are, in fact, arbitrary. These can be chosen in a number of ways, but for now we will just set An = 1 for convenience. So now this auxiliary problem has given us an infinite set of solutions if^n(x). What good are they? Well, they have a couple of very useful properties. First notice that the product of any two of these functions will vanish when integrated over the slab unless the functions are identical: ±
0,
2
2
iim^n
/
This property is known as orthogonality and proves to be of very considerable usefulness, as we will see in a moment. The second property of the eigenfunctions x(^n(x) is that they form a complete set in the mathematical sense that any reasonably "well-behaved" 1 function f ( x ) can be represented as a linear combination of the 00
/(*)- 2
«= l
cntH(x).
(5-90)
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/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Of course, such a representation is only of formal interest unless some scheme is available for determining the expansion coefficients c n , but this is where orthogonality comes in handy. Multiply Eq. (5-90) by *pm(x) and integrate over x to find a
a
a
2
2
2
~2
2
2
or a c
m
= j f
2
, d x f ( x ) t j x ) .
(5-91)
2
Hence given any function f ( x ) , we can evaluate the appropriate expansion coefficients cm by a simple integration. It should be mentioned that for the specific example of a finite slab we have been considering, an eigenfunction expansion is simply a Fourier series expansion and is probably already quite familar to most students. 1 However, the properties of orthogonality and completeness characterizing such trigonometric functions also hold for much more general eigenfunction expansions. With this background, we are finally ready to return to solve our original boundary value problem. As we mentioned, the essential idea is to seek the solution as an expansion in the eigenfunctions *(*)= 2
cM*)-
(5-92)
n=1
We will also expand the source term in a similar fashion 00 SW=
2
smrp„(x).
(5-93)
n=1
Notice that since S (x) is known, we can use orthogonality to determine the source expansion coefficients [as in Eq. (5-91)] as a 2
s
r2
n=~
(5-94)
adxS(x)lP„(x)
a. J _ _ 2
Of course since <£(*) is unknown, we cannot determine the cn in a similar fashion, but that is just what we can use the original equation [Eq. (5-77)] to accomplish. If we substitute Eqs. (5-92) and (5-93) into Eq. (5-77), we find 00
1
S i n=1
dx
1 n= 1
However using Eq. (5-78), we can eliminate d2$n/dx2 00 2 n—\
c
n(Bn +
(5-95)
L2^n
2
L
77 2 '
to find
Mv
(5-96)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
175
Thus we are left with one equation for an infinite number of unknowns, the cn. Fortunately orthogonality once again comes to our rescue. Multiply by integrate over x, and use orthogonality to find
Thus we find the flux for any source distribution as 00
n=1
where i//rt(x) = cos(wrx/<5), n odd and ^ ( x ) = sin( nmx /a), n even. We can rewrite this in a bit more familiar form if we substitute Eq. (5-94) into Eq. (5-98) and rearrange things a bit: a
,/ \
C ' ,
J
l
V
».(*)».(*')
S(x
')
2
a 2
= f_adx'Gplix,x')Six').
(5-99)
2
Hence we have found an explicit representation of the plane diffusion kernel or Green's function for a finite slab as an eigenfunction expansion r
(
,,
2
V
mm
Note in particular that the Green's function for a finite geometry is no longer a displacement kernel, that is, a function only of x — x\ as it was for infinite geometries. This intimate relationship between Green's functions and eigenfunction expansions is actually a very general result. Suppose we consider the diffusion equation characterizing any homogeneous geometry: 1
S(t)
subject to the usual vacuum boundary conditions on the extrapolated boundary: Boundary conditions: <J>(rs) = 0, rs on surface. As before, we first construct the eigenfunctions as the nontrivial solutions to the associated homogeneous problem V V ^ W - o , Boundary condition: i//n(rs) = 0.
(5_io2)
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
One can demonstrate that these eigenfunctions are orthogonal in the sense that f d 3 r\p m (r)\p n (r) = 0, jv
(5-103)
Since Eq. (5-102) is homogeneous, any solution ^ ( r ) may be multiplied by a constant. We will scale each \pn so that the eigenfunctions are normalized such that (5-104) J y
[Our earlier slab eigenfunctions \pn(x) would be normalized if we multiplied them by (2/<J) 1/2 .] The set of orthogonal and normalized eigenfunctions { ^ ( r ) } is said to be orthonormal. It can also be shown to be complete. Hence we can expand
where
" * d3r'\pn(r')S
sn=(
(r').
J V
If we substitute these expansions into Eq. (5-101), we can use orthogonality as before to solve for n/
"
1+
(5-106)
L2B}
Hence we find (r)= f d3r'G(r,r')S(r%
(5-107)
J y
where ^(r)^(rQ a
n
1
+
L2B}
(5-108)
as a general result for any geometry (although the eigenfunctions or spatial modes $„(r) may be very hard to construct in practice).
B. Numerical Methods for Solving the Neutron Diffusion Equation 1. INTRODUCTION Thus far we have confined our attention to neutron diffusion in homogeneous (or perhaps regionwise homogeneous) media since in this case the one-speed diffusion equation could be solved analytically. However in any realistic reactor
THE ONE-SPEED DIFFUSION THEORY MODEL
/
177
calculation the heterogeneous nature of the core must be taken into account. One not only must consider nonuniformities corresponding to fuel pellets, cladding material, moderator, coolant, control elements, but spatial variations in fuel and coolant densities due to nonuniform core power densities and temperature distributions as well. Such complexities immediately force one to discard analytical methods in favor of a direct numerical solution of the diffusion equation. In fact , even when an analytical solution of the diffusion equation is possible, it is frequently more convenient to bypass this in favor of a numerical solution, particularly when the analytical solution may involve numerous functions that have to be evaluated numerically in any event, or when parameter studies are required that may involve a great many such solutions. The general procedure is to rewrite the differential diffusion equation in finite difference form and then solve the resulting system of difference equations on a digital computer. It is perhaps easiest to illustrate this approach by a very simple example (sufficiently simple, in fact, to enable analytical solution). Suppose we wish to solve (5-109) dx
subject to the boundary conditions characterizing a finite slab of width a: <£(0) = 4>(a) = 0. (For convenience we will ignore the extrapolation length.) We first discretize the spatial variable x by choosing a set of N + 1 discrete points equally spaced a distance A = a/N apart (for convenience).
4V X
N
We now want to rewrite Eq. (5-109) at each of these discrete points xt, but to do so we need an approximation for d2^/dx2. Suppose we Taylor expand
=h + A
dx d<j>
A 2
+ 1
dx
A 2 d2
(5-110)
If we add these expressions, we find d2<j> 2
(5-111)
dx
to within order A2. Hence if A is chosen sufficiently small, this three-point central
178
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
difference formula should be a reasonable approximation to the value of d2$f dx2 at the point xr If we now use this difference formula to write Eq. (5-109) at any mesh point xiy we find
^
— J+S ^ S , ,
where again we have defined St=S(xWe to rewrite it as
/ = 1,2,...
(5-112)
can rearrange this difference equation
D 1 =
A: a
u - \
a
u
S
i
(5-113)
a,
or au_
!,_! + a ^
+ ait i+ !i + 1 = S f
(5-114)
/=1,...,7V-1
Hence we now have reduced Eq. (5-109) to a set of N— 1 algebraic equations for N + 1 unknowns ( ^ o ^ i ' f e - * - ' ^ ) - ^ w e ac*d on the boundary conditions at either end, say
THE ONE-SPEED DIFFUSION THEORY MODEL
/
179
subject to boundary or interface conditions that we will leave arbitrary for the moment. Actually we should remark here that this form of the diffusion equation is even a bit too general for most reactor applications. One rarely encounters reactor configurations in which the composition varies in a continuous way from point to point [i.e., D(x) and 2 a (x)]. Rather the system properties are assumed to be essentially uniform in various subregions of the reactor core (or can be suitably represented by spatially averaged or "homogenized" properties within each subregion). Hence the far more common situation is one in which the diffusion equation [Eq. (5-109)] with constant Dj and 2 . must be solved in a number of regions j. We will develop the difference equations for the general diffusion equation [Eq. (5-11$)], however, since they are not really any more difficult to derive or solve, and in certain cases they are useful in avoiding technical difficulties arising in less general approaches (such as the handling of region interfaces). As in our simple example we begin by setting up our discrete spatial mesh as shown below, although we will now allow for nonuniform mesh spacing.
A! V
A/"
i
W-1
+1
X
N
There are a variety of schemes that can be used to generate a difference equation representation of Eq. (5-115) on this mesh. We have already considered a simple problem in which a Taylor series expansion was used to derive a central difference formula for d1^/ dx1. A more common scheme is to integrate the original differential equation over an arbitrary mesh interval, and then to suitably approximate these integrals (after an occasional integration by parts) using simple mean values or difference formulas. By way of illustration, suppose we integrate Eq. (5-115) over a mesh intervalx t — ki/2<x<xi+ A / + lv /2 surrounding the mesh point xr
•V*!"]
I
"x"-
|
x*+!
Let us choose the simplest scheme to approximate the integrals by expressing them as the value of the integrand evaluated at the meshpoint xt times the integration interval. For example,
f J
A.
dx 2a(x)
/T •J V
A. "l
A-a. I+
T
~T
(5-116)
k
dx S (x) = Sj
T
+
i+ 1
(5-117)
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/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
The derivative term requires a bit more work. First write
/
+
+1 2
A,
V m
2
d<j> , d d x - ^ D v( x7 ) ^ = D (v x ) dx dx dx
A/ '
(5-118)
To handle d<j>/ dx, we can use a simple two-point difference formula [which can be derived by subtracting Eqs. (5-110)1:
*»
d<$> dx
Xi
+
—
+
dx;
Furthermore we will use a centered average for D: A..^, \
i
/
A
(5-119) Then we find that Eq. (5-118) can be written as 1 f *,•+- 2 ^ — Z ) ( X ) d<j> — dx
1
+
A;
dx
2
(5-120) If we now combine Eqs. (5-116), (5-117), and (5-120), we arrive at a set of difference equations very similar to our earlier results a
(5-121)
+ ai,i+ 1& + 1 = Si>
Ui-l
where a
u-1
=
A + A->\
_
A,.
a, .• = 2„ + ' \
/A,. + A,.+ 1 '
A+i + A +. A-i + A l A/+1
'
/A + 1 + A \ \
1
A,.
A,.
l
(5-122)
/ A , + A, + 1 '
l
j A , + A, + 1 -
Hence once again we have arrived at a set of N — 1 three-point difference equations for the iV+1 unknown discretized fluxes, $0,<j>v...,4>N. In the particular case in which the mesh size A(- is constant and the coefficients D(x) and 2 a ( x ) do not i
THE ONE-SPEED DIFFUSION THEORY MODEL
/
181
depend on x, we return to our earlier results [Eq. (5-113)] derived via a Taylor series expansion. Actually we should note that the coefficients aSJ depend only on a single subscript, j. However double subscripting is useful, for we will rewrite these algebraic equations as a matrix equation. Our final task is to append to these equations two additional equations taking into account the boundary conditions. Of course we could simply use the vacuum extrapolated boundary conditions, d>0=0,
and J
N,N-\§N-
1
+
a
NtN§N
~
(5-123)
S, "N-
Such sets of three-point difference equations are characteristic of onedimensional diffusion problems (indeed of any ordinary differential equation of second order). The coefficients a{J will depend upon the scheme used to derive the difference equations. Fortunately if the mesh spacing A is small, these differences will be insignificant in actual calculations. Since the spatial variation of the flux is essentially characterized by the diffusion length L, one generally chooses a mesh spacing A less than L. Similar three-point difference equations will also arise in curvilinear geometries with one-dimensional symmetry. For convenience we will assume regionwise uniform properties. Then in cylindrical coordinates, the diffusion equation becomes d2
—D
dr
i dtp
2
(5-124)
r dr
while in spherical coordinates, we find d24>
D
dr2
2 +
r
(5-125) dr
Hence we can derive difference equations corresponding to these geometries, using either of the earlier techniques, to find for uniform mesh spacing 17 a
19,-1 + au$i
u
+ % + I
s
n
(5-126)
where now D A2 L
a
u-\
2D
a
u+1
=
D_ A2
2/-1 v
1+
(5-127)
21 — 1
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/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
where c = 0,1,2 for plane, cylindrical, or spherical geometries, respectively. To complete the specification of these difference equations, we will consider the case of vacuum boundary conditions. For reference we first consider the slab geometry with uniform mesh spacing A:
Boundary conditions: (p0 =
0.
XN
Notice if we were to apply Eq, (5-126) to the case / = 0 (assuming 6 _ j = 0). then this boundary condition would imply , = 0 . However this is inconsistent with the 1= 1 equation. Hence we must ignore the < = 0 (and i=N) cases in applying the difference equation, and use the boundary conditions to eliminate the unknowns <J>0 and tpN from the i = 1 and i = N—\ equations. For example, the i = 1 equation is .0 (5-129)
Now consider the case of cylindrical geometry. Our boundary conditions are now somewhat different. At rN we still have the usual vacuum boundary condition
d
= 0=^0 = ^,
(5-130)
Again we ignore the cases /'= 0 and i = N. At 1=1, we find that a 10 = 0 and hence a , ^ ! + &i2(p2=:
(5-131)
The i= N— 1 equation is simplified by <j)N = 0. Very similar considerations hold for the case of spherical geometry. Thus we have now derived the general form of the difference equations characterizing one-dimensional neutron diffusion in plane geometry, and the difference equations characterizing regionwise homogeneous cylindrical and spherical geometries. (The extension to nonuniform media and mesh intervals for cylindrical and spherical geometries will be left as an exercise for the reader.) Our
THE ONE-SPEED DIFFUSION THEORY MODEL
/
183
next task is the determination of a suitable prescription or algorithm for solving this system of algebraic equations. 3. SOLUTION OF THREE-POINT DIFFERENCE EQUATIONS Suppose we have developed an appropriate set of difference equations similar to Eq. (5-121). We must now solve for the discretized fluxes To be more explicit, let's first write out these equations in detail a
a
+
u$l
= Si
l2^2
l
a
^23^3
a
a
22^2 32
=
S2 = S3
33
a
N- \,N-2N-2+
a
N - l ^ V - 1
=
(5-132)
-1
It is useful to rewrite these equations in matrix form
a
\2
a
2\'
v*.
^"23 ^33 ^43
A$=S_,
(5-133)
where A is an (N— l)X(jV — 1) matrix, and
1
S_.
(5-134)
Such tridiagonal matrices can be inverted directly using Gaussian elimination (the "forward elimination—backward substitution" method). 18 The general scheme is to subtract au u _ j times the (/— l)th row from the (/)th row to eliminate the au,_, element in the (<)th row. At each step, the (/)th row is then divided by its diagonal element, and the procedure is continued. For example, we have indicated
184
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
schematically the "forward elimination" on the first two equations in Eq. (5-132): au
0
al2
«12 a
\\
v a
2l
a
22
a
6
*I
a
\
a
21
a
22
21
0 a 23
(5-135) a
2\a\l\
/
such that we eventually arive at a system of equations of the form
0
r
i 0 0 ( 9 \ a \o 0
r
0 0
1 g
(5-136)
A !•
0
•
where . _
a
n,n+l aa n,n ^+aan,n-l^n-A
a
n,n
_
A n,n-\An-\
a
_
a
l2
1'
"l|
'
a
il
We can now substitute back up the matrix to find
<»V-1=«V l> ^V-2 = ~ =
- / l ,
,V - 2^,V- I + v
_ 2 « V - |
+
a
N
-
« V - 2 <
2
(5-138)
and so on. Thus Gaussian elimination consisting of forward elimination and backward substitution can be used to directly solve the difference equations [Eqs. (5-132)]. This scheme is particularly important since it frequently appears as an integral part of the iterative methods used in two- and three-dimensional diffusion problems. For this reason it is useful to formalize Gaussian elimination a bit by noting that what we have in fact accomplished by forward elimination is the factorization of
THE ONE-SPEED DIFFUSION THEORY MODEL
/
185
the matrix A into a product of a lower (L) and upper (U) triangular matrix: 19
L
U
(5-139)
Hence our sequence of steps in Gaussian elimination begins with A9= LUq=S.
(5-140)
First we perform a forward elimination sweep to construct and invert L U
(5-141)
followed by a backward substitution to invert U and solve for ± = U ~ l l ~ l $ = U~ X «•
(5-142)
As an aside we should observe that while such methods for solving systems of linear algebraic equations are most easily understood and analyzed (mathematically) in matrix notation, they are most easily programmed when written as a simple algorithm such as Eq. (5-138). For example, one could simply construct a loop to generate and store all An and an using Eq. (5-137) and then evaluate all
Most detailed neutron diffusion calculations characterizing nuclear reactors require either two- or three-dimensional treatments. Such details are particularly important in studying power profiles in large reactors subject to nonuniform fuel loading and depletion. Hence we now must consider the numerical solution of the more general diffusion equation ~VD
(r)V
(5-143)
Once again the geometry of interest is discretized into a mesh of cells such as the rectangular grids illustrated in Figure 5-11. Perhaps the most general way to derive
186
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR J
FIGURE 5-11.
Rectangular two- and three-dimensional grids
difference equations for the mesh is to integrate the diffusion equation [Eq. (5-143)] over the spatial volume of a given mesh cell, using this to define the spatially averaged cell properties. In general one can write 20 17 v
(5-144)
f
i Jvs
(5-145)
,7 i c P r [ - V D ( r ) V ^ L l ( p -
±
' i Vj
i f
j V s K
(5-146)
J= 1
(5-147)
' «
Here the sum is taken over the adjacent mesh point neighbors j=\ J = 2, 4, or 6 in 1-, 2-, or 3-dimensional Cartesian geometries, while
,J where
(5-148) j= i
where the mesh coupling coefficients are determined by the particular mesh geometry and finite-difference scheme one chooses. For example, in Cartesian coordinates using essentially the approximation schemes represented by Eqs. (5116) and (5-120), one would find (5-149) where we define /• = i / n , n i >• ~ 2 1 ' '
h
distance between mesh » ~ points i and j.
(5-150)
The difference equations representing Eq. (5-143) then take the form D-
I J
Du
\
(5-151) where i runs over all of the mesh points.
THE ONE-SPEED DIFFUSION THEORY MODEL
/
187
EXAMPLE: As an example, let's derive the difference equations characterizing the two-dimensional diffusion equations for a uniform medium with a uniform mesh in rectangular geometry d2
d2
J
dx
,
,
,
v
(5-152)
2a<J>(x,.y) = ^ (xjO-
Our mesh will be defined such that the mesh points are denoted by x 0 , x,, ..., x , . . . , xN; y0, y„ ..., yJr ..., yM with mesh spacing Ax and Ay respectively. The most direct manner in which to derive the difference equations is to use a central difference formula to approximate d2
9, -Lj-
+
2
dx
(Ax)
2
(5-153)
82<j>
frj-i^.-j
+ frj-n (Av)
2
m
--
x., y
N *
Ji
1!
yo
» X
*N-1
FIGURE 5-12.
N
The two-dimensional spatial mesh
(We will defer to an exercise at the end of the chapter the more general derivation for nonuniform media using mesh cell integration.) Substituting these expressions into Eq. (5-152) evaluated at the mesh point (x^yr) we find
(Ax)
(4k) + 2a + 2D V (Ax) / = \ ,...,N
2
+
1
(Ay)2
(5-154)
— 1
As before, we can use boundary conditions to specify $oj>fy.o'
fy\m-
188
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
5. ITERATIVE SOLUTION OF MULTIDIMENSIONAL DIFFERENCE EQUATIONS We now turn our attention to the solution of these difference equations. Our first task is to cast the set of equations into matrix form. This requires first assigning a single index to each mesh point. For example, in a two-dimensional mesh we could label the mesh points as (ij)^k=i+( j - l ) ( N - l ) .
For a two-dimensional problem, the matrix structure takes the following form-in this case, for a five-by-four mesh point array
I (A7,2)2A (A7,6)2A^
(A7,2)2 +(A7,6)2 +(A7i8)2
/
(5-155) Notice that the tridiagonal form we encountered in the one-dimensional case has now been augmented by two additional side-band diagonals—as we might have expected, since on closer examination we find the two-dimensional case yields a five-point difference equation. Similarly, assigning a single index to each mesh point of a three-dimensional problem yields the matrix structure which corresponds to a seven-point difference equation.
Now let's consider how we might solve such systems—that is, invert such matrices. Since Gaussian elimination can be applied to any matrix (formally, at least), we might first consider applying this technique to obtain a direct inversion. Recall that for a one-dimensional diffusion equation, the forward elimination
THE ONE-SPEED DIFFUSION THEORY MODEL
/
189
sweep reduced the original tridiagonal matrix to a form with only two diagonals, while the backward substitution step completed the matrix inversion
Backward
(5-157)
substitution
After a bit of examination, it becomes apparent that when a similar forward sweep is conducted on the five-diagonal matrix characterizing two-dimensional problems, the result is to fill in all of the zero entries between the main and outer diagonal.
(5-158)
This implies that one will require considerably more computer memory to allow such a direct inversion of the matrix. Such a direct algorithm is also rather complicated to program and leads to problems resulting from computer round-off error. For these reasons, it is far more efficient to use an iterative procedure to invert such matrices when N is large, since such schemes attempt to preserve the sparse structure of the original matrix in their operations. Let's illustrate the basic idea with a simple example: Suppose we wish to invert a matrix A—that is, we wish to solve (5-159)
S-
d±=
We first compose A into its diagonal and off-diagonal elements
A
=
\
D
K
~
—
\
B 1
(5-160)
Now D can be easily inverted:
2221
D
=
(5-161)
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/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Hence suppose we use Eq. (5-160) to rewrite Eq. (5-159) first as Q±=
+ S
(5-162)
and then invert D to find ±=
(5-163)
Now is where the iterative philosophy comes in. Suppose we guess on the right-hand side—call the guess
1
S_>
(5-164)
We can continue this iteration, calculating the m + 1 guess as 4>(m+1)=
D '
l
B ±im)
+D~lS.
(5-165)
Hopefully, then, as m becomes large, we converge to the true solution
(5-166)
Hence the general idea behind such iterative schemes is to generate improved guesses or iterates by solving the original system of equations in an approximate, but efficient, manner. We continue such an iterative process until two successive iterates and <£ (m+1) are sufficiently close together, at which point the iteration is stopped and <|>(m+1) is regarded as the solution. Notice that throughout the iterative process, we maintain the sparse structure of the original five-diagonal matrix A, thereby significantly reducing storage and calculational requirements. The particular scheme we have presented is known as the Jacobi-Richardson or Point-Jacobi method, and although it is a very simple scheme, it has the drawback that it converges very slowly. The reader might very roughly think of the convergence rate of such iterative processes as being determined by how big a chunk of the original matrix he is willing to invert on each iteration. (More precisely, the convergence rate is determined by the size of the matrix norm of D~lB_.) In the Point-Jacobi method, only a relatively small bit of the matrix, its main diagonal, is inverted on each step and hence we might expect convergence to be slow. (As an extreme example, one bites off a much bigger chunk in Gaussian elimination—the whole matrix A—and hence only a single iteration is needed.) One can accelerate this convergence in several ways. First, one could attempt to invert a bigger chunk of A^ on each iteration. It is also possible to use information about the next flux iterate during an iterative step. Finally, one can extrapolate from earlier flux iterates in order to more rapidly approach the true solution. To understand how to improve the Jacobi iterative scheme, let's write it out
THE ONE-SPEED DIFFUSION THEORY MODEL
/
191
explicitly in terms of the algebraic system
= Sx
= SN
(5-167) Hence we can solve for the m + 1 flux iterate immediately as
(m+l)_ _J_ S.. a,
/ = 1,2, ...,7V.
2 <wm) 7=1
(5-168)
It should be noted here that the Jacobi scheme does not use all of the available information during each iteration. For example, if the equations are solved in sequence from / = 1 to / = N, as they would be on a computer, then the solution of the first equation yields ^>!(m+1); but to find using the second equation, is used rather than the improved estimate
or successive
relaxation
method
is obtained. In this case, the system of equations in each iteration is solved as (m+l)
+ axMmy+
*iMm)-+
' * * + *w
+ a23
(m) = S +a2N4>T> 2 (5-169)
+ • " +V3N
= 5N and the solution is <#) (m+l) =
2 N
J_ a:
j-t+ijj
From solution of
From previous
previous equations
(mth) iteration
in current (m + 1) iteration
(5-170)
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/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
This can be rewritten in matrix form by decomposing A into the sum of an upper and lower triangular matrix:
Here L contains elements of the main diagonal and below it, while U contains elements above the main diagonal. Now we write Eq. (5-159) as L±=
JI±+
(5-172)
The Gauss-Seidel scheme described above amounts to inverting L by forward elimination, stepping row by row down the matrix. Hence our iterative scheme is
The fact that the Gauss-Seidel method utilizes the latest iterate elements of <£>(m+1) when solving successive equations yields a factor of two better in error reduction per iteration than the Jacobi method. It is possible to accelerate the convergence of the iteration scheme even further by introducing an acceleration parameter to extrapolate the iterative flux estimate. This procedure, known as the successive overrelaxation (SOR) method, can be illustrated by considering how one utilizes the / for convenience. N
v
a:
Gauss-Seidel estimate
S - ' ^ a ^ j j
(m+l)
2
j=i +
From SOR in current iteration [(m+l)st]
Now <j>f-m+1) is calculated as a linear combination of iterate
From SOR in previous iteration [mth] +
(5-174)
* ) and the previous SOR
^<™+1>-cl)^/m+1) + (l-co)^(M).
(5-175)
Here the extrapolation or acceleration parameter ranges between 1 and 2. Of course for o) = 1 we return to the Gauss-Seidel method in which no extrapolation is used. The iterative algorithm for each element can then be written as
2 = /+ 1 N
(m+l)
CO
a,
1-1
+ (1 -u)
(5-176)
THE ONE-SPEED DIFFUSION THEORY MODEL
FIGURE 5-13.
/
193
Flux extrapolation in the S O R method 20
We have sketched in Figure 5-13 how the flux extrapolation can enhance convergence to the true solution The optimum value of co giving the maximum rate of convergence can be related to characteristics of the original matrix A . In certain cases one can achieve a convergence rate as much as two orders of magnitude larger than the Jacobi method. It should be noted, however, that the estimate used for w can strongly affect the convergence rate of this method, and it frequently must be determined by experience. Very similar methods can be applied to three-dimensional diffusion problems. In this case the diffusion matrix A has seven diagonal elements as indicated below
(5-177)
Again iterative methods are utilized in which the outer diagonal elements are handled in a manner similar to those used in two-dimensional problems. However there is some reduction in iterative convergence rates due to a loss of procedure implicitness caused by the additional diagonal elements. Such iterative algorithms for the solution of the finite difference equations characterizing two- or three-dimensional diffusion problems are frequently referred to as inner iterations. This terminology arises from the fact that in n u c l e i reactor criticality calculations, the solution of the diffusion equation is itself imbedded in yet another iterative scheme—the so-called outer or source iterations—necessary to handle the presence of a fission term. We will study this latter scheme in Section 5-IV.
194
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THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
6. NODAL METHODS There are many instances in nuclear reactor analysis in which one requires a full three-dimensional calculation of the neutron flux, for example, in core fuel depletion or control rod ejection studies. Although a direct numerical solution of the diffusion equation can be performed on a modern digital computer, it is extremely expensive to do so, particularly when a series of such calculations would be required for a parameter study. We desire a scheme for determining the three-dimensional core flux distribution that avoids the large storage and execution time requirements of a direct finite difference treatment of the diffusion equation. Such a scheme is provided by so-called nodal methods22-23 The general idea is to decompose the reactor core into relatively large subregions or node cells in which the material composition and flux are assumed uniform (or at least treated in an average sense). One then attempts to determine the coupling coefficients characterizing node cell to node cell leakage and then to determine the node cell fluxes themselves. To develop this approach in more detail, consider the neutron diffusion equation in its general time-independent form given by Eq. (5-143). Now we know that we can formally write the solution to this equation as (5-178) where G(r,r') is the diffusion kernel or Green's function for the particular geometry of interest that satisfies — VmD (r)VG
(r,r') + 2 a ( r ) G ( r , r ' ) = S (r —r').
(5-179)
Notice, in particular, that G(r,r') can be interpreted physically as the flux resulting at a position r from a unit point source at r'. Of course we usually cannot construct G(r,r')—if we could, we would have already solved our problem. But suppose we ignore this annoyance for the moment. We will instead introduce the principal aspect of nodal methods by dividing the reactor core (or, more typically, one quadrant or octant of the core, since some symmetry is usually present) into N node cells as shown schematically in Figure
l
•»>.
Node cell n'—_
FIGURE 5-14.
Nodal cell division of a reactor core
^-Node cell n
THE ONE-SPEED DIFFUSION THEORY MODEL
/
195
5-14. We now integrate Eq. (5-178) over the volume Vn of the /?th node cell f d3r
JV r
Jv
n
J
n N
= ri^ =f1 dV"3 r fV d V C ( r , r')S(r').
(5-180)
If we define the spatial averages over the nodal cells,
S„ = y- j n
(5-181)
d3rS(r),
(5-182)
Vn
d3rf
j r f
^ ^
^H(r), Vn
K
"
" ,
d3r'G (r,r')S V
T ~ J
:
(r')
, V
d h f S
(5-183)
^
then we can rewrite Eq. (5-180) as
*„=
2
Knn,Sn„
(5-184)
n'= 1
or in matrix form KS,
(5-185)
where
matrix
while Knn, a r e k n o w as t h e nodal coupling
coefficients.
Thus if we know K, we can easily determine the flux resulting from a given source S by a single matrix multiplication. But of course we don't know K since we don't know G(r,r'). The key to such nodal methods therefore lies in our ability to approximate or guess the coupling coefficients Knn,. Of course from a formal point of view, if the number of nodal cells N is large, the nodal method becomes equivalent to the finite difference scheme and hence loses any calculational advantages. The real power of the nodal approach is realized only when the number of node cells N is small, since then the cells are large enough that they become coupled via neutron diffusion only to nearby cells—that is, the transfer matrix K is sparse (i.e., it has many zero elements). However choosing large node cells places the burden of the calculational effort on an estimate of the coupling coefficients Knn,. The determination of these coefficients is usually accomplished in a most empirical fashion (a nice way of saying they are fudged). Typically the Knn, are determined by assuming a flat nodal source and uniform composition in each cell
196
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
and allowing neutron transfer to only the six nearest neighbor cells. The transfer coefficients are represented as linear combinations of several simple trial functions (e.g., from one-dimensional slab geometry calculations). The blending coefficients in this representation are then determined by comparison with more accurate finite difference benchmark calculations and lots of experience, fiddling, and fudging. If the transfer coefficients are properly chosen, then such nodal methods can be extremely useful in generating three-dimensional flux distributions when only limited accuracy is required. Unfortunately such empirical schemes for choosing the Knn. are quite problem-sensitive and require a good deal of experience on the part of the reactor analyst. We will leave numerical methods for solving diffusion equations until later when we must generalize these methods to account for fission processes and energydependence. The above discussion has been an admittedly curse description of numerical methods for solving differential equations. There is a vast literature on this subject that provides the details of the methods we have so briefly outlined. 12-16 And perhaps the most valid argument for presenting only a brief sketch of such numerical methods lies in the recognition that these topics are of such vital importance to the practicing nuclear engineer, that he almost certainly will have had or will take further courses on numerical analysis in any event.
III. ONE-SPEED DIFFUSION THEORY OF A NUCLEAR REACTOR A. Introduction Thus far we have studied the diffusion of neutrons in nonmultiplying media as described by one-speed diffusion theory. We now wish to apply this theory to the study of nuclear reactors in which fissile material is present. Hence we must determine how to include nuclear fission in the one-speed diffusion equation (5-16). To this end, let us first recall the sequence of events involved in a fission chain reaction. To be specific, we first consider the processes occurring in thermal nuclear reactors (see Figure 5-15). Fission neutrons are born at high energies in the MeV range. It is possible that such fast neutrons induce fission in either fissile ( 235 U or 239 Pu) or fissionable isotopes ( 238 U). It is far more likely that the fast fission neutrons will be moderated to lower energies by elastic scattering collisions with light moderator nuclei (e.g., }H or ^C). As the fission neutrons are slowed down, they pass through energies comparable to the absorption resonances in heavy nuclei such as 238 U and hence experience an appreciable probability of being absorbed. They may also leak out of the reactor core during this slowing down process. In a thermal reactor, however, over 85-90% of the neutrons will manage to slow down to thermal energies. They will then diffuse about the reactor core until they either leak from the core or are absorbed. If they are absorbed in the fuel, then they may induce a new fission, thereby repeating the cycle. The processes involved in fast reactors are somewhat similar (see Figure 5-16). In such reactors an effort is made to prevent the fission neutrons from slowing down before they will have had a chance to induce fission. Low mass number material is avoided to reduce the energy loss via elastic scattering. However some moderation will occur, due both to elastic scattering from materials such as oxygen (remember, most fast reactor fuels are oxides) as well as inelastic scattering from materials such
THE ONE-SPEED DIFFUSION THEORY MODEL
/
197
107 106
Fission
Scatter Fission ^
105 \>
E
104
Moderation
103
Resonance absorption
Conversion
101
10"1
Thermal diffusion
Absorption—Fission
10"2
F I G U R E 5-15.
A schematic of the various processes involved in a thermal reactor
as sodium. In particular there will be a tendency for some of the neutrons to slow down to the energy range in which appreciable resonance absorption may occur, although most of the fission reactions will be induced by neutrons with energies above this range. Hence neutron moderation, leakage, and resonance absorption all play an important role in fast reactor physics, just as they do with thermal reactors. It should be evident that the various processes occurring during this sequence are strongly energy dependent. We will apply our one-speed diffusion model to study such processes, however. Our motivation is partly pedagogical since this model is by far the simplest description of nuclear reactor behavior and allows us to introduce many concepts of nuclear reactor analysis in the simplest possible framework. However as we have noted earlier, the one-speed diffusion model can also provide a very useful qualitative description of certain reactor types (notably very thermal or very fast reactors) provided one uses the correct values for the cross sections ( 2 a , 2 f , 2 t r ) which appear in the model.
B. The Fission Source Term We now direct our attention toward determining a way to include fission in the one-speed diffusion equation. We will assume that diffusion, absorption, and fission all occur at the same energy. Then a term to represent fissions can easily be derived by noting that if 2f
(5-186)
198
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
10 7
106
Scatter
Leakage
Fission
105
Fission^
Fission
Resonance absorption
Breeding
104 103 FIGURE 5-16.
Neutron processes Involved in a fast reactor
If this is the only source of neutrons in the reactor, + then the appropriate diffusion equation becomes
V
dt
- V-DV
(5-187)
Note here that we can identify the various components of the macroscopic absorption cross section which appear in Eq. (5-187) as: ^moderator
2structure
2 coo ^ ant + -J- 2"Vfuel
and y fuel _ yfuel . y fuel •^a y f
(5-188)
C. The Time-Dependent "Slab" Reactor 1. GENERAL SOLUTION We will begin our study of nuclear reactor behavior as described by the one-speed diffusion equation by considering a uniform slab of fissile material characterized by cross sections 2 a , 2 t r , and 2 f . This unrealistic appearing "slab reactor" is chosen to introduce many of the concepts of nuclear reactor analysis, since its one dimensional geometry greatly facilitates the detailed solution of the one speed diffusion equation.* The appropriate mathematical description of the
t Actually we should hedge here a bit. Equation (5-187) actually represents only "prompt" neutrons, that is, those born instantaneously in fission. The "delayed" neutrons arising from fission product decay require a slightly different treatment. We will defer this modification until Chapter 6. 1
In this sense it is somewhat akin to the "vibrating string" or "simple harmonic oscillator" problems in physics that also get beaten to death since they contain most of the interesting physics—and yet are easy to solve.
THE ONE-SPEED DIFFUSION THEORY MODEL
2
2
FIGURE 5-17.
/
199
The slab reactor
neutron flux in such a reactor is 1
t; d/ "
D
T7
with initial condition:
+
') = =
(5"189)
=
and boundary conditions: ^ y , t^ =
(symmetric), —
, t^ = 0.
Notice that we have assumed that our initial flux is symmetric. We will find later that such an assumption will imply similar symmetry for all times, <£(*,/) = <£>(— x,t). This will simplify our manipulations somewhat. Unlike our earlier studies of time-independent neutron diffusion, we are now faced with a partial differential equation to solve. There are a number of ways to attack such equations, but perhaps the simplest is to use separation of variables1-3 by seeking a solution of the form
(5-190)
If we substitute this form into Eq. (5-189) and divide by ip(x)T(t), J_ T
d^
_ v_
dt
we find
= constants -X.
(5-191)
$
Here we have noted that since we have a function only of x set equal to a function only of t, both terms must in fact be equal to a constant. We have named this constant — A. However X is as yet unknown. Hence the separation of variables given by Eq. (5-190) has reduced the original partial differential equation in two variables to two ordinary differential equations:
f
= -A r(/),
dV
\
1
v
dx
(5-192)
We can easily solve the time-dependent equation r(0=7X0)e-x',
(5-193)
200
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
where T(0) is an initial value which must be determined later. To solve the space-dependent equation, we must tack on the boundary conditions:
J > 0 + ( £ + »'Zf-z>)iK*)=o, Boundary condition: ^ ( | ) = t ( - | ) = 0.
(5-194)
Here A is still to be determined. However we recall the eigenvalue problem d \ dxz
5195
^(fHKH
(- )
has symmetric solutions (we are only interested in symmetric solutions since is symmmetric): eigenfunctions:
ipn(x) —
eigenvalues:
cosBnx
=
,
a?—1,3,5,...
^
^
If we identify Eq. (5-194) as the same problem, it is apparent that we must choose X = t>2a + vDBf - vv2f = Xn,
n = 1,3,5,...
(5-197)
These values of Xn are known as the time eigenvalues of the equation, since they characterize the time decay in Eq. (5-193). The general solution to Eq. (5-189) must therefore be of the form (x,t) = 2 A e x p ( - A „ / ) c o s ^ .
(5-198)
odd
This solution automatically satisfies the boundary conditions. To determine the An, we use the initial condition to write Initial condition:
(*, 0) = 0(jt) =
• n odd
(5-199)
^
Using orthogonality, we find a
1
r2
3dx
~
(5-200)
2
Thus we have found that the flux (for any symmetric initial distribution) can be represented as a superposition of modes, each mode weighted by an exponential
THE ONE-SPEED DIFFUSION THEORY MODEL
/
201
factor:
$ /
adx'4>0(x')cosBnx'
exp( - Xn t) cos Bn x,
(5-201)
n
odd
where the time eigenvalues Xn are given by A„ = t ) 2 a + vDB2-
t»2f,
B
n
=?f.
(5-202)
Before we proceed to examine this solution in more detail, it is useful to make a few comments about the separation of variables approach. First notice that the separation parameter that arose was in fact identified as an eigenvalue. Thus separation of variables is essentially equivalent to an eigenfunction expansion. Indeed if we had the foresight to expand the spatial dependence of the flux in the eigenfunctions for the slab (using symmetry to restrict this expansion to odd n), *(*,') = 2
Tn(t)cos^,
(5-203)
n
odd
where we have noted that the expansion coefficients now must be time-dependent, then we could have immediately arrived at an equation for the Tn dT
=
(5-204)
by substituting Eq. (5-203) into the original equation (5-189) and using the orthogonality property of the eigenfunctions. This alternative approach is frequently useful when encountering problems in which sources are present, because the separation of variables approach we have presented applies only to homogeneous equations. Finally, note that although we initially sought solutions i//(x)r(/) which were separable in x and /, these solutions were eventually superimposed to yield a nonseparable function of space and time [cf. Eq. (5-201)]. Hence separation of variables does certainly not imply a separable solution. Interestingly enough, however, there is one very important situation in which such separability will occur, that involving the behavior of the neutron flux for very long times. 2. LONG TIME BEHAVIOR Notice that one can order B?
as
oo.
(5-205)
This implies that regardless of the initial shape of
202
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
A
•/
0 >o
•
\
N.
, X
FIGURE 5-18. Time decay of higher order spatial modes in the slab reactor
fundamental mode shape. Of course, we have implicitly assumed that A x will not be zero. The coefficient of the fundamental mode is just ^
r
'
a
J
a = a~ J f a dx'4>uv0hc')cos^ ' a
.
(5-206)
Since
(f f ^
^ S ^ o m e t r i c buckling.
(5-207)
This nomenclature is used since B„ is a measure of the curvature of the mode shape 1
Since there will be a larger current density J and hence leakage induced by a mode with larger curvature or buckling, we might expect that the mode with least curvature will persist in time the longest. 3. THE CRITICALITY CONDITION Let us now see what is required to make the flux distribution in the reactor time-independent—that is, to make the fission chain reaction steady-state. We will define this situation to be that of reactor criticality: Criticality = when a time-independent neutron flux can be sustained in the reactor (in the absence of sources other than fission).
THE ONE-SPEED DIFFUSION THEORY MODEL
/
203
Notice that we have qualified this definition by specifically demanding that the flux be time-independent in the absence of a source. As we have seen in Chapter 3 (and will see later in the problem set at the end of this chapter), a source present in a critical system will give rise to an increase in flux that is linear in time. If we write out the general solution for the flux 00 <j>(x, t) = Ax exp( - X x t ) c o s B x x +
^
An e x p ( - X n t ) c o s B n x ,
(5-209)
n= 3
n odd
it is evident that the requirement for a time-independent flux is just that the fundamental time eigenvalue vanish Xx = 0 = t > ( 2 a - ?2 f ) + vDBf,
(5-210)
since then the higher modes will have negative Xn and decay out in time, leaving just <j>(x, t)->Ax
cosBxx
f u n c t i o n of time.
If we rewrite this "criticality condition" using the notation BX=Bg2, we must require =
(5-211)
then we find
(5-212)
It has become customary to refer to —
=B^=
material
buckling
(5-213)
since it depends only on the material composition of the reactor core (whereas Bg depends only on the core geometry). Hence our criticality condition can be written very concisely as (material composition) B 2 = B 2 (core geometry).
(5-214)
Thus to achieve a critical reactor, we must either adjust the size (Bg) or the core composition ( B s u c h that B 2 = B„2. We also note &m >
i< 0
=
supercritical, critical,
B2
(5-215)
subcritical.
In particular notice that by increasing the core size we decrease B g , while by increasing the concentration of fissile material we increase 2 f and hence B 2 . Both of these modifications would therefore tend to enhance core multiplication. Yet recall that in Chapter 3 we expressed the criticality condition in terms of the multiplication factor h. We can make the connection between these two criteria if we write the time eigenvalue as
+
(5-216)
204
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Now recall that (t?2) _ 1 is the mean lifetime for a given neutron-nuclear reaction to occur. Hence ( u 2 a ) _ 1 must be just the mean lifetime of a neutron to absorption (ignoring leakage—that is, in an infinite medium). Furthermore, we can identify *>2f f2fuel 2auel 1 2 a a
•If-*"
Now the only remaining task is to identify (1 + L2B2)~l. neutron leakage is given by L e a k a g e = fdS-J=
f d3rV-J=-
rate
Jv
^s
(5
'
"217)
Recall that the rate of
f d3rDV2
(5-218)
**v
where we have used both Gauss's theorem and the diffusion approximation. Hence we can write
f J i/
Rate of neutron absorption Rate of neutron absorption plus leakage f J yv
d3rZa
d3rDV2
f
J y
r
a
f
d*r$ + D B f f
1 3
d r<j>
1 +
(
(5-219) L2
*»)
However we can identify this ratio as just Nonleakage probability = P N L =
— .
1+
L2B2
(5-220)
Therefore we can interpret 1 (t)2 a )(l + L2B2)
_p nl
/ 1 | — / — neutron lifetime \ »2a / in a finite reactor,
(5 221) ; ^ "
since we have just reduced the lifetime to absorption in an infinite medium to take account of neutron leakage. If we now combine Eqs. (5-217) and (5-220), we find that the multiplication factor k for this model becomes just
?2 f /2 <5
-222)
Thus we can identify our fundamental time eigenvalue as just the inverse of the reactor period -A, =
=
(5-223)
If we also recall from Eq. (5-202) that A l = vD(Bg2-B^),
(5-224)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
205
then it is apparent that the various forms of the criticality condition are indeed equivalent: \ x = 0 & B 2 = B 2 ^ k = 1.
(5-225)
In particular notice that by using -PNL = (1 + L2B2)~l we can avoid the analysis of the initial value problem and proceed directly to the criticality condition k=
r2 f /2 a —r—r = 1. 2 2
1+
(5-226)
LB
We will return later to consider how these results can be applied to reactor criticality studies, but first we will extend them to more general reactor geometries.
D. The Criticality Condition for More General Bare Geometries Note that the only quantity characteristic of the reactor size or geometiy that appears in k or /*NL is the geometric buckling, B2. For the case of a slab reactor of width a we found B2 = (ir/a)2. We might suspect that for more general geometries we need only replace this by the geometric buckling characterizing the specific geometry under consideration. This suspicion is in fact easily verified, but only for so-called "bare" geometries in which the reactor composition is uniform. For the more complicated multiregion geometries, such as reactors composed of a core surrounded by a reflecting material, one can no longer derive simple expressions for P N L or k in terms of the reactor geometry and composition. Consider, then, a bare reactor of uniform composition surrounded by a free nonreentrant surface characterized by vacuum boundary conditions. If the reactor is critical then the neutron flux must satisfy the steady-state diffusion equation - D V2<J> + 2a<J>(r) = *>2f<J>(r),
(5-227)
subject to the boundary condition <£(fs) = 0 for r s on the extrapolated surface. Of course in general there will be no solution to this equation unless we have happened to hit on just the right combination of composition and system size. To see this more clearly, divide Eq. (5-227) by — D so that it can be written as /i>2*-2 \ V2
(5-228)
(rs) — 0.
Sometimes Eq. (5-228) is written in a somewhat different form as V^ + l ^ - ^ r ^ O , boundary condition:
(5-229)
(?s) = 0.
Now notice that this equation is identical to that which generates the spatial eigenfunctions for this geometry V \ + B„\( r) = 0, boundary condition:
^(r s ) = 0.
(5-230)
206
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
We know that this latter equation has nontrivial solutions t//„(r) only for certain values of the parameter B 2 , the eigenvalues B 2 . Hence by comparing Eqs. (5-228) and (5-230) we find that the steady-state diffusion equation for the flux <£(r) will only have nontrivial solutions when the core composition is such that (p2 f — 2 J / Z > is equal to an eigenvalue Bn2, and then the flux <J>(r) will be given by the corresponding eigenfunction i//n(r). However since there are an infinite number of possible eigenvalues B2, we might be tempted to think that there are an infinite number of values (?2 f — 2 a ) / Z ) = B2 for which the reactor is critical. However it should be recalled that in the case of a slab geometry, only the lowest eigenvalue B\ = (fn/af=B2 had a corresponding eigenfunction \pl(x) = cositx/a that was everywhere positive. The eigenfunctions or spatial modes \pn(x) corresponding to higher eigenvalues oscillated about zero. This same feature also characterizes the eigenfunctions of more general geometries. Only the eigenfunction ^,(r) corresponding to the smallest eigenvalue B 2 is everywhere nonnegative. Since the neutron flux can never be negative, it is apparent that the only solution to the eigenvalue problem Eq. (5-230) physically relevant is that corresponding to the smallest eigenvalue, B2=B2. Hence for the reactor to be critical we require
just as for the slab. Thus we can continue to use P N L = (1 + L2B2)~X and Jk = ( r 2 f / 2 a ) ( l + L2B2)~X for more general bare geometries provided we identify the geometric buckling B2 as the smallest eigenvalue B2 of the Helmhotz equation V2
(5-232)
subject to the boundary conditions that <£(r) vanish on the extrapolated boundary of the reactor. The corresponding critical flux distribution <J>(r) is then given by the fundamental eigenfunction ^ ( r ) , which is everywhere nonnegative. It should be pointed out that although the Helmhotz equation (5-232) will provide us with the flux shape in a critical reactor, it will tell us nothing about the magnitude of the flux. Since it is a homogeneous equation, if <£(r) is a solution, then any multiple of <>(r) is also a solution. Of course the magnitude of the flux was determined for us by the initial condition
(5-233)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
207
This is just the local thermal power density at position r in the core. Hence the total power generated by the core is just the integral of the power density over the core volume f d3rwfZf( r).
(5-234)
Jy
This relation can be used to determine the magnitude of the flux in terms of the core thermal power level. Thus we now have developed a rather simple scheme to study the criticality of a nuclear reactor—at least a bare, uniform reactor. The only mathematical effort required is the solution of the Helmhotz equation characterizing the geometry of interest for the geometric buckling Bg2 (the fundamental eigenvalue By) and the critical flux shape <£(r) [the fundamental spatial eigenfunction ^ ( r ) ] . To illustrate how these quantities are determined, we will consider a simple yet very important example: EXAMPLE: A Right Circular Cylindrical Core The most common reactor core shape is that of a right circular cylinder of height H and radius R. (Actually a sphere would be the more optimum geometry from the aspect of minimizing neutron leakage, but spheres are very inconvenient geometries to pass coolants through.) The appropriate form of the Helmholtz equation is then
subject to boundary conditions
Since this is a homogeneous partial differential equation, we can seek its solution using separation of variables
Then if we substitute this form into Eq. (5-235), we arrive at two ordinary
FIGURE 5-19.
Finite cylindrical reactor core
208
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
differential equations r f / ^ t
^
+
=
'
^ f - + A2 %(z) = 0,
)
(5"236a)
= °>
%( ± ^ ) = 0,
(5-236b)
where the separation constants a 2 and A2 are constrained by the relationship B 2 = a 2 + A2. Each of these equations represents a separate eigenvalue problem that one can use to determine a and X (and hence B2). The eigenfunctions and eigenvalues of the axial equation are well known to us
2„(z) = c o s ( ^ ) ,
A> = ( f )
2
,
„-i>3,...
(5-237)
To construct the eigenfunctions of the radial Eq. (5-236a), we first identify its general solution in terms of zeroth order Bessel functions (see Appendix D) <&(r) = AJ0(ar)
+ CY0(ar).
(5-238)
Since y0(ar)—»oo as r—>0, we must set C = 0 . Applying our boundary condition at r— R, we find J? ) = AJ0(aR
) = 0 ^>aR = pn,
(5-239)
where vn are the zeros of J0. In particular, the smallest such zero is v0 = 2.405... (kind of like m to a Bessel function). Hence we find the eigenfunctions and eigenvalues generated by the radial equation (5-236a) are just )2>
^('WO(Y)'
" = 0,1,...
(5-240)
Therefore, consistent with our prescription of seeking the smallest value of B 2 as our geometric buckling, we find
corresponding to a spatial flux shape =
(5-242)
Since this is the geometry most frequently encountered in reactor design, it is useful to calculate the normalization factor A in terms of the core power level P by noting 3
P = Jd rwslf<(>(r)
= wf2i27TA
J
drrJJ^)
f
H
=
w^fA4VJ{(v0)
(5-243)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
209
Thus we find ^ =
w{zfV
»
7TR 2 H.
(5-244)
It should perhaps be mentioned that since reactor cores are fabricated from either square- or hexagonally-shaped fuel assemblies, one can only approximate such cylindrical geometries. However for most purposes one can assume the reactor core is essentially a right circular cylinder. One can proceed in a very similar manner to analyze other bare core geometries. For convenience, we have tabulated the geometric buckling and critical flux profile in other common geometries in Table 5-1.
TABLE 5-1
Geometric Ducklings and Critical Flux Profiles Characterizing Some
Common Core Geometries
Geometric Buckling B i
1TX
Slab
Infinite Cylinder
cos — a
rt§ L[i
Finite Cylinder
m lsin
Sphere
Rectangular Parallelepiped
Flux profile
( j ) '
1
(f)
2+ 2+ 2
i (f) (f) (f) «K¥HFH¥)
7
Hjjl
The reader should not be deceived into believing that such criticality calculations are always so straightforward. For we must remember that the expression we have derived for the nonleakage probability / > NL = (1 + L 2 B 2 )~ X holds only for uniform, bare reactor geometries (i.e., single-region cores). As we will find later, it is no longer possible to derive simple expressions for P N L or k in terms of the reactor geometry and composition for multiregion (e.g., reflected) reactors. These results can be used to determine the core geometry or composition that
210 / THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
will yield a critical reactor. For example, if the material composition is specified, one can compute the material buckling B 2 in terms of the macroscopic cross sections using Eq. (5-213). Then by using the criticality condition B 2 = B 2 (along with Table 5-1), one can infer the core dimensions that will yield a critical system. In the more usual situation the nuclear designer will be given Bg2 (rather than B 2 ) since the core dimensions are determined by limitations on core thermal performance (and not nuclear considerations). That is, the core must be built a sufficiently large size to avoid excessively high temperatures for a desired power output. The nuclear designer must then determine the fuel concentration or loading (i.e., B 2 ) that not only will result in a critical system, but will also allow the core to operate at a rated power for a given time period. EXAMPLE: As a specific example we will study the one-speed diffusion model of a bare, homogeneous cylindrical reactor with material composition representative of that of a modern PWR such as described in Appendix H. We will use "homogenized" number densities corresponding to a PWR core operating at full power conditions and containing a concentration of 2210 ppm of natural boron (as boric acid) dissolved in the water coolant for control purposes. The fuel is taken as U 0 2 enriched to 2.78% 235 U. In Table 5-2 we have listed the number densities and microscopic one-speed cross sections for this core. (Here the cross sections are actually averages over the neutron energy distribution in such a reactor core.)
TABLE 5-2 Material H O Zr Fe
235u 238u 135
Xe io B
Number Density and Microscopic Cross Sections N{\/b-cm) 2.748 x l O " 2 2.757 X l O " 2 3.694 x l O " 3 1.710X10" 3 1.909 x l O " 4 6.592 x l O " 3
0.787 0.554
of(b)
V
0.650
0.294
0.260
0.000
1.62 1.06 1.21
1.78X10"4 0.190 2.33 484.0 2.11 2.36 XlO 6
0 0 0 0 312.0 0.638 0
0 0 0 0 2.43 2.84 0
1.001 x l O " 5
0.877
3.41 x l O " 3
0
0
This data can be used to calculate the macroscopic cross sections tabulated in Table 5-3. Here we have also included the relative absorption rates in each material which serve as a measure of neutron balance within the core. TABLE 5-3 Material H O Zr Fe
235u 238
U io B
Macroscopic Cross Sections ^(cm"1)
j»2 f (cm~')
1.79 X l O " 2 7.16X10"3 2.91 X l O " 3 9.46 x l O " 4 3.08 x l O " 4 6.95 x l O " 3 8.77 X 1 0 " 6
8.08X10" 3 4.90 X 10" 6 7.01 X l O " 4 3.99X10"3 9.24 X l O " 2 1.39X10"2
0 0 0 0 0.145 1.20X10"2 0
3.62 X l O " 2
0.1532
2tr(cm"1)
3.41 X l O " 2
0.1570
Relative
Absorption
0.053 0 0.005 0.026 0.602 0.091 0.223 1.000
THE ONE-SPEED DIFFUSION THEORY MODEL
/
211
It should be noted that the transport cross sections used in this example have been artifically adjusted (reduced by almost an order of magnitude) to take some account of fast neutron leakage which would normally not be described by a one-speed model. We can use these cross sections to calculate a number of important parameters characterizing the PWR core: Diffusion coefficient:
D = 9.21 cm
Infinite multiplication constant: Material buckling:
k^ =
= 1.025
5 2 = ( r 2 f - 2 a ) / Z ) = 4.13X 10" 4 c m " 2
Extrapolation distance:
z 0 = 0.71X tr = 19.6cm
Leakage fraction for a critical core:
1 — PNL — 0.025
(5-245)
Next we will compute the critical core dimensions. If we assume that the core height is fixed at 370 cm by thermal considerations, then we can determine the radius at which a core with such a composition will be critical. First calculate the axial buckling
B2
5z2 = ( - | ) 2 = 6 . 0 0 x l 0 - 5 c m - 2 .
(5-246)
Then using Eq. (5-241), we can determine the radial buckling B2 = b2-B2
= 3.53xl0"4cm"2.
(5-247)
Hence we can solve for the critical radius as
R = [^-)-z0=
108 cm.
(5-248)
(It should be noted that this is somewhat smaller than the radius of 180 cm for a typical PWR core. This illustrates the limitation of such a one-speed model for obtaining quantitative estimates in reactor analysis.)
E. Reflected Reactor Geometries To illustrate the complications that arise with multiregion core geometries, we will return to our slab reactor and add a reflector of nonmultiplying material of thickness b to either side (see Figure 5-20). For the purposes of this analysis we will characterize the reactor core by superscript " C " and the reflector by "R." Rather than repeat our earlier analysis of the initial value problem for this geometry, we will proceed directly to examine the time-independent diffusion equations that must be satisfied by the fundamental mode flux shape. As in our earlier analysis of the nonmultiplying reflected slab, we will seek a solution in each region of the
212
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
FIGURE 5-20.
Reflected slab reactor
reactor and then use interface conditions to match these solutions. Hence we must solve Core:
d2<j>c -D
2
+ ( ^ - p % f ) < t > c ( x ) = 0,
dx a,R
Reflector:
0<x
,
a , r
dx&
subject to the set of boundary conditions:
(a)
^(f)-^(f)
(b)
y«tf)-y{f)
(c)
*"(!+t)-0.
Note that we have used the reactor symmetry to narrow our attention to the range of positive x. As we noted earlier in Section 5-III-D this problem will have no solution unless we choose the proper combination of core composition and size. We would anticipate that a criticality condition relating these core characteristics would emerge in the course of our analysis. The general approach, as always, is to determine the general solutions in the core and reflector and then use the boundary conditions to determine the unknown coefficients. In the core the general solution will be <J>c(x) = ^ c c o s f i S c ,
(5-250)
where we have utilized the symmetry of the core to discard the sine term. Here the material buckling characterizing the core is defined by
(5-251) D
THE ONE-SPEED DIFFUSION THEORY MODEL
/
213
while in the reflector we will seek a solution satisfying the vacuum boundary condition (c)
<j>K(x) =
ARsinh
a
, r
2+
b
(5-252)
LK
where the reflector diffusion length L R = ( Z ) R / 2 f ) 2 . We now apply the interface boundary conditions to find
D
^
S
I
N
^
)
-
L ^ F C O S H F I ) .
(5-253)
Dividing these expressions, we can cancel Ac and AK to find
Notice that this equation represents a relation between reactor composition (DciB^,DK,LK) and size (a,b) that must be satisfied if a solution to the steadystate diffusion equations (5-249) is to exist. Hence this is just the reactor criticality condition for this particular geometry. Admittedly, it doesn't look anything like our earlier condition, B 2 = Bg2, that characterized a bare reactor. In fact the criticality condition for a reflected reactor is transcendental-one cannot obtain an explicit solution for the critical size or composition. Instead, either numerical or graphical techniques must be used. The latter technique is more useful for our present discussion. Rewrite Eq. (5-254) as (B^a
(—
M
^
\
j
DRa
( g
=
\
•
(5-255)
If we plot the LHS against ( B ^ a / 2 ) , we can then determine the solution of this transcendental equation graphically by noting where it intersects the value of the RHS, as shown in Figure 5-21. [Actually since there will be many such intersections, we are only interested in the lowest value of (B^a/2).] From this graph we notice that the critical value of B ^ must be such that BSa mu
„ 7T
2~
or
T
_->
T\i
B ?
in contrast to the bare (unreflected) core in which = ( t t / 5 ) 2 . Hence we see that the width a required for criticality is somewhat smaller when a reflector is added, but we would expect this since a reflector is added primarily to reduce neutron leakage. It is conventional to define the difference between bare and reflected core d i m e n s i o n s a s t h e reflector
savings
8:
8 = a (bare) - a (reflected).
(5-257)
214
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR I1 r1
h
1
i
i Z/ 1I
1 J/ |
RHS
/ '
/
1
1
1 1 1 1 7r 2
/ / /
1 1
/
/ 1 / 1 / 1 1 | 3tt 1 T
/ / / '
i 1 1 1
/ Bc a
/
2 FIGURE 5-21. Graphical solution for reflected reactor criticality
For example, the reflector savings for our slab reactor can be written as S=-Ltan bS
dcbSlr D
tanh
(5-258)
For a thick reflector 6 > L R this simplifies to (5-259)
Dk
which is essentially a measure of the maximum reflector savings that can be realized. Reflectors serve another function besides reducing neutron leakage. They tend to flatten the flux and hence the power distribution in the reactor core. Unfortunately one-speed diffusion theory is not adequate to describe this effect, which results in a peaking of the thermal flux in the reflector region (see Figure 5-19), so we must defer a further discussion of reflected cores until we have developed multigroup diffusion theory.
IV. REACTOR CRITICALITY CALCULATIONS A. Introduction Let us now turn to the very important topic of determining the composition or size of a reactor that will yield criticality. It should be apparent after the last example that in most practical reactor designs one cannot simply determine the geometric buckling for a core geometry and then use B 2 = B 2 to arrive at criticality. One "brute force" procedure would be to determine the lowest time eigenvalue
- V D (r)V<J> + ( 2 a - *2 f )*(r) = ^ *(r), v
(5-260)
and then keep adjusting things until A = 0. However this is rather awkward, and
THE ONE-SPEED DIFFUSION THEORY MODEL
/
215
moreover would tend to introduce errors in an unnatural manner when we generalize our analysis to include energy dependence. Instead suppose we write our diffusion equation as - V DV + 2a<J>(r) = boundary condition:
(5-261) 0(r s ) = 0
(which, of course, is the steady-state equation we solved analytically in the earlier simple examples). Unfortunately this equation has no solution in general—unless we just happen to hit on the exact combination of core composition and geometry such that the reactor is critical (a highly unlikely possibility on a computer). What we can do is introduce an arbitrary parameter "k" into this equation as: -V-Z>V* + 2 a * ( r ) = - ^ ^ ( r ) .
(5-262)
Then for some value of k, we assert that this equation will always have a solution. The idea is to pick a core size and composition and solve the above equation while determining A:. If k should happen to be unity, we have chosen the critical size and composition. If k=?-1, however, we must choose a new size and composition and repeat the calculation. As one might expect, k turns out to indeed be the multiplication factor we defined earlier in Chapter 3, as we will demonstrate later. We could give a formal mathematical proof that Eq. (5-262), or its generalizations will always have a solution for some k, but it is more convenient to simply argue physically that since varying k will vary the effective fuel concentration NF->N¥/k, one can always achieve a critical system by making k sufficiently small. Sometimes a slightly different formulation is used in which one pretends that v, the number of neutrons emitted per fission, is in fact variable. (Of course it isn't, but it is a useful device to regard it as adjustable for the moment.) Now physically we know that there must be some value of v, call it v c , that will yield a nontrivial solution to - V • DV + 2a =
(5-263)
regardless of what composition or geometry we have chosen. Hence the idea is to determine this vc, then readjust composition and geometry until we have forced Actual=v-
(5-264)
If we compare this approach to our earlier scheme in which we calculate k, it is evident that k=v/vc.
(5-265)
From a mathematical point of view, each of these approaches introduces a new parameter into the steady-state diffusion equation, either k or vc, which can then be regarded as an eigenvalue in a subsequent analysis. Once this eigenvalue has been calculated, one can return and readjust composition and geometry in an effort to force this eigenvalue to a desired value (e.g., A:—> 1 or vc->p). Hence the criticality
216
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
calculation is converted into a sequence of eigenvalue problems for the criticality eigenvalue
( s o m e t i m e s a l s o c a l l e d t h e multiplication
eigenvalue).
Of course, in general there will be a set of criticality eigenvalues kn corresponding to the eigenvalue problem represented by Eq. (5-262). For example, our earlier analysis for the bare slab reactor indicated the existence of the set of eigenvalues: ^ ( f S f / S J O + L 2 ^ 2 ) - 1 where B2 = (mr/a)2, n = 1 , 2 . . . . Only the largest such eigenvalue (in this case, kx) will correspond to an everywhere-nonnegative flux distribution
multiplication
factor
a n d d e n o t e it b y
As we will see below, kejj can be identified as the multiplication factor k for the reactor core defined earlier in terms of fission neutron generations in Chapter 3.
B. Numerical Criticality Searches We have seen in Section 5-III how one can obtain a criticality condition for a bare, uniform reactor. Let us now see how the criticality search is conducted in practical reactor calculations in which numerical methods must be used to solve the one-speed diffusion equation. To simplify our manipulations, let us first rewrite the criticality eigenvalue problem (5-262) in operator notation as M
(5-266) K
where we identify M° = — V D (r)V° + 2 a ( r ) ° = F° = p 2 f ( r ) ° =
Production
Destruction (leakage plus operator absorption) operator
(fission) We will leave the boundary conditions on <£(r) as understood. Of course in any numerical solution, finite-difference methods will lead to a representation of the neutron diffusion equation (5-266) as a matrix eigenvalue problem for the eigenvalue k~l. The solution of such eigenvalue problems can be accomplished using a common technique from numerical analysis known as the power method. We will introduce this scheme using physical arguments. First notice that if we assumed that the "fission source" term S = F
k as k(0).
(5-267)
We next solve for the flux <£(1) resulting from this source estimate: (5-268)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
217
using our earlier procedures. With this solution, we can now explicitly calculate the fission source resulting from this flux
(5-269)
This can then be taken as a new estimate of the fission source and used to generate a new flux, <j>(2\ and so on—provided we can also generate improved estimates of k. That is, we can iteratively solve for an improved source estimate S ( n + 1 ) from an earlier estimate S b y solving +
(5-270)
for (n+1) and then computing +
=
(5-271)
However we also need a prescription for generating improved estimates of k^ n \ This prescription can be obtained by returning to our original eigenvalue problem (5-266). As n becomes large, we would anticipate that (if our fission source iteration scheme really works), (w + 1) will converge to the true eigenfunction <^>(r) that satisfies Eq. (5-266). That is, for large n
The convergence of (n) to <£>(r) and k^ to k can be proven mathematically. It can also be motivated physically by recognizing that if indeed we have adjusted k such that a steady-state or self-sustaining flux profile were possible, then regardless of the initial fission source estimate, successive fission neutron generations will eventually fall into this distribution. Now for finite n, it is highly unlikely that <£(n + 1) and /c (n+1) will satisfy Eq. (5-266) exactly. Nevertheless if we integrate Eq. (5-272) over all space, we should be able to obtain a reasonable estimate for k^n + v> as fd3rF4>(H+1) k (n
+1) _
(5-273)
<
J
d3rM
However F<j>(n+ is just the (n + l)st estimate of the fission source, while we can use Eq. (5-270) to write Af(rt+1) in terms of the nth estimate of this source to find J k (n
+1) _
d3rS(n+l\r) t
We can now use this relationship to compute a new guess of
(5-274)
from <£>(rt + 1) and
218
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
We should note that this prescription is quite consistent with our earlier interpretation of k as the multiplication factor—that is, the ratio of the number of neutrons in two consecutive fission generations—if we note that a factor of A:(w) must be inserted in the denominator since is in fact the effective fission source that generates 5 < n + 1). We can now use Eqs. (5-270), (5-271), and (5-274) as the basis of an iterative algorithm to determine both k and
1)
and/or
g(n) —
max
gin-I)
< £2.
(5-275)
Notice that by scaling the source term appearing in the diffusion equation (5-266) by a factor of 1 / k ^ in each iteration, we will prevent the rapid growth or decrease of successive source iterates (causing possible overflow or underflow) in the event that a number of iterations are required when k is not close to unity. That is, dividing the source term by A:(n) removes the dependence of the flux iterate (rt+1) on n [at least as approaches the true solution]. This iterative scheme to determine the effective multiplication factor keff and the corresponding flux <J>(r) is known as the power iteration or source iteration method. T h e i t e r a t i o n s t h e m s e l v e s a r e k n o w n a s outer o r source
iterations.
In addition to such outer or source iterations, one will also be required to perform inner iterations to solve the diffusion problem M
(5-276)
when two- or three-dimensional calculations are necessary. The general strategy then takes the form sketched in Figure 5-22.
C. Source Extrapolation Needless to say, there is strong incentive to perform as few iterations as possible in converging to the desired accuracy. For that reason, one usually attempts to accelerate the source iteration convergence by extrapolating ahead to a new source guess. This is accomplished by introducing an extrapolation parameter (much as is used in relaxation methods). For example, in a one-parameter extrapolation, one would use as the source definition g(n)=
1
k(n)
F(f>(n)
+
(5-277)
A two-parameter extrapolation takes the form +
(5-278)
The extrapolation parameters a and (3 range between 0 and 1, and can be chosen by using methods based on Chebyshev polynomial interpolation. 13
THE ONE-SPEED DIFFUSION THEORY MODEL
/
219
Guess core geometry and composition ! Guess initial fission source S<°) and jfe<°)
Inner iterations
k
Criticality search
S< n + 1 >=F<*><" + 1> / d 3 r S < n + 1>( r) *<» + !> =
^/c^rS^r) Outer iterations
Convergence test k(n)
<€l &
S
<€2
-No—
± Yes
k
eff =
•No-
1
Yes
1
FINISHED! F I G U R E 5-22.
Calculation strategy for reactor criticality calculation
V. PERTURBATION THEORY It is frequently of interest to compute the change in core multiplication caused by a small change in the core geometry or composition. Fortunately if this change or "perturbation" is sufficiently small, one does not have to repeat the original criticality calculation, but instead can use well-known techniques of perturbation theory to express the corresponding change in multiplication in terms of the fluxes characterizing the unperturbed core. By way of example, consider a very simple one-speed diffusion model of a bare, homogeneous reactor in which the criticality relation is k—
?2f/2
^—^ =
1+
L2B2
(5-279)
220
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Now suppose we were to uniformly modify or perturb the absorption cross section to a new value =
+
(5-280)
where we will assume that the perturbation S 2 a is small—that is, S2a«2a.
(5-281)
Then the value of k' corresponding to the perturbed core can be written as
=
—-— a k 1 -
1 + LaB
\
2a
1+
L2B2J
,
(5-282)
where we have expanded k' in S 2 a / 2 a a n d have neglected all terms of higher than first order in the perturbation ( S 2 J . This has allowed us to express the perturbed multiplication factor k' in terms of the unperturbed multiplication k and the perturbation S 2 a . These general features appear in applications of perturbation theory to more general problems in nuclear reactor analysis in which the perturbations may be localized or in which the multigroup diffusion equations are used as the basic model of the core behavior. Although the general ideas are essentially as simple as those in the example above, it is necessary to introduce a few mathematical preliminaries. (For more details, the reader is referred to Appendix E.) We will describe the multiplication of the core by the criticality eigenvalue problem [Eq. (5-266)]:
M ^ ^ - V - . D W V ^ W + ^ W ^ r ) = ^2 f (r)<#>(r) =
(5-283)
where we will leave it as understood that the solution of this equation,
(5-284)
where f*(r) denotes the complex conjugate of / ( r ) , and V is the core volume. We can now use this inner product to define the operator M* adjoint to the operator M as that operator M^ for which (M*f,g)
= (f,Mg)
(5-285)
for every / ( r ) and g(r) satisfying the boundary conditions f ( r s ) = 0 = g (r s ). We can use this definition to explicitly construct the adjoint of an operator. Consider for example the operator F° = which simply corresponds to multiply-
THE ONE-SPEED DIFFUSION THEORY MODEL
/
244
we can use Eq. (5-300) to find the change in reactivity due to the addition of an absorption cross section 8 2 a (r)
(•'•ss.f') "
-
W
<5 303)
'
-
As it stands, this expression is still quite exact, but also still quite formal since it involves the perturbed flux, <j>', which we usually don't know (and usually don't want to calculate). However this is where the idea of "perturbation theory" comes in. For if the perturbation S 2 a is small, then presumably the corresponding perturbation in the flux 8
(
A p = - ——
S
2aS<J>)
- -—
(
(
F8
+
—
(
+ •••
(5-304)
($\F4>)
Then neglecting second and higher order quantities in the perturbation—that is, u s i n g first order perturbation
theory—we
find
(t'.ss.*) (5
-305)
Since the one-speed diffusion operator is self-adjoint, we know
(5-306)
J y
It should be noted that all of this analysis was exact until we neglected secondorder terms in Eq. (5-304). Thus, we have calculated a first-order estimate of the reactivity change Ap due to introducing a localized absorber 8 2 a (r) in terms of the unperturbed flux distribution. E X A M P L E : Consider a bare slab reactor characterized by one-group constants Z), 2 a , and ?2 f . We will perturb this reactor by imagining that an additional absorber is uniformly inserted in the region 0 < x < h . One might consider this to be a model of a bank of control rods inserted to a depth h in the core. Of course to allow the application of perturbation theory, we must assume this absorption to be relatively small. Hence our perturbation is 52a(x)=f52I
0,
°
< X < h
h<x
(5-307)
245
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
If we note that the unperturbed flux in this reactor is <>(x) = 4>0sin
17X a
(5-308)
then we compute the reactivity change due to an insertion of the absorber to a depth h as: SSa
f A
r
p(h)=~
a
vSf
J
irh 2a
7T
1
4
27rh a
sin
(5-309)
<j>2dx
o
It is customary to refer to the reactivity change due to such an absorber as the "worth" of the absorber. (This concept will be defined more precisely in Chapter 14.) Hence the reactivity worth can be sketched for various insertion depths h as shown in Figure 5-23. It is also of interest to compute the "differential worth" defined as dp dh
av1,{
2 irh 1 — cos
(5-310)
a
Note that the differential worth is at a maximum when the edge of the absorbing region (e.g., the tip of the control rods) is in the region of largest flux in the center of the core (see Figure 5-23). Such an analysis, while certainly of interest in illustrating general trends, is of limited usefulness in detailed control studies because of the highly absorbing nature of most control elements. Such elements very strongly perturb the flux in their vicinity, hence invalidating the use of perturbation theory. We will consider alternative methods required for computing control rod worth in Chapter 14. One can obtain more general expressions for the reactivity change induced by perturbations in the core parameters. For example, if we were to simultaneously perturb 2; = 2 f + S 2 f ,
=
+
D' = D + 8D,
(5-311)
the corresponding reactivity change then would be /
Ap = —
d3r[(v82f-82a)
.
f d rv^
Jy
3
(5-312)
2
The adjoint flux <£+(r) has a rather interesting physical interpretation. Suppose we imagine an absorber inserted into the reactor core at a point r 0 such that S2 a (r) = « S ( r - r 0 ) .
(5-313)
Here a is the effective strength of the absorber. (If we were to imagine that the S-function was, in fact, a mathematical idealization of an absorber of volume F a , then a — VA.) Now strictly speaking, perturbation theory should not be valid for such a singular perturbation, but we will dismiss such concerns with a wave of
THE ONE-SPEED DIFFUSION THEORY MODEL
/
246
ing a function by p2f(r). If we write (f,Fg)=
f d3rf*i>I,tg
J y
= f
d3r(v2J)*g
J y
= {v^fg) = (F%g\
(5-286)
where we have merely shuffled 2 f (r) around in the integral (noting that 2 f is real) to identify = p2f(r)°.
(5-287)
Notice that in this case, F^ and F are in fact identical. We refer to such operators as b e i n g
self-adjoint.
For a more complicated example, consider the spatial derivatives in the diffusion operator M : q, ( f V D V g ) = f d3rfv-DVg,
(5-288)
• ir
Now if we use the vector identity V ab = a V b + b Va,
(5-289)
we can rewrite this as
= jd3r
( / , V-DVg)
- jd3r ^
\
/) k
[ v f f ' ^ g } I ,
>
V
J '
rr ^ ^
(5-290)
h
Using Gauss's law, we can convert the first term into an integral over the surface: f d3rV-[fDVg]
= fdS-f*DVg.
J y
(5-291)
JS
However since we require that / and g vanish on the surface, this term vanishes. If we repeat this procedure we find we can rewrite (f,VDVg)=
fd3r[V-DVf\*g
= (VDVf,g).
(5-292)
JTS
Hence we find that VDV*° = V D V o
(5-293)
Thus we have again encountered a self-adjoint operator. From these examples, it is apparent that the operator A f ° = — + 2 a ° is also self-adjoint, M^ = M. We will continue to distinguish between the adjoint and direct operators M^ and M however, since for more general multigroup diffusion calculations, M will not be self-adjoint (as we will find in Chapter 7). We will
247
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
define the adjoint flux
as the corresponding solution of (5-294)
(Although again we keep in mind that M^ = M and F^ — F implies that = for the one-speed diffusion model of a reactor.) To understand the application of these concepts, let us go back to the criticality equation (5-283). Now suppose we were to perturb the macroscopic absorption cross section, say by adding a localized absorber, to a new value 2 ; ( r ) = 2 a (r) + S2 a (r).
(5-295)
We will assume that this perturbation 8 2 a (r) is small and attempt to calculate the corresponding change in k as governed by the perturbed criticality problem (5-296)
/c
Note here that the perturbation in the core absorption appears as a perturbation 8M in the diffusion operator M ' = M + 8M,
8M° = S2 a (r)°.
(5-297)
To calculate the change in k, first take the scalar product of Eq. (5-296) with the adjoint flux characterizing the unperturbed core, that is, satisfying Eq. (5-294), M
8M§')
= ± (
(5.298)
Now using the definition Eq. (5-285) of the adjoint operator, we find
Hence we find M \k
k!
(4>+,F<£')
We could now calculate 8k = k' — k. However it is far-more convenient to define the core
reactivity
9= ^
,
(5-301)
which essentially measures the deviation of the core multiplication from unity. Then since the perturbation in reactivity is just (5.302)
THE ONE-SPEED DIFFUSION THEORY MODEL
/
225
FIGURE 5-23. Relative and differential control rod bank worth
the hand and use our earlier result to find the corresponding reactivity change as f rf3/-t(r)S2a(r)(r)
Jl/
A p=-
f
J y
dW{r)»2f(r)
= - -§-*VoMro).
(5-314)
where we have denoted the denominator by a constant 6 (since it is independent of the perturbation). If we recognize that a
(5-315)
a
is simply proportional to the change in reactivity per neutron absorbed at r 0 per second. In this sense, then, the adjoint flux ^ ( r ) is a measure of how effective an absorber inserted at a position r is in changing the reactivity of the core. Evidently if (^(r) is large at r, the core multiplication will be quite sensitive to the absorption of neutrons at that point. Hence *(r) is sometimes referred to as the neutron importance
o r the importance
function.
We can see this from a somewhat different perspective if we consider the flux induced in a subcritical reactor by an arbitrary source S (r) as governed by (M — F)
— V*Z>V + ( 2
a
— p2f)= S .
(5-316)
226
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Consider the adjoint problem (
t
M
-
+
(5-317)
(Of course for one-speed diffusion theory, M = M * is self-adjoint, but we will retain the generality for a bit.) Notice that we have allowed the source S \ r ) appearing in the adjoint equation to differ from that in Eq. (5-316). Now suppose we multiply Eq. (5-316) by and integrate over r, then multiply Eq. (5-317) by and integrate, and then subtract these two results to find (5-318) However by the definition of the adjoint M F ^ , the LHS is zero. Hence we find: f d3r^(r)S(r)
= f d3rSf(r)<#>(r).
Jy
Jy
Since this must hold for any choice of S ( r ) and advantage by specifying S(r) as a unit point source at r0: S(r) = S(r-r0),
(5-319) we will use it to our
(5-320)
and S*(r) as the cross section 2 d (r) characterizing an imagined detector placed in the core. Then we find
<*>t(r0)= f d3r 2d(r)<J>(r).
(5-321)
J y
Hence in this instance the adjoint flux is simply the response of a detector in the core to a unit point source inserted at a position r 0 . Once again we find that
REFERENCES 1. J. W. Dettman, Mathematical Methods in Physics and Engineering, McGraw-Hill, New York (1969). 2. B. Friedman, Principles and Techniques of Applied Mathematics, Wiley, New York (1956). 3. G. Arfken, Mathematical Methods for Physicists, 2nd Edition, Academic, New York (1970). 4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I, McGraw-Hill, New York (1953), p. 173. 5. Ibid., pp. 115-117. 6. B. Davison, Neutron Transport Theory, Oxford U. P. (1958), pp. 51-55.
THE ONE-SPEED DIFFUSION THEORY MODEL
/
227
7. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, Mass. (1967). 8. G. Arfken, Mathematical Methods for Physicists, 2nd Edition, Academic, New York (1970), pp. 733-739. 9. Ibid., pp. 748-768. 10. E. A. Coddington, Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N. J. (1961), p. 67. 11. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods, Wiley, New York (1969). 12. M. Clark, Jr., and K. F. Hansen, Numerical Methods of Reactor Analysis, Academic, New York (1964). 13. E. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, N. J. (1966). 14. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J. (1962). 15. H. Greenspan, et al., Computing Methods in Reactor Physics, Gordon and Breach, New York (1968). 16. G. E. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, N. J. (1967). 17. T. Craig, SLODOG, A Modified One-group, One-dimensional Diffusion Code, University of Michigan Nuclear Engineering Report (1968). 18. E. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, N. J. (1966), p. 22. 19. Ibid., p. 24, 25. 20. R. G. Steinke, A Review of Direct and Iterative Strategies for Solving Multi-dimensional Finite Difference Problems, University of Michigan Nuclear Engineering Report (1971). 21. W. R. Cadwell, et al., P D Q - A n IBM-704 Code to Solve the Two-dimensional Fewgroup Neutron Diffusion Equations, WAPD-TM-70 (1957) and related reports, WAPDTM-179, WAPD-TM-230, WAPD-TM-364, WAPD-TM-678. 22. D. L. Delp, et al., FLARE, A Three Dimensional Boiling Water Reactor Simulator, General Electric Company Report, GEAP 4598 (1964). 23. E. G. Adensam, et al., Computer Methods for Utility Reactor Physics Analysis, Reactor and Fuel Processing Technology, Vol. 12, No. 2 (1969). 24. E. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Englewood Cliffs, N. J. (1966). p. 83. 25. R. Froehlich, in Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, USAEC CONF-730414-P2 (1973), p. V I M .
PROBLEMS 5-1
5-2
5-3
Compare the derivation of the one-speed neutron diffusion equation with that for the equation of thermal conduction, taking care to point out the assumptions and approximations used in each case. Refer to any text on heat transfer such as those listed at the end of Chapter 12. By considering a plane source or absorber of neutrons located at the origin of an infinite medium, derive the interface condition Eq. (5-15) on the neutron current density by modeling the source term in the one-dimensional diffusion equation as S3 (x) and then integrating this equation over an infinitesimal region about origin. Compute the rms distance « x 2 ) ) 1 / 2 a neutron will travel from a plane source to absorption using one-speed diffusion theory. Compare this result with the rms distance to absorption in a strongly absorbing medium (in which neutron scattering can be neglected). In particular, plot the rms distance to absorption in water in which boron has been dissolved against the boron concentration to determine whether the
228
5-4
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
diffusion theory result for ( ( x 2 ) ) 1 / 2 ever approaches the result characterizing a purely absorbing medium. (Use the thermal cross section data in Appendix A.) It is possible to derive an expression for the relaxation or diffusion length L from the one-speed transport equation characterizing a homogeneous medium 3
dx
^
/
\
2S
r + 1
+ 2t
dfi'
(we have assumed isotropic scattering for convenience). Seek a solution of the form
= 0.
5-5
Using the assumption that 2 a < 2 s , expand L as a power series in 2 a / 2 t , substitute this expansion into the equation above, and evaluate the coefficients of the expansion in order to derive Eq. (5-32) and obtain the transport corrections to the diffusion length L = ( / ) / 2 a ) ' / 2 . 5-6 Determine the neutron flux in a sphere of nonmultiplying material of radius R if an isotropic point source of strength S0 neutrons per second is placed at the center of the sphere. Assume the sphere is surrounded by a vacuum. 5-7 The Milne problem: Imagine a diffusing medium in the half space x > 0 with a source of infinite magnitude at infinity such that the boundary condition on the flux is that
5-8
5-9
5-10
5-11
5-12
(a) Using one-speed diffusion theory and the boundary condition of zero reentrant current, determine the flux in the medium. (b) Repeat the solution of this problem using the extrapolated boundary concept. (c) Determine the conditions under which these two boundary conditions might be expected to yield similar results. Consider a slab of nonmultiplying material with a plane source at its origin emitting S0 neutrons/cm 2 -sec. By solving this problem first with the condition of zeroreentrant current and then extrapolated boundaries, compare the absorption rate in the slab predicted by these two approaches. Also calculate the rate at which neutrons leak from the slab in each case. Consider a thermal neutron incident on a slab-shaped shield of concrete 1 m in thickness, and determine the probability that: (a) the neutron will pass through the shield without a collision, (b) it will ultimately diffuse through the shield, and (c) it will be reflected back from the shield. (For convenience, treat the concrete as if it had the composition of 10% H 2 0 , 50% calcium, and 40% silicon.) Consider an infinite nonmultiplying medium containing a uniformly distributed neutron source. If one inserts an infinitesimally thin sheet of absorber at the origin, determine the neutron flux throughout the medium. Derive the expression given by Eq. (5-47) for the albedo characterizing a slab of material of thickness a. In particular plot this albedo for a slab of water for various thicknesses. (Use thermal cross section data.) Comment on the behavior of the albedo as given by Eq. (5-47) for both very thin and very thick slabs. One defines the blackness coefficient characterizing a region as J+(a)~J-{a) J+(a)
9
THE ONE-SPEED DIFFUSION THEORY MODEL
/
229
where a denotes the surface of the region. Yet another useful parameter characterizing interfaces is the ratio of the current density J to the flux at the interface
y
5-13
5-14
5-15
5-16 5-17 5-18
. 5-19
5-20 5-21
~ f ( a ) '
Determine the relation between these parameters and the albedo, assuming that diffusion theory can be used to describe the material adjacent to the region of interest. (It should be remarked that one frequently uses these concepts to characterize very highly absorbing regions such as fuel elements or control rods in which diffusion theory will usually not be valid in the highly absorbing region.) In reactor analysis it is frequently of interest to determine the neutron flux in a so-called unit fuel cell of the reactor, that is, a fuel element surrounded by a moderator. As a model of such a cell characterizing a cylindrical fuel element, consider a fuel pin of radius a surrounded by a moderator of thickness b. For reasons that will become more apparent in Chapter 10, one assumes that the fission neutrons that slow down to thermal energies appear as a source uniformly distributed over the moderator—but not directly in the fuel. Furthermore it is assumed that there is no net transfer of neutrons from cell to cell—that is, the neutron current vanishes on the boundary of the cell (although the neutron flux will not vanish there). Determine the neutron flux in this cell geometry. In particular, determine the thermal utilization characterizing the cell by computing the fraction of those neutrons slowing down that is absorbed in the fuel. Consider a one-dimensional slab model of a fuel cell in which the center region consists of the fuel, and the outer regions consist of a moderating material in which neutrons slow down to yield an effective uniformly distributed source of thermal neutrons S0 neutrons/cm 3 -sec. Determine the neutron flux in this cell. In particular, compute the so-called self-shielding factor fs defined as the ratio between the average flux in the fuel to the average flux in the cell. Consider two isotropic point sources located a distance a apart in an infinite nonmultiplying medium. Determine the neutron flux and current density at any point in a plane midway between the two sources. Determine the infinite medium Green's functions or diffusion kernels characterizing cylindrical and spherical geometries. By representing a plane source as a superposition of isotropic point sources, construct the plane source kernel Gpl(x,x') by using the point source kernel Gpt(r,r'). Obtain an expression for the plane source diffusion kernel characterizing a finite slab of width a by solving for the neutron flux resulting from a unit plane source at a position x' in the slab. This can be most easily accomplished by seeking a separate solution on either side of the source plane which satisfies the vacuum boundary conditions at either end, and then matching these solutions at the source plane using the interface condition of continuity of flux and a discontinuity in the current density given by the source strength. Consider a neutron source emitting a monodirectional beam of neutrons into an infinite medium. Using one-speed diffusion theory, calculate the neutron flux in the medium. For convenience, locate your coordinate system with its origin at the source and align the x-axis along the source beam. Since the source is highly anisotropic, you cannot apply diffusion theory directly. Rather, compute the distribution of first scattering collisions of the source neutrons along the x-axis, and then assume that each of these collisions acts in effect as an isotropic point source of neutrons for the subsequent diffusion theory analysis (assuming that such scattering is isotropic). Use the method of variation of constants to determine the flux in a finite slab that contains a uniformly distributed neutron source. Construct the spatial eigenfunctions of the Helmholtz equation in spherical geometry.
230 5-22 5-23 5-24 5-25
5-26
5-27
5-28
5-29 1
5-30
5-31
.v \ !\ 5-32
5-33
5-34
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Construct the spatial eigenfunctions of the Helmholtz equation in infinite cylindrical geometry. Construct the spatial eigenfunctions of the Helmholtz equation characterizing a parallelepiped geometry. Demonstrate explicitly that the eigenfunctions for the slab geometry are indeed orthogonal. Consider a slab of nonmultiplying material containing a uniformly distributed neutron source. Determine the neutron flux in the slab: (a) by directly solving the diffusion equation and (b) using eigenfunction expansions. Demonstrate that these solutions are indeed equivalent. Determine the neutron flux in a long parallelepiped column of nonmultiplying material caused by a uniform source distributed across the face of the column. In particular determine the spatial behavior of the flux far from the source plane. Describe how one might measure the neutron diffusion length in the column by studying the spatial behavior of the flux. The objective of this problem is to write a computer code to calculate the flux in a uniform nonmultiplying slab containing an arbitrary source distribution. Perform the following steps: (a) Derive the finite difference form of the appropriate one-speed diffusion equation using vacuum boundary conditions. Use 30 mesh points and equal mesh spacing. (b) Write the equations in matrix form, A
THE ONE-SPEED DIFFUSION THEORY MODEL
/
231
Determine the thickness a for criticality when: (a) inner region is vacuum and (b) inner region is a medium with ko0 = 1 and same D and L as the outer slabs. 5-35 A bare spherical reactor is made with 2 3 5 U uniformly dispersed in graphite (p = 1.7) with an atomic ratio Nc/N25 = 104. For the cross section values given below, calculate the critical size and mass of the reactor according to one-group diffusion theory. If the reactor is modified by placing a cavity (vacuum) of half the total radius in its center, find the critical size for this case. Recalculate the critical radius if the center void is filled instead by a perfect absorber. Use as data: a f = 4.3 b, a a =0.003 b, a 2 5 = 105 b, a 2 5 = 584 b, and D = 0.9 cm. ^J-\5-36 A bare spherical reactor is to be constructed of a homogeneous mixture of D 2 0 and 235 ^ 1 U. The composition is such that for every uranium atom there are 2000 heavy water ^ molecules (i.e., o /iV25 = 2000). Calculate: (a) the critical radius of the reactor using one-speed diffusion theory (Data: tj 2 5 = 2.06, Z>d2o = 0.87 cm, ^ ° = 3 . 3 x 10" 5 c m - 1 , a f 2 ° = 0.001 b, and a 2 5 = 678 b.) and (b) the mean number of scattering collisions made by a neutron during its lifetime in this reactor. 5-37 There is strong motivation to obtain as flat a power distribution as possible in a reactor core. One manner in which this may be accomplished is to load a reactor with a nonuniform fuel enrichment. To model such a scheme, consider a bare, critical slab reactor as described by one-speed diffusion theory. Determine the fuel distribution Nf(X) which will yield a flat power distribution P (x) = wf2f(jc)(.*) = constant. For convenience, assume that fuel only absorbs neutrons and that it does not significantly scatter them. Also assume that all other materials in the core are uniformly distributed. 5-38 A one-dimensional slab reactor system consists of three regions: vacuum for x < 0 ; a multiplying core for 0<x
where Ax is the mesh spacing. Again use 30 mesh points and require a convergence criterion on k ^ of
k(n+l)
<€=10
After each criticality calculation, readjust the slab width a and recalculate After several such calculations, plot k e jj against a to determine the critical slab width a*. Compare this with the analytical expression for a*. 5-40 Investigate the convergence of the inner and outer iterations in the one-speed diffusion code developed in Problem 5-39 for the following modifications:
232
5-41 5-42
5-43 5-44 5-45 5-46
5-47
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
(a) Use the outer iteration source S^n\x) as the initial estimate in the inner iterations. (b) Determine the sensitivity of both inner and outer iterations to the convergence criteria. (c) Attempt to accelerate the outer iterations by source extrapolation. Verify the general first-order perturbation theory result Eq. (5-312). Why is the adjoint system introduced in developing the perturbation equations? Illustrate your answer with an example showing that only the use of the adjoint system will yield the desired result. Describe a reasonable experimental procedure by which one could measure the variation of neutron importance within a reactor core. Calculate the error in the critical mass of a bare homogeneous spherical reactor due to a 1% error in k. Assume that only the core size would be adjusted to give criticality. Calculate the relative worth of a control rod bank inserted axially into a cylindrical reactor core. Consider a critical bare slab reactor of thickness a which is composed of a homogeneous mixture of fuel and moderator. Estimate the reactivity change if a thickness 8x of the fuel-moderator mixture at a position x is replaced by pure moderator. What 8x at a distance from the centerline of x = 0.4 a is required to give the same reactivity change as a perturbation thickness S;e0 at the center of the slab? Variational methods can be used in a manner very similar to perturbation theory to estimate the multiplication of a given core configuration using only crude guesses of the flux in the core. For example, a useful variational expression for the multiplication of a core described by a one-speed diffusion theory is f dsr<}>(r)M
.
k~x=-
f d3r
Compare the accuracy of such a scheme for a slab of width a where the estimates of the flux or "trial functions"
6 Nuclear Reactor Kinetics
For a nuclear reactor to operate at a constant power level, the rate of neutron production via fission reactions should be exactly balanced by neutron loss via absorption and leakage. Any deviation f r o m this balance condition will result in a time-dependence of the neutron population and hence the power level of t h ^ reactor. This may occur for a number of reasons. For example, the reactor operator might desire to change the reactor power level by temporarily altering core multiplication via control rod adjustment. Or there may be longer term changes in core multiplication due to fuel depletion and isotopic buildup. More dramatic changes in multiplication might be caused by unforeseen accident situations, such as the failure of a primary coolant p u m p or a blocked coolant flow channel or the accidental ejection of a control rod. It is important that one be able to predict the time behavior of the neutron population in a reactor core induced by changes in reactor multiplication. Such a topic is known as nuclear reactor kinetics. However, we should recognize that the core multiplication is never completely under the control of the reactor operator. Indeed since multiplication will depend on the core composition, it will also depend on other variables not directly accessible to control such as the fuel temperature or coolant density distribution throughout the reactor, but these variables depend, in turn, on the reactor power level and hence the neutron flux itself. The study of the time-dependence of the related processes involved in determining the core multiplication as a function of the power level of the reactor is known as nuclear reactor dynamics and usually involves a detailed modeling of the entire nuclear steam supply system. Although we briefly discuss several of the more important "feedback" mechanisms involved in determining core multiplication later in this chapter, our dominant concern is with predicting the time behavior of the neutron flux in the reactor for a given change in multiplication. 233
234
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
The principal applications of such an analysis are not only to the study of operating transients in reactors, but also to the prediction of the consequences of accidents involving changes in core multiplication, and to the interpretation of experimental techniques measuring reactor parameters by inducing time-dependent changes in the neutron flux. One can roughly distinguish between two different types of analysis depending on the time scale characterizing changes in the neutron population or core properties. For example, one is interested in relatively shortterm changes possibly ranging f r o m fractions of a second up to minutes in length when analyzing normal changes in reactor power level (e.g., startup or shutdown) or in an accident analysis. By way of contrast, changes in core composition due to fuel burnup or isotope buildup usually occur over periods of days or months. Needless to say, the analysis required for each class of time behavior is quite different. The reader will recall that we have already considered a particularly simple example of nuclear reactor kinetics when we discussed the time behavior of the neutron flux in a slab reactor model in Chapter 5. Although this earlier analysis was useful for deriving the condition for reactor criticality, it is not valid for an accurate description of nuclear reactor kinetics, since it assumed that all fission neutrons appeared promptly at the instant of fission. As we demonstrate in the next section, it is essential that one explicitly account for the time delay associated with delayed neutrons in describing nuclear reactor kinetics. Fortunately the one-speed diffusion model we have been using to study reactor criticality is also capable of describing qualitatively the time behavior of a nuclear reactor, provided we include the effects of delayed neutrons. Indeed such a model is frequently too detailed for practical implementation in reactor kinetics analysis due to excessive computation requirements, particularly when the effects of phenomena such as temperature feedback are included. For that reason, we will begin our study of nuclear reactor kinetics by reducing the one-speed diffusion model still further with the assumption that the spatial dependence of the flux in the reactor can be described by a single spatial mode shape (the fundamental mode). Under this assumption, we can remove the spatial dependence of the diffusion model and arrive at a description involving only ordinary differential equations in time. This model is sometimes known as the point reactor kinetics model, although this is somewhat of a misnomer since the model does not really treat the reactor as a point but rather merely assumes that the spatial flux shape does not change with time. Although we will rely heavily on the point reactor kinetics model in our study of nuclear reactor time behavior, we will indicate its generalizations to include a nontrivial spatial dependence as well as feedback mechanisms. There are two other aspects of nuclear reactor kinetics that we will not touch upon in this chapter. The first topic involves the study of nuclear reactor control and includes not only the analysis of the various schemes used to adjust core multiplication but those mechanisms by which changes in the core power level can affect multiplication as well. The second topic involves the study of long-term changes in the core power distribution due to fuel depletion and the buildup of highly absorbing fission products in the reactor core. However the study of both of these subjects involves only a steady-state analysis of the neutron flux in the reactor—or, at most, a sequence of steady-state criticality calculations. Only the time-dependence of the slowly varying changes in core composition such as those due to fuel depletion must be explicitly considered. Therefore such topics can be more appropriately developed in later chapters.
NUCLEAR REACTOR KINETICS
/
235
I. THE POINT REACTOR KINETICS MODEL A. The Importance of Delayed Neutrons in Reactor Kinetics In our earlier treatment of the simple bare slab reactor (Section 5-III-C), we found that the neutron flux in such a system could be written as a superposition of spatial modes (or eigenfunctions) characteristic of the reactor geometry, each weighted with an exponentially varying time-dependence: * ( M ) = 2 X « P ( - V ) *„(>•)•
(6-i)
n
Here the spatial eigenfunctions were determined as the solution to the eigenvalue problem [Eq. (5-232)]: V \ + B n \ ( r) = 0,
^„(f s ) = 0,
(6-2)
while the time eigenvalues Xn were given by Xn = vDB2 + t ; 2 a -
(6-3)
These eigenvalues are ordered as — Xx > — A 2 > • * *. Hence for long times the flux approaches an asymptotic form 4>(r, /) ~ A x exp( — Xx t)
(r) = A, exp
Ur),
(6-4)
where we identify / = [ t > 2 a ( l + L 2 i? g 2 )] k=
—r =
1+ Li?2
* = mean lifetime of neutron in reactor,
/
—-
—— ^multiplication factor.
1+
L2B*
It would be natural to inquire as to just how long one would have to wait until such asymptotic behavior sets in. We can determine this rather easily by assuming that the reactor is operating in a critical state such that Xx = 0, and then estimating Xn. If we recall that for a slab B2=n2 (ir/&)2, then \n=-vD
(BZ-B*
)=-vD
{Bt-B*
)=-vD{n*~
1)(|)2'
(6-6)
Now in a typical thermal reactor, 5—300 cm, 3 X l O 5 c m / s e c , and D~ 1 cm. Hence the higher order Xn are of the order of 100-1000 s e c - 1 , which implies that the higher order spatial modes die out very rapidly indeed. We can utilize this fact to bypass our earlier separation of variables solution of the one-speed diffusion equation [Eq. (5-187)] by merely assuming a space-time separable flux of the form
(6-7)
where ^ ( r ) is the fundamental mode or eigenfunction of the Helmholtz equation,
236
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Eq. (6-2). If we substitute this form into the one-speed diffusion equation, we find that n(t) satisfies dn dt
In this sense, n(i) can be interpreted as the total number of neutrons in the reactor at time t [if we normalize /d 3 r^/j(r)= 1]. Actually since the normalization of n(t) is arbitrary, we could also scale the dependent variable n(t) to represent the total instantaneous power P ( t ) being generated in the reactor core at any particular time. That is, we could let n(t)^P(t)
= w{v2fn(t\
(6-9)
where w{ is the usable energy released per fission event. Since the reactor power level is usually a more convenient variable to monitor, we will frequently express the reactor time-dependence in terms of P(t). Equation (6-8) represents a somewhat simplified form of a more general set of equations, which we shall derive in a moment, that are commonly referred to as the point reactor kinetics equations. This term arises since we have separated out the spatial dependence by assuming a time-independent spatial flux shape ^ ( r ) . In this sense we have derived a "lumped parameter" description of the reactor in which the neutron flux time behavior is of the form of the product of a shape factor ^(r) and a time-dependent amplitude factor n(t), 1 ,
r =
<£>(r, /) = t?w(/) / i( ) ^"o
ex
P
(V)
Mr),
(6-10)
characterized by a time constant T = -j-^—r = reactor period. /c i
(6-11)
This model is of course identical to that developed in our qualitative discussion of chain-reaction kinetics in Section 3-I-B except that one-speed diffusion theory has now given us an explicit expression for the neutron lifetime I [Eq. (6-5)]. However there is, of course, something very important missing f r o m this model. For by assuming a fission term in Eq. (5-186) of the form p S ^ r , / ) , we have implied that all fission neutrons appear promptly at the time of fission. But we know that a very small fraction 0.7%) of such neutrons are emitted with appreciable time delay (recall the discussion of Section 2-II). Although these delayed neutrons are only of minor significance in steady-state critical reactors, they are extremely important for reactor time behavior. For if we recall that the prompt neutron lifetime / is typically of the order 10" 4 sec in a thermal reactor (10~ 7 sec in a fast reactor), then it is apparent that the reactor period predicted by this model would be far too small for effective reactor control. We can give a crude estimate of the influence of delayed neutrons on the reactor time behavior by noting that the effective lifetime of such neutrons is given by their prompt neutron lifetime / plus the additional delay time characterizing the /?-decay of their precursor. If we weight both prompt and delayed neutrons by their respective yield fractions, we can estimate an average neutron lifetime
NUCLEAR REACTOR KINETICS
/
237
characterizing all fission neutrons as 6
(6-12)
Using the delayed neutron data given in Table 2-3, we find that this average lifetime is typically •—0.1 sec, which is considerably longer than the prompt neutron lifetime 10~6— 10~ 4 sec. Hence delayed neutrons substantially increase the time constant of a reactor so that effective control is possible. This fact suggests a related idea; suppose we consider a reactor that is very slightly subcritical when only prompt neutrons are considered. Suppose further that the fraction /? of delayed neutrons provides just enough extra multiplication to achieve criticality. This fraction will, in fact, control the criticality—and hence the time constant. However if k — 1 > /?, the reactor will be critical (or supercritical) on prompt neutrons alone, and the reactor period should become very short, since the delayed neutrons are not needed to sustain the chain reaction. Obviously we should design a reactor such that this situation will never occur.
B. Derivation of the Point Reactor Kinetics Equations In actual fact, one cannot proceed so heuristically. We must first set up a set of equations describing the time dependence of the delayed neutrons. To this end, we must define the precursor atomic number density: Ct{r,t)d3r
= expected number of "fictitious" precursors of /th kind in d3r about r that always decay by emitting a delayed neutron.
(6-13)
Note that C(-(r,/) is only some fraction of the true precursor isotope concentration, since only a fraction of the /th isotope nuclei eventually decays by delayed neutron emission. For example, the 87 Br precursor described in Section 2-II is characterized by a fictitious precursor concentration (6-14)
C B r ( r , 0 = (-029) (.7) Br(r, t).
One can immediately write down a balance relation for these precursor concentrations by referring to our earlier discussion of radioactive decay to identify Number of precursors decaying in =
q (r, t)d3r,
(6-15)
3
d r / sec
Number of precursors being produced =/^S^r, in
t)d3r.
(6-16)
3
d r/sec
[This latter relation assumes the precursors don't migrate or diffuse before decay-
238
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
ing.] Hence our precursor concentration balance equation is just 9C,
= -\C,(r,f) +
(6-17)
N o w our old friend, the one-speed diffusion equation, can still be used to describe the flux—provided we treat the delayed neutron contribution to the fission source explicitly by writing it as
Sf(r, 0 = 0 - 0 > 2 ^ , 0 + 2 \ C , ( r , * ) . /=i
(6-18)
Hence our system of equations describing the neutron flux in a reactor including delayed neutrons is v
dt
- Z) V2<J> + 2a<J>(r, t) = ( \ - f i
t) + 2
\C,(r,/),
itm{
9C,
=
+
i=l,...,6.
(6-19)
We will apply these once again to the asymptotic situation in which both the flux and precursor concentrations can be written as separable functions of space and time
vn(t)xpl(r),
(6-20)
(*)*,('),
where ^,(r) is the fundamental mode of Eq. (6-2).If we substitute these forms into Eq. (6-19) and use our expressions for the prompt neutron lifetime / and multiplication factor k for a bare, uniform reactor as given by Eq. (6-5), we arrive at a set of ordinary differential equations for n(t) and C;(t):
d n
k ( l - D - i
4M
(6-21) dC
These are known as the point reactor kinetics equations and represent a generalization of Eq. (6-8) to include the effects of delayed neutrons. One frequently rewrites these equations in a somewhat different form by introducing two definitions. First we must define the mean neutron generation time: _ I generation time between birth of neutron A= — = k and subsequent absorption inducing fission.
(6-22)
If k~ 1, then A is essentially just the prompt neutron lifetime /. Next we define a very important quantity known as the reactivity, which essentially measures the
NUCLEAR REACTOR KINETICS
/
239
deviation of core multiplication from its critical value k= 1, f)(t) —
k(t)-l ^ — = reactivity.
(6-23)
Notice here that we have explicitly indicated that k and hence p may be functions of time. We will comment more on this in a moment. These definitions allow us to rewrite Eqs. (6-21) in perhaps their most conventional form dn dt dQ
—
p{t)-$ =i
(6-24)
[i,
= ^n{t)-\iCi(t),
i=l
6.
Hence we now have a set of seven coupled ordinary differential equations in time that describe both the time-dependence of the neutron population in the reactor and the decay of the delayed neutron precursors. Unfortunately the solution of this system of equations is not as straightforward as it might first appear for several reasons. First the reactivity p(t) is usually a function of time and in fact frequently depends on the neutron population n(t) itself. Hence the equations will generally be nonlinear. Furthermore the time constants characterizing the nuclear processes represented by the equations range all the way from A — 1 0 " 6 - 1 0 ~ 4 sec to the lifetime of the longest lived precursor, usually about 80 sec. These widely different time scales complicate even a direct numerical solution of Eqs. (6-24) since the time step size allowed in most standard numerical schemes (e.g., R u n g e Kutta or predictor-corrector) is primarily controlled by the smallest time constant —in our case, the prompt neutron lifetime. Since reactor dynamics usually occur on a time scale characterized by the delayed neutrons, such direct approaches tend to be quite inefficient and more sophisticated methods are frequently required.
C. Limitations of the Point Reactor Kinetics Model A number of questionable assumptions have entered into the derivation of these equations, such as the one-speed diffusion approximation, and a timeindependent spatial shape. Fortunately the point reactor kinetics equations can be derived in a much more general fashion in which such assumptions are not necessary. 1 Such derivations usually proceed from the transport equation itself and are also usually very formal. They lead, however, to the set [Eq. (6-24)] in which only the definitions of A, and p are changed. Hence provided one uses the more general expressions for these parameters, the point reactor kinetics equations can be regarded as having a much broader domain of validity. One major modification that must be introduced into the equations is to take some account of energy-dependent effects. These arise primarily because the delayed neutrons appear with somewhat lower energies than do the prompt fission neutrons (recall Figures 2-21 and 2-23). Hence in a thermal reactor, they do not have to slow down quite so far and therefore are characterized by a somewhat higher probability of inducing thermal fission (by as much as 20%). Of course, this may work in just the opposite direction in fast reactors, since delayed neutrons
240
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
usually appear with energies below the fast fission threshold. We can take account of these effects by characterizing each delayed neutron group by a slightly different fast nonleakage probability and a resonance escape probability, say - P f n l a n c * P l Then we would merely modify the definition of the parameters appearing in the point reactor kinetics equations as (6-25)
FNL>
(6-26)
\ ^ > \ p l P FNL> fitP'PiwL A — =
—
ftp'Pi — =
P ^ P = 2 AZ-l
NL 7 — r >
(6-27)
(6-28)
In large thermal power reactors, /?. is typically several percent larger than the physical value /?,. (although in small research reactors, may be as much as 20-30% larger than /?,). It should also be kept in mind that the delayed neutron yield fractions must be calculated as suitable weighted averages over the relevant mixture of fuel isotopes. In fact one will find that in most thermal spectrum reactors these averaged delayed neutron fractions will decrease with core life since the most common bred materials, 2 3 9 Pu and 2 3 3 U, have somewhat lower delayed neutron yields than 2 3 5 U. In 2 3 9 Pu/ 2 3 8 U-fueled fast reactors, the situation becomes much more complicated since there are as many as six fissioning isotopes, each characterized by six different precursor groups. T o handle 36 delayed neutron precursor concentration equations would be very complicated indeed. Fortunately the recent trend toward direct experimental measurement of isotopic yields as opposed to precursor group data should allow one to eventually solve directly for the isotopic concentrations of the major precursor isotopes (e.g., 87 Br, 88 Br, and 137 I) and only lump the minor isotopes into effective precursor groups. This would then eliminate the problem of isotopic averaging for several fissionable isotopes, and decouple the data used in the kinetics calculation more effectively from the particular reactor core composition. Perhaps the most serious approximation involved in the point reactor kinetics equations involves the assumption that the flux can be adequately represented by a single, time-independent spatial mode ^ ( r ) . It should first be noted that this shape function is actually not the f u n d a m e n t a l mode characterizing a critical system, but rather the fundamental mode characterizing the reactor core that has been subjected to a reactivity change away from critical. Nevertheless it is common to utilize a shape function ^ ( r ) characterizing a critical core configuration if the reactor is close to a critical state or on a truly asymptotic period. W h e n the changes in core composition are sufficiently slow, as in fuel depletion or fission-product poisoning studies, one can perform an instantaneous steady-state criticality calculation of the shape function ^ ( r ) , even though this shape will slowly change with time. Such a scheme is known as the adiabatic approximation? More elaborate procedures exist for including a time variation in the shape function. 3 However for rapidly varying transients in which spatial effects are
NUCLEAR REACTOR KINETICS
/
241
important, one is usually forced to solve the time-dependent neutron diffusion equation directly (at considerable expense). W e will return to discuss such spatially dependent kinetics problems later in this chapter. Of course one could proceed in a much more empirical fashion by noting that the point reactor kinetics equations hold for more general situations than we have considered, but then simply to postulate that the correct values for the parameters /?, A, and p are available (perhaps from experimental measurement). In this sense, all detailed considerations of the flux spatial shape are avoided (or, rather, swept under the carpet). Let us now briefly examine the reactivity p(t) = [k(t) — I]/k(t) appearing in the point reactor kinetics equations. We know that the multiplication factor k, and hence p, depends on the size and composition of the reactor. In our specific one-speed diffusion model of a bare reactor core, we have found an explicit form for k [Eq. (6-5)]. Hence by changing the size or composition—say by inserting or withdrawing a rod of absorbing material or adjusting a poison concentration—we can change p and hence control the reactor. In this sense, p will in general be a function of time partly under the control of the reactor operator. However for any reactor operating at power, p will also depend on the flux itself due to several factors. First, the power level will influence the temperature of the components of the reactor core. However the atomic concentrations of materials in the core depend sensitively on their temperature. As the temperature changes, they may contract or expand or change phase. This in turn will cause a change in the macroscopic cross sections—and hence in the reactivity. Furthermore temperature changes may directly affect the microscopic cross sections (e.g., via the Doppler effect). Finally, the atomic concentration of materials in the core will vary as fission products are produced or fuel nuclei are fissioned and depleted. This will also strongly influence the reactivity. Such processes whereby the reactor operating conditions will affect the criticality of the core are known as feedback effects and play an extremely important role in reactor operation. Stated mathematically, such feedback effects imply that the reactivity must be regarded as a nonlinear function of the power level p[n(t),t]. Hence the point reactor kinetics equations are actually a coupled set of nonlinear ordinary differential equations that are extremely difficult to analyze analytically, with the exception of certain very simplified model cases. Although we will eventually discuss the physics of several of the more prominent feedback mechanisms, we will initially limit our study to those cases in which p(t) is a specified function of time (and hence the point reactor kinetics equations are linear). Such a situation is commonly referred to as the zero-power point reactor model, since it ignores the feedback that would occur due to variations in the reactor power level. Although limited in this sense, this model does reveal quite a bit about the time behavior of the neutron population in a reactor.
II. SOLUTION OF THE POINT REACTOR KINETICS EQUATIONS A. Solution With One Effective Delayed Group We will first apply the point reactor kinetics equations to the situation in which the reactivity is a prescribed function of time. In fact we will begin with the
i
r
242
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
very simple situation in which we imagine a reactor operating at some given power level P0 prior to a time, say ? = 0, at which point the reactivity is changed to a nonzero value p 0 . Rather than attempting to solve the full set of point reactor kinetics equations for this situation, we will first consider the case in which all delayed neutrons are represented by one effective delayed group, characterized by a yield fraction (6-29)
and an averaged decay constant A
(6-30)
The point reactor kinetics equations for this simplified case become
P(t) +
Po-P
dP dt
\C(t), (6-31) t > 0.
f - P p U - X C U ,
Note that we have chosen to work with the instantaneous reactor power level P ( t ) as our dependent variable. The precursor concentration is then slightly modified to C = C n e w = w f 2 f C o l d . N o w prior to t = 0, the reactor is operating at a steady-state power level P0. Hence we find that for t < 0 we must require dP
dC
n
P _
r
(6-32)
This relationship yields the appropriate initial conditions that accompany Eq. (6-31): P(0) = Po,
C(0) = ^ - P 0 .
(6-33)
This simple initial value problem can be solved in a variety of ways (the easiest being by Laplace transforms as discussed in Appendix G). We will use a more pedestrian approach by merely seeking exponential solutions of the form P(t)
= Pes',
C(t)=Ces',
(6-34)
where P, C, and s are to be determined. Then if we substitute these forms into Eq. (6-31), we find the algebraic equations sP
P + AC,
(6-35) sC=
VP-AC. A
NUCLEAR REACTOR KINETICS
/
243
This set of homogeneous equations has a solution if and only if
t+y-^-o. or As2 + (AA +
(6-36)
- p 0 ) j - PqK = 0.
Hence we have arrived at a characteristic equation for the parameter s. In particular we find that there are two possible values for s 1
- ( ^ - p
0
+ A A ) ± V ( i 8 - p 0 + A A ) 2 + 4AAp (
(6-37)
and hence our general solutions will be of the form P (t) = PY exps{t + /*2 exps 2 /,
(6-38)
C ( ? ) = C j e x p j ^ - I - C 2 exps 2 /. To determine the unknown coefficients we can apply both the initial conditions and the equations (6-35) to obtain four algebraic equations for the four unknowns Pu P2, C\, and C 2 . Since these results are still rather cumbersome, we will examine the solution in the case in which ( / J - p 0 + AA) 2 »4AAp 0 (or A A / / ? < 1 ) and |p 0 |
fi-Po
while the power P(t)
becomes A
p(t)*p
(6-39)
9
p (
exp
fi-Po
Po
p-Po
11-
(
Po
\
(
-P-Po\
.
(6-40)
We have sketched this solution in Figure 6-1 for both positive and negative reactivity insertion of an amount |p 0 |—0.0025 into a reactor characterized by £ = 0.0075, A = 0.08 sec" 1 , and A = 1 0 " 3 sec. In particular, it should be noted that after a very rapid initial transient C$ 2 - 1 ~ .2 sec), the reactor time response becomes exponential with a period of 25 sec. This time constant characterizing the more slowly varying asymptotic behavior is occasionally referred to as the stable reactor period.
B. The Inhour Equation The reciprocal time constants sx and s2 characterizing the single delayed group model were given as the roots of a quadratic equation (6-36) in terms of p 0 , /?, A, and A. W e can generalize this result to the situation in which there are several groups of delayed neutrons by first rewriting Eq. (6-36) in a slightly different form.
244
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
t{ sec)
FIGURE 6-1.
Reactor power level behavior following a positive step reactivity insertion
Using our definitions of p and A in Eqs. (6-22) and (6-23), we can rewrite Eq. (6-36) as
^
, It/'
_
'
+
^T+TIT+a)'
<W1>
' \ '
This equation will determine the decay constants s for any constant reactivity p 0 . T o generalize to six delayed groups, we write
P o =
7 + T
+
^7+T 2 /-I
1
=p(s)-
(6-42)
This equation, which expresses the decay constants s- as the roots of a seventhorder polynomial, is known in reactor theory as the inhour equation. [This terminology arises from the very early attempts to express reactivity in units of "inverse hours" or inhours, which were defined as the amount of reactivity required to make the reactor period equal to one hour. Reactivity is more commonly expressed today either in decimals or percentages or pcm (per cent mille= 10~ 5 ) of A k / k . W e will later introduce another unit called the dollar, in which p is measured in units of the delayed neutron fraction /?.] The roots of Eq. (6-42) are most conveniently studied using graphical techniques. In Figure 6-2 we have plotted the right-hand side of Eq. (6-42) for various values of s. The intersection of these curves with the line p = p 0 yields the seven decay constants j. characterizing the time-behavior 7
P(t)=
2 PjOipsjt.
(6-43)
7=1
The root lying farthest to the right, s0, is identified as the reciprocal reactor period, sx=T~l. It is apparent from Figure 6-2 that only this root will ever be positive. The remaining roots sJ9 j>l, can then be identified as transients that die out rapidly after a reactivity p 0 is inserted into the reactor.
NUCLEAR REACTOR KINETICS
/
245
It should be noted that the range of p 0 is bounded by unity, / k - i - cc < p = — - — < 1.
(6-44)
In the limiting cases we find Po
(critical)
= 0^> 1 y 1 = 0
p 0 -> 1 = > 5 ^ 0 0
(supercritical)
oo = > 5 , ^ —A!
(subcritical)
This last limit is particularly interesting because it implies that no matter how much negative reactivity we introduce, we cannot shut the reactor down any faster than on a period T— 1 /\X determined by the longest-lived delayed neutron precursor. In 235 U fueled thermal reactors, Af ^ 8 0 sec. Several other limiting cases are of particular interest: (1) Small reactivity insertions (p 0
A/V /=i
Thus the reactor period is given by 6
T=
J- = -I
s,
Po
/+" /= 1
<0
Po
k - 1 '
(6-45)
246
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
However this is of course identical to the result suggested by our earlier heuristic argument. Hence for small reactivities the reactor period is determined essentially by the average neutron lifetime including delayed neutrons. (2) Large positive reactivity insertions ( p 0 » / 3 ) : In the opposite extreme of large reactivity insertion, we can assume to write the inhour equation (6-42) as S,
P o - — —
/ - I
+ - 7 7 T 7
x
Sx + I
6
2
S. +
BL~L
A - — — r '
Sx + /
sl +1
(6"46)
1
Hence we can solve for the reactor period as r =
~ « Tk 7 — (Po~P)
k
~
l
>
(6"47)
which is just the result we would have obtained by totally ignoring delayed neutrons. Thus for large positive reactivity insertions, the reactor response is determined essentially by the prompt neutron lifetime /. (3) p 0 = / 3 : This is essentially the "break point" between a reactor kinetic response controlled by delayed neutrons and that governed by prompt neutrons alone. If we refer to the point reactor kinetics equations (6-24), it is apparent that for the reactor to be critical on prompt neutrons alone, we would require p 0 = A - For p 0 < / 2 the delayed neutron contribution is needed for reactor criticality, and hence the time response of the reactor will be determined to a large extent by the time delay characterizing the precursor /?-decay. Some common terminology is to refer to the range 0 < p < fi as delayed critical, while p > (3 is referred to as prompt critical or prompt supercritical. Obviously it is very important to avoid this latter situation in reactor operation, since the reactor time response would be very rapid if prompt criticality were exceeded. Actually the transition between these two classes of time response is not quite so abrupt. In Figure 6-3 4 we have shown the asymptotic reactor period 7 1 = j f 1 plotted versus p 0 for positive reactivity insertions in 2 3 3 U, 2 3 5 U, and 2 3 9 Pu-fueled cores. The rather dramatic decrease in the reactor period in the neighborhood of prompt critical p 0 = (3 should be noted. The significance of the prompt critical condition has led to the custom of measuring reactivity in units of ji. More precisely, a p = ft is referred to as one dollar ($) of reactivity. For example, a reactivity of $.40 would correspond to p = 0.4^ = 0.0028 A&/& = 280 pcm for a 2 3 5 U-fueled thermal reactor.
C. The Inverse Method5 There are very few problems for which it is possible to obtain an exact solution for P(t) given a specific p(t). Actually it is frequently more appropriate to invert the problem by determining that p(t) which will yield a desired P(t) behavior, since this is more in line with the philosophy of reactor control.
NUCLEAR REACTOR KINETICS 10 - I
IV 11 I iV|i] i l y u
t ivjn i rat ] i i 111 i i II
N
" / = 10~ 8
NO"7
So-6
10-5
MO" 4
247
i i 1 | * 1 111 1233
-
/
U :
s e c ^ \
10" 1
I
10-
2 I Mil I I III I I III I I III I Mil 1 Mil 1 Mil 1 1 11 i 111 K i 1cr6 10"5 10"4 10"3 10"2 1o~ 1 1 10 1o 2 1o 3
Asymptotic period (T), sec
10-6
10-5
10" 4
10"3
10"2
10 _ 1
1
10 2
10
10 3
Asymptotic period (T), sec 10 - 1 1 VI 1 1 \ l | 1i
• i i VT" i—r\nj i i 111 i i II | i 1 III 1 1 111 1239pu "
-
I ^ J
- / = 10"8
S o - 7 V 1 0 - 6 \ 1 0 - 5 \ -j Q—4 \ 1 0 - 3 s e c ^ S
-
10_1
"S CO 0)
CC
10 10"6
FIGURE 6-3.
I I11 1 1 l\l , .,l . . i.l , I 1 1 I 1 Mil 1 1 I 1 1 1 1 I 1 1 1 1 Mil 1O" 5 10"4 10"3 10" 2 10" 1 1 10 10 2 1o 3 Asymptotic period (T), sec Reactor period versus reactivity for fissile isotope delayed-neutron data 4
In order to solve for p(t) in terms of P(t), let us derive yet another form of the point reactor kinetics equation. Suppose we begin by formally solving the equations for the precursor concentrations in terms of P(t):
C,.(0= f
d f - ^ P ( O e x p - M t - 0 = f°V|Uxp(-A,.T)i>(/-T),
(6-48)
J - oc
where we have implicitly assumed that C,(7) exp\?—>0 as t-*—cc and then let r = t — tf to obtain the second integral. [Note that if we wish to take explicit account of initial conditions on C(.(f)s say at a time t—t0y then we would merely separate out the contribution in the first form of the integral in Eq. (6-48) from t = — cc to t = t0 to represent C^t^).] We will now substitute this into the first of the point reactor kinetics equations to write dP dt
p(Q-jS
A
P(t)+
/ I
dr Z j ~ r
e x
P
- a
<
t
P(t-r).
(6-49)
248
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
If we define the delayed neutron kernel D (r) 6
= 2
XB "Vexp-V,
i-i
(6-50)
p
(noting that D (r) dr is just the probability that a delayed neutron will be emitted in a time dr following a fission event at t = 0 ) then we arrive at an "integrodifferential" form of the point reactor kinetics equations dP dt
P(0~j8 A
P
We can now rearrange this equation to yield p(t) in terms of
r
d
= P+
P(t):
P(t~r)
x
(6-52)
This particular relationship is important for two reasons: (a) it can be used in principle to determine the time-dependence of the applied reactivity required to yield a specific power variation—that is, to program the control rod motion and (b) the interpretation of the measured power responses in transient analyses of reactivity changes can be used to provide information about the feedback mechanism in the reactor. Several specific applications of this relationship are of considerable interest: 1. PERIODIC POWER VARIATION
Suppose we seek the reactivity that will yield a sinusoidally varying power of the form P (t) = P0+ Px sin cot.
(6-53)
(Note that we must require P0>PX since a negative power would offend our physical intuition.) Inserting this expression into Eq. (6-52) and performing a few manipulations (left for the reader's enjoyment to the exercises at the end of the chapter), we arrive at
Px
P ( ' ) = p 1 ! Y(i<*)\
sin(wf — <j>) p
(6-54)
1 + — sin o)t "o
where
(6-55)
and Y(ios) is a function which looks suspiciously like a chunk of the inhour
NUCLEAR REACTOR KINETICS
/
249
equation
y (/«) = /« a + 2 7/co + X, a
(6-56)
(It will cross our path again.) Notice in particular that the reactivity insertion that gives rise to a purely sinusoidal power variation is periodic—but not sinusoidal (at least for large power variations). One can show, in fact, that p ( 0 has a negative bias. This fact proves of some importance in reactor oscillator experiments in which an absorbing material is oscillated within the core and the core power response is then measured in an effort to infer system parameters (e.g., /? and A). 2. REACTIVITY A F T E R A POSITIVE POWER TRANSIENT
As a second application of Eq. (6-52), suppose we determine the reactivity present following a power transient in which the reactor power increases from its initial value P0 at t — 0, and then decreases back to P0 at a later time t0
p(t0) = A
'o
1 dP
i
£ I
dt
dr D ( t )
n'o-O-^o
(6-57)
Now we know that for a positive transient, P(t0— r)> P(t0) = P0 and hence the integral is always positive. Furthermore it is apparent that at t = /0, the slope dP/dt | , o < 0 . Thus we can infer that the reactivity following a positive power excursion must, in fact, be negative. That is, the reactor must be taken subcritical to return the power to its original level. 3. R A M P REACTIVITY INSERTION
Suppose the reactor power level is found to increase very rapidly from an initial level P0 as P 0 exp(a* 2 ) for f > 0 . Then using Eq. (6-52) we can find that for long times /, p(t)^f]
+ 2Aat.
(6-58)
Hence p(t) approaches a linear function of time—that is, a ramp insertion of reactivity above prompt critical. We can turn this calculation around to infer that the response of the reactor power P(t) to a ramp insertion p(t)=*yt should behave as e x p [ y f 2 / 2 A ] for long times. This conclusion has particular relevance for certain classes of fast reactor accident models in which fuel melting is assumed to lead to core reassembly in a prompt supercritical configuration with a ramp reactivity insertion. Obviously the power increase resulting from such an occurrence would be very rapid indeed.
D r Approximate Solutions As we have seen, exact solutions of the point reactor kinetics equations are known for only a few special reactivity insertions. Hence we now turn our attention to approximate schemes for solving these equations in the absence of feedback.
250
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR 1. C O N S T A N T D E L A Y E D N E U T R O N P R O D U C T I O N R A T E APPROXIMATION
In certain problems, such as when the reactor is shut down by rapid insertion of the safety rods, we are interested in the response of the reactor power to a given reactivity insertion in short time intervals following a time t0. During such short time intervals we can ignore the change in the rate of production of delayed neutrons, replacing C,(/) by C^O). Hence in this approximation, the point reactor kinetics equation becomes dP dt
P{t)+Q{t),
(6-59)
where the effective source of delayed neutrons Q(t) is now presumed known and given by g ( 0 -
2\c,(0). i=\
(6-60)
However we now just have a first-order inhomogeneous differential equation that we can easily integrate for any given p(t). A rather interesting application of this result is to the case of a reactor scram in which control rods are rapidly inserted into the reactor core to shut the reactor down in an emergency situation such as the loss of primary coolant flow. Since the rod insertion takes a finite time, we cannot really treat the reactor scram as a step reactivity change (as in Section 6-II-1). A more reasonable model is to assume a negative ramp insertion, that is, p ( f ) = ~~ yt. If we substitute this into Eq. (6-59) and solve this equation subject to the initial condition imposed by steady-state power operation prior to the scram, 2\.C,.(0)=f/>0,
(6-61)
then we find that the power level decreases as
(6-62) After the scram rods have been fully inserted, the negative reactivity becomes constant, and one can determine P(t) for subsequent time using our earlier solutions of the point reactor kinetics equation for constant p(t) = —p0. 2. T H E P R O M P T J U M P A P P R O X I M A T I O N
In our earlier study of the response of the reactor power P(t) to a step change in reactivity, we found that there was initially a very rapid transient behavior on a time-scale characteristic of the prompt neutron lifetime, followed by a more slowly varying response governed by delayed neutron behavior. Since this transient is so rapid, a very useful approximation to make for systems below prompt critical is
NUCLEAR REACTOR KINETICS
/
251
one in which the prompt neutron lifetime is essentially taken to be zero such that the power level jumps instantaneously to its asymptotic behavior (recall Figure 6-1). This so-called prompt jump approximation is effected by merely neglecting the time derivative dP/ dt in the point reactor kinetics equation
0=[PM-/?]i>(0+Ai\.c,.(0 1-1
(6-63) dC,
B,
Since the delayed neutron production cannot respond immediately to a step change in reactivity, this model predicts that a reactivity j u m p from px to p 2 causes an instantaneous change in reactor power from Px to P2 as given by
i
r
=
M
i
p
r
—
-
Pi
/
;
-
?
"
,
(
;(
6
"
" '
6
4
)
' (
The prompt j u m p approximation is frequently used in numerical studies of the point reactor kinetics equations since it eliminates the very short time scale due to A which plagues finite difference methods. However it is also of use in simplifying analytical estimates based on the one-effective delayed group model, since in this case the precursor concentration C(t) can be eliminated to find a simple first-order differential equation for the power P(t)
M O - 0 ] f
+
MO
P(t) = 0.
(6-65)
If p(t) is given, we can again solve for P(t) as P(t) = P(0)eA°\
(6-66)
where
A(t)= dT j0 l^^r f
rp(r) + Ap(r)l
For example, for a ramp insertion p(t)=yfit, implies
P
•
"
the prompt j u m p approximation x
P(t) = P(0)e~Xt[l-yt]-{X+y).
(6 67)
' '
"
(6-68)
The prompt j u m p approximation is frequently found to yield an adequate description of reactor kinetic behavior. Numerical solutions of the point reactor kinetics equations have demonstrated the approximation is valid to within about 1% up to reactivity insertions of p 0 = $0.50. 6
252
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
3. SMALL A M P L I T U D E APPROXIMATION (LINEARIZATION)
Suppose that we assume that small reactivity variations will produce only small changes in the reactor power f r o m its equilibrium value P0. We already know that this assumption is not true for a critical reactor since even a slight positive step in reactivity gives rise to an exponentially increasing power response that eventually grows beyond any bound. However the assumption will still be valid if we consider only short times following the step insertion. Furthermore for certain classes of reactivity changes such as a periodic reactivity insertion with an appropriate negative bias, the resulting power variations remain small for all times. In this case, the point reactor kinetics equations reduce f r o m a set of linear O D E s with variable coefficients to a set of linear O D E s with constant coefficients. Consider again the integrodifferential f o r m of the point reactor kinetics equations dP dt
(6-69)
If we now let p(t) denote the power variations about a reference level P0 P(t) = P0+p(t),
(6-70)
then substitution of Eq. (6-70) into Eq. (6-69) and use of the identity
J yields (after a bit of manipulation) dp p(t) i = —
p(*)p(t)
+
r 00
drD(r)=
1
(6-71)
0
8 roo + fjT * D ( r ) [ p ( t - r ) - p m
(6-72)
Thus far our analysis has been exact, merely consisting of a bit of algebraic manipulation. We now introduce our key approximation by assuming that p(t) and p(t) are sufficiently small that we can neglect the second-order term p(/) p(t) to write
idp ~p ( 0 o UB <*rl>(r)[p('-r)-p(0l =
P
+
(6-73)
This approximation is sometimes (incorrectly) referred to as the linearization approximation. Actually both the original point reactor kinetics equation as well as Eq. (6-72) are already linear, provided we are given p{t). All we have done is to remove p(/) as a time-dependent coefficient in the equation and replace it by a time-dependent inhomogeneous term p(t)P0/A. Equation (6-73) forms the basis of nuclear reactor stability analysis, since it relates the response of the reactor power to a small change in reactivity. At this point one can essentially turn the mathematical analysis of this equation over to the systems (or control) engineer who will then proceed to hack away at it using the powerful tools of linear systems analysis. 7 It is not our intent here to venture very far into this topic.
NUCLEAR REACTOR KINETICS
/
253
However with the recognition that many readers of this text (particularly electrical engineers) will already be acquainted with methods of linear systems analysis, we do feel it useful to provide a very short discussion of this topic in an effort to bridge the gap to reactor kinetics. 8 ' 9 (Other readers may wish to bypass the remaining discussion of this section.) The key tool used in linear systems analysis is the Laplace transform (defined and discussed in Appendix G). W e will define the Laplace transform f(s) of a f u n c t i o n / ( 0 by f(s)=f a e - y C f ) .
(6-74)
Then we can easily Laplace transform the "linearized" reactor kinetics equation (6-73) and solve for p(s) =
(6-75)
P0Z(s)p(s)
where we have defined the zero power transfer function Z (s) -l ' 1 v
'
s
1
A
j- 1
S
+ X: J
(6-76)
Y(s)'
Hence to determine the behavior of the reactor power, we need only study the poles of Z(s) and £(.?). However there is a great deal more we can do by employing the very powerful methods of linear systems analysis. In particular we can study the stability of the reactor when it is operating at power (i.e., when we introduce feedback). The concept of a transfer function is much more general. The response (or output) of any physical system to a signal (or input) applied to it can be expressed in terms of such transfer functions. More precisely, we define Laplace transform of response Transfer function = — — — = Z (J). Laplace transform of input
(6-77)
This can be conveniently represented by a "block diagram."
Input (cause)
Z(s)
Output (effect)
In our case, the input is the reactivity p(t), while the output is the fractional power change p(t)/P0. Of course other choices are possible, such as the inlet coolant temperature and pressure. The transfer function defined for a linear system in this way is sometimes called the "open-loop" transfer function, since we have assumed that the output (power level) does not affect the input (reactivity) in any way. Later we will consider the
254
/
T H E ONE-SPEED DIFFUSION M O D E L O F A NUCLEAR REACTOR
case in which we allow feedback effects:
Input
(That is, we "close" the feedback loop.) It should be noted that the concept of a transfer function allows one to use the principle of superposition to write the total output resulting f r o m several inputs as the sum of the individual output f r o m each of the inputs. In particular since £ {§(/)} = 1, the response to a S-function reactivity input is just the inverse Laplace transform of the transfer function itself.
/
exp s t
PO
= 2(/).
(6-78)
7=2
A +
Here, %{t) is the so-called unit impulse response, that is, the power response to a unit impulse reactivity insertion. The Sj are the roots of the inhour equation, while we have noted that for a critical system, s 0 = 0. One can now use the convolution theorem (Appendix G) to reexpress Eq. (6-75) in the "time domain" as p(0-rof'**(<-r)f>(r).
(6-79)
Note that Z ( t - r ) is just the Green's function for the linearized point reactor kinetics equation. Hence it is not surprising that Z(t) plays an extremely important role in the study of nuclear reactor kinetics. In particular one can infer the stability properties of a linear system by studying %(t). One defines such a linear system to be stable if its response to any bounded input is also bounded. It is possible to show (see Problem 6-27) that a necessary and sufficient condition for stability is for r oo
/
\%(t)\dt
(6-80)
In this regard it is interesting to note that since the zero power reactor response function £ ( / ) - * A _ 1 as t->oo, the integral of %(t) over all time is not bounded. This then implies that a critical reactor without feedback is unstable with respect to bounded reactivity inputs. However as we will see later, there are always sufficient negative reactivity feedback mechanisms present in reactors to render them stable under almost any conceivable operating conditions.
NUCLEAR REACTOR KINETICS
/
255
As an example of the use of the reactor transfer function, let us determine the response of the reactor to a sinusoidal reactivity input p(t) =
(6-81)
Spsinut.
Then if we note that the Laplace transform of this input is just Spa) p(s) =
(6-82)
* 2 +
we can easily invert Eq. (6-75) to find p(0
exp Sjt = 8p\Z (/to)| sin(atf +
^
(6-83)
where <£(
p(t) ~
= 8p\Z(io))\ sin(co* + )+
So
.
(6-84)
Notice in particular that there is a shift in the average power level of P0Sp/o)A. If a reactor operating at a steady-state power level is subjected to a sinusoidal perturbation in reactivity, the power will oscillate with the source frequency, but with a phase shift <J>(to) = a r g { Z } (actually a phase lag) and an amplitude proportional to G(
F. Some General Comments on Applications of the Point Reactor , Kinetics Equations The point reactor kinetics equations are commonly applied to analyze most short-term transients in nuclear reactor behavior. The popularity of the point reactor kinetics model is not due necessarily to its validity (which is limited), but rather the fact that alternative methods that also treat the spatial dependence of the flux are usually prohibitively expensive to use in the large variety of problems typically encountered in a core design effort.
256
/
T H E O N E - S P E E D DIFFUSION M O D E L OF A NUCLEAR REACTOR
0.01
0.1
1
10
100
1000
O) FIGURE 6-4.
Gain and phase shift of the zero-power reactor transfer function
Most experimental measurements on reactors involving a reactivity change can be analyzed via the point reactor kinetics equations. Indeed since such experiments are usually performed at zero power, it is frequently not necessary to append to the point reactor kinetics equations suitable models of reactivity feedback. Although we will briefly review a variety of experimental techniques commonly used to measure reactor kinetics parameters later in this chapter, we would merely mention here as an example experiments in which the reactivity worth of control rods are measured by withdrawing a rod and then determining the resulting asymptotic reactor period. The inhour equation can then be used to relate this period to the worth of the control rod. Normal operating transients in nuclear power reactors are usually very easy to analyze, since they are typically rather slow. This is because the rate of change of reactor power level in a large plant is limited not by neutronic considerations, but rather by the rate of temperature change allowable in nonnuclear plant components. For such slow transients (typically a 5% change in power per minute or less for maneuvering) reactivity insertions are kept far below prompt critical, and hence the prompt j u m p approximation, in which the mean generation time A is set equal to zero, is adequate. Furthermore on such a time scale the spatial power shape usually remains in a fundamental mode and hence the point reactor kinetics equations are valid. The solution of the point reactor kinetics equations for such slow transients does necessitate consideration of reactivity feedback, however. Frequently one finds the point reactor kinetics equations augmented with transient
NUCLEAR REACTOR KINETICS
/
257
models of the remainder of the plant suffices for on-line system performance studies in the plant itself (using hybrid digital-analog process computers). The most stringent application of the point reactor kinetics equations is in the analysis of hypothetical reactor accident situations in which reactivity insertions may be quite large (several $) and power transients very rapid. Indeed one frequently encounters situations in which such equations are not valid. Typically one uses them to provide a parameter study of the effects of various input data such as reactivity insertions on reactor behavior under the postulated accident conditions. Occasionally spatially dependent kinetics calculations are performed to check the predictions of the point reactor model. Obviously an effort is always made to generate conservative predictions. For most severe transients in thermal reactors, such as a loss of coolant accident or the ejection of a control rod, the usual transient thermal-hydraulics models are adequate to supplement the core neutronics calculations. However in fast systems very severe transients may lead to fuel melting and slumping with an appreciable change in core configuration. Indeed such core rearrangements may produce large positive reactivities, leading to very large energy release, followed by core disassembly (that is, a small nuclear explosion). Such severe accidents, although quite remote in probability, must nevertheless be studied in reactor analysis. They require not only models of the core neutronics (e.g., the point reactor kinetics equations) and thermal-hydraulics, but as well models of the physical motion of the core due to melting and disassembly. Needless to say, such analyses can become very complicated indeed.
III. REACTIVITY FEEDBACK AND REACTOR DYNAMICS A. Mathematical Description of Feedback Thus far we have assumed that the reactivity p(t) appearing in the point reactor kinetics equations is a given function of time. However we know, in fact, that it depends on the neutron flux (or power level) itself. This dependence arises because the reactivity depends on macroscopic cross sections, which themselves involve the atomic number densities of materials in the core:
2(T,t)
=
N(T9t)o(T,t).
(6-85)
Now it is easily understandable how the atomic density N (r, t) can depend on the reactor power level, since: (a) material densities depend on temperature T, which in turn depends on the power distribution and hence the flux and (b) the concentrations of certain nuclei is constantly changing due to neutron interactions (buildup of poison or burnup of fuel). However it should also be noted that we have explicitly written the microscopic cross sections as explicit functions of r and t. This must be done since the cross sections that appear in our one-speed diffusion model are actually averages of the true energy-dependent microscopic cross sections over an energy spectrum characterizing the neutrons in the reactor core. And
258
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
this neutron energy spectrum will itself depend on the temperature distribution in the core, and hence the reactor power level. Such reactivity variation with temperature is the principal feedback mechanism determining the inherent stability of a nuclear reactor with respect to short-term fluctuations in power level. Evidently our first task in constructing a model of temperature feedback is to determine the temperature distribution in the core. Of course, one could begin by writing the fundamental equations of heat a n d mass transport characterizing the reactor. That is, one could write the equations of thermal conduction and convection using the fission energy deposition as the source of heat generation. The motion of the coolant through the core would then be described by the equations of hydrodynamics. As we will find in our detailed development of thermal-hydraulic core analysis in Chapter 12, this fundamental approach results in a formidable set of nonlinear partial differential equations unless further simplifications are introduced. Hence it has become customary to avoid the complexity of a direct solution of the full set of thermal-hydraulic equations by replacing the spatially dependent description by a "lumped parameter" model similar in philosophy to the point reactor kinetics model. The reactor core is typically characterized by several average temperatures, such as an average fuel temperature, moderator temperature, and coolant temperature. One then models a simple dependence of the reactivity on these temperature variables. Of course one must append to the original point reactor kinetics model equations describing the changes of these core temperatures produced by changes in the reactor power level. Typically such calculations are based on a model of a single average fuel element and its associated coolant channel, and describes the conduction of fission heat out of the fuel element and into the coolant, and then its subsequent convection out of the reactor core. We will consider such models in some detail in Chapter 12. Our more immediate concern in this chapter, however, is to study how the results of such thermal-hydraulic models, namely the core component average temperatures such as TF (fuel) and TM (moderator or coolant), can be used in suitable models of reactivity feedback. T o this end let us return to consider our point reactor kinetics model. W e will write the reactivity p(t) as a sum of two contributions p(t) = 8peJt)
+
8p{[PL
(6-86)
The 8p notation signifies that the reactivity is measured with respect to the equilibrium power level P0 [for which p = 0]. Furthermore, 8pext(t) represents the "externally" controlled reactivity insertion such as by adjusting a control rod. §p f [.P] denotes the change in reactivity corresponding to inherent feedback mechanisms. This latter component is written generally as a functional of the reactor power level. When the reactor is operating at a steady-state power level PQ, then there will be a certain feedback reactivity p f [P 0 ]. Since in almost all cases this will correspond to a negative reactivity, it is customary to refer to p f [P 0 ] as the power defect in reactivity. T o sustain the criticality of the system, we must supply a counteracting external reactivity p 0 (such as by withdrawing control rods) such that (6-87)
NUCLEAR REACTOR KINETICS
/
259
In this sense then, we define our incremental reactivities as 5
Pext(0 = Pext(0-Po>
8p{[P]
= ps[P]-Pr[Po\-
(6-88)
It is also useful to recall our definition of the incremental power p(t) = P(t)-P»
(6-89)
From Eq. (6-88) we note that Spf[/? = 0] = Sp f [P 0 ] = 0. We can now sketch the block diagram characterizing a reactor with feedback as shown in Figure 6-5. In particular, it should be recalled that we have already analyzed the "black box" describing the reactor kinetics, since this is just given by the point reactor kinetics equation without feedback, which can be most conveniently written in the form Eq. (6-72). This equation can be regarded as d e t e r m i n i n g p ( i ) for any p(t). Of course we commonly analyze this equation under one of a variety of approximations, for example: (a) ignoring delayed neutrons ( D = 0 ) for very large reactivity insertions, (b) prompt j u m p approximation, and (c) linearization for small reactivity insertions, in which case we can solve Eq. (6-73) in terms of the open loop or zero-power transfer function to find p\p] = P0fdT%(t-T)p(T).
(6-90)
We therefore turn our attention to the study of the black box describing the feedback functional Spf[/?].
B. Models of Temperature Feedback 1. TEMPERATURE COEFFICIENT OF REACTIVITY
Of most concern in the study of short-term reactivity feedback is the effect of the core temperature T on the multiplication of the core. One usually expresses this in terms of a temperature coefficient of reactivity aT defined as dp =
(6-91)
pit) §
Pext
External reactivity
-o
Net reactivity
Spf
pit) = pip] Neutron kinetics
5pf [p] Feedback mechanisms F I G U R E 6-5.
Closed-loop block diagram
o > Incremental power
260
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
This should more properly be referred to as an isothermal temperature coefficient, since it assumes that the core can be characterized by a single uniform temperature T.
Such a simple model illustrates quite clearly the effect of temperature feedback on reactor stability. For if a reactor were to possess a positive aT, then an increase in temperature would produce an increase in p, hence the power would increase, causing a further increase in temperature, a n d so on. In this sense, the reactor would be unstable with respect to temperature or power variations. The more desirable situation is that in which aT is negative, since then an increase in temperature will cause a decrease in p, hence a decrease in reactor power and temperature which tends to stabilize the reactor power level. The temperature coefficient of reactivity depends on a great many processes occurring within the core. Since the details of these processes depend on the specific reactor type under consideration, we will defer a discussion of just how such temperature coefficients are evaluated to Chapter 14. However we should remark at this point that the concept of such an isothermal temperature coefficient has a very limited range of usefulness in reactor analysis. In a heterogeneous reactor, most of the fission energy release is confined to the fuel elements. This energy must then be transferred by thermal conduction through the fuel element, the clad, and then into the coolant before it can be withdrawn from the core. Hence one has a very nonuniform temperature distribution. For example, fuel centerline temperatures may range as high as 2700°C, while coolant temperatures are as low as 250°C. Hence one usually introduces several such temperature coefficients of reactivity, each of which characterizes the reactivity feedback due to a variation in the effective or average temperatures of each major component of the core, such as fuel, moderator, and structure
j
j
J
However one must also keep in mind that the various thermal processes in the core are characterized by widely different time behaviors. The fuel temperature responds relatively rapidly to any power level changes. However it takes an appreciable amount of time to transfer this energy to the coolant, and hence its temperature response is much slower. For this reason, it is convenient to divide up the temperature coefficient of reactivity into prompt and delayed components. Effects that depend on the instantaneous state of the fuel—for instance, resonance absorption (Doppler effect) or thermal distortion of fuel elements—may be regarded as prompt, while effects that depend primarily on the moderator or coolant—neutron energy spectrum and thermal expansion of moderator material— are delayed. Prompt feedback mechanisms are of particular importance in reactor safety studies, since they play a very important role in limiting any reactor transients that may occur. For these reasons, the isothermal temperature coefficient of reactivity is not really a very useful quantity. A more useful quantity is the change in reactivity caused by a change in reactor power. W e will consider this in detail in the next section. There is one instance in which the concept of an isothermal temperature coefficient is of some use, however. When the reactor is at zero power, there is no
NUCLEAR REACTOR KINETICS
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261
fission energy being released in the fuel, and hence the entire reactor core can essentially be characterized by a single temperature. As the temperature of the core is increased, say by heating the primary coolant, there will be a decrease in core multiplication due to temperature feedback. One defines the so-called temperature defect of reactivity, Ap TD , as the change in reactivity that occurs in taking the reactor core from the fuel-loading temperature (i.e., the ambient temperature) to the zero power operating temperature
A
P t d
=
r T zero power g 0 / -£dT.
(6-93)
Jt ambient
The magnitude of Ap XD is primarily determined by the coolant temperature coefficient of reactivity, and depends sensitively on the moderator-to-fuel ratio and soluble poison concentration. In a L W R Ap T D ~.02-.04(AA;//:). 2. POWER COEFFICIENTS OF REACTIVITY
A far more useful parameter characterizing feedback is the power coefficient of reactivity defined by
where the Tt are again the effective temperatures associated with each core component. Such a parameter takes account of the temperature differences occurring in a reactor while it is operating at power. The reactivity due to power feedback can then be written as p = fPdPaP(P).
(6-95)
J
o
Obviously if a reactor is to be inherently stable against power-level fluctuations, it must be designed with a p < 0 . A closely related quantity is the power defect, which is defined to be the change in reactivity taking place between zero power and full power C
P
APpd= I
f u l l power 3 O
-£dP.
(6-96)
•A)
The power defect can be quite sizable. For example, in the L W R the power defect is typically of the order Ap PD ->.01 — .03(A/c/£:). Thus far we have only discussed how the reactivity depends on temperature. The remaining problem of how the temperature depends on the power level is strongly relate^Lto the reactor type and involves a thermal-hydraulics analysis of the core. For slow power changes, one can use the steady-state analysis developed in Chapter 12 to determine the core temperatures for a given power level and hence the power coefficient of reactivity in terms of the temperature coefficients of the various components of the core. Such a steady-state thermal analysis will no longer be valid for more rapid power transients. For example, the thermal time constant of the fuel is frequently as large
262
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
as 10 seconds. For power transients on shorter time scales the fuel temperature coefficient of reactivity (principally determined by the Doppler effect) will be the dominant factor in determining the power coefficient of reactivity. Although a realistic description of temperature feedback involves a transient thermal-hydraulics model of the reactor core such as those we will develop in Chapter 12, several alternative lumped parameter models of the reactor temperature dependence are occasionally used for qualitative studies of reactor dynamics. One common model assumes a single effective coolant temperature Tc, and models the fuel temperature TF by Newton's law of cooling =KP(t)-Y(TF-
Tc),
(6-97)
where K and Y are thermal constants characterizing the core. At the opposite extreme would be an adiabatic model, in which the heat loss is assumed to be negligible (such as in a very rapid transient) = KP (/).
(6-98)
Still another model assumes constant power removal such that dTF = K(P-P0).
(6-99)
All of these models can be considered as special cases of a general linear feedback functional: Pf(0= Jr —
drh(t-r)[P(r)-P0l
(6-100)
00
A little inspection should convince you that the feedback kernel h(t) in each of these cases takes the form: aKe~yt
Newton's law of cooling:
h(t) =
Adiabatic model:
h(t) = aK[P0 = 0]
Constant power removal:
h(t) — aK.
(6-101)
If more reliable design information is required, then one is usually forced to go to a multigroup diffusion calculation coupled with a detailed thermal-hydraulic analysis of the core. One can then calculate the multiplication for several different power levels, and hence determine directly the dependence of reactivity on power level. The corresponding power and temperature coefficients are frequently precalculated and stored in tabular form for use in determining the reactivity feedback to use in point reactor kinetics models.
C. The Transfer Function of a Reactor with Feedback 1. CLOSED-LOOP TRANSFER FUNCTION
Let us now return to consider the dynamic behavior of a reactor with feedback. In particular we will consider the effects of feedback on the reactor
NUCLEAR REACTOR KINETICS
/
263
r e s p o n s e t o a n e x t e r n a l reactivity i n s e r t i o n . F i r s t w e will c o n s i d e r t h e p o w e r oscillations r e s u l t i n g f r o m a p e r i o d i c r e a c t i v i t y i n s e r t i o n of small a m p l i t u d e , since this i n e f f e c t m e a s u r e s t h e r e a c t i v i t y - t o - p o w e r t r a n s f e r f u n c t i o n . A t y p i c a l series of reactivity p o w e r t r a n s f e r f u n c t i o n g a i n m e a s u r e m e n t s 1 0 , 1 1 is s h o w n i n F i g . 6-6. I n p a r t i c u l a r n o t i c e h o w d i f f e r e n t t h e t r a n s f e r f u n c t i o n at p o w e r is f r o m t h e z e r o p o w e r t r a n s f e r f u n c t i o n . T h e m a r k e d r e s o n a n c e b e h a v i o r in t h e vicinity of .03 cps a t 7 ^ = 550 k W in t h e first p l o t is a p p a r e n t . Such b e h a v i o r is d u e t o t h e p r e s e n c e of f e e d b a c k . A s t h e p o w e r level increases, t h e r e s o n a n c e p e a k b e c o m e s n a r r o w e r a n d h i g h e r . A s w e shall see, this implies t h a t f o r s u f f i c i e n t l y large p o w e r s , t h e r e a c t o r is u n s t a b l e . L e t u s go b a c k a n d d e v e l o p a m a t h e m a t i c a l e x p r e s s i o n f o r t h e t r a n s f e r f u n c t i o n c h a r a c t e r i z i n g a r e a c t o r w i t h f e e d b a c k . 1 W e will restrict ourselves t o small p o w e r v a r i a t i o n s a b o u t t h e e q u i l i b r i u m level P0 so t h a t t h e f e e d b a c k c a n b e a d e q u a t e l y r e p r e s e n t e d a s a l i n e a r f u n c t i o n a l similar t o E q . (6-100) C 00
Spf[p]
= j
drh(r)p(t~r).
(6-102)
N o t e t h a t w i t h this sign c o n v e n t i o n t h e n e g a t i v e f e e d b a c k n e c e s s a r y f o r r e a c t o r
Frequency (cps)
+ 12
I rsM llllj
+8
1 M i l l I|
+4
7/in // i
-
0
\
X
V
/
0.01
i i i i mil 0.1
1 1 1 1 Mill 1
'//
1
—
1 III! ll
10
I I I I III!
w, rad/sec FIGURE 6-6.
—
—
^ yy /
A\
—
-16
-
/ / !
1
A ij i \
-12
—
i i i
o o -4 o
I 1 i mi!
60 MWV ry—71 MW — 20 M W \ 4 0 MW ~ 10 MW\ A l / \Y\ _
Zero\ power V
-
<3
I 1 t Mill
Examples of measured closed-loop transfer functions 10 ' 11
100
264
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
stability will imply that / * ( t ) < 0 . It should also be kept in mind that the feedback kernel h(t) actually depends on the equilibrium power level P0. If we substitute Eq. (6-102) into Eq. (6-72), we find dp_
=
dt
f oo SPext(0+/ drh(r)p(t-r) J A 0
_
\
[P0+p(t)]
^lf^dTD(r)[p(t-r)-p(t)].
(6-103)
Of course this equation is still nonlinear. We will consider only small power variations p(t)<&P0 such that we can linearize Eq. (6-103) to write dp_ dt
1
Now, as before, we will assume the reactor is operating at a steady state power level P0 prior to f = 0. Then by Laplace transforming Eq. (6-104) we find
SP ( ' ) = X
P
05Pex, + X
D
( S ) P (*)
+
A
P
°
H ( S ) P (S)
H ~ A
p
or p(t) Po
Z(s) 8
1
Pen(s)
-P0N(s)Z(s)
=
L s
( )$PtAs)>
(6-105)
where Z(s) is the usual zero power transfer function given by Eq. (6-76) while H(s)= £{h(t)} is the feedback transfer function. We have further defined the reactivity-to-power or closed-loop transfer function L(s) Z(s) L(s)
1
-P0H(s)Z(s)
(6-106)
Notice that as ^ o ^ O , L(s)^>Z(s), the zero power transfer function. This notation is consistent with our earlier block diagram that has been relabeled in Figure 6-7. Just
FIGURE 6-7.
Closed-loop Transfer function
NUCLEAR REACTOR KINETICS
/
265
as in the zero power case, the inverse of / ( / ) = £ ^ L ^ ) } , is the Green's function for the point-reactor kinetics equation with feedback pit^pJ'dTlit-T^p^T).
(6-107)
•'O
Let us examine L(s) in a bit more detail. Unlike Z(s), L(s) is analytic at 5 = 0 with a value L(0)=-[P0H(0)]~l=
/
l(t)dt.
(6-108)
•'o Hence the long time response to a positive step reactivity insertion of magnitude 8
Pext-^dPo
is
Just p(t)
= P0dpJtdr!(T)^P08p0L(0).
(6-109)
This implies that the reactor approaches a new equilibrium power level P (t^oo)
= P0[ 1 + S p 0 L ( 0 ) ] .
(6-110)
This is in sharp contrast to the zero power (i.e., no feedback) reactor whose power level grew exponentially for long times. The equilibrium state occurs when the power level reaches a value such that the feedback reactivity just compensates for the step reactivity insertion 8Pf(t)=
fdTh(t-T)p(T)^p(x>)H(0)=-8p0.
(6-111)
2. RESPONSE TO A SINUSOIDAL REACTIVITY INSERTION
Let us once again consider a sinusoidal reactivity variation 8pext(t)
= 8p0smc>t.
(6-112)
Then rfhe long time response is given in analogy to the zero feedback case [Eq. (6-84)fby Pi*) "o
= | L (id) | sin (o>t + 4>),
(6-113)
N o w if we recall the form for L(s) given by Eq. (6-106), we can see that a resonance in the gain G(co)= |L(/
(6-114)
where fl(«) = a r g { - / / ( / w ) Z ( i « ) } .
(6-115)
266
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
However for the resonance condition [Eq. (6-122)] to be satisfied, we require that both of the conditions 0(o))=
180°
and
P0\H( /co)Z ( / « ) | = 1
(6-116)
be simultaneously satisfied. These two conditions will determine a critical power level PCTit and a resonance frequency cocrit at which a true resonance situation will arise. As long as the reactor power P0 is kept below P c r i t , only a finite resonance peak such as those illustrated in Figure 6-6 will occur. At this peak, the gain |L(/«)| will assume a value
OIII L O cm/1
|Z
//t0;rit\.
(6-117)
In the light of this discussion we can more readily understand the transfer function gain measurements given in Figure 6-6. In almost all cases of reactor design, the critical power level Pcrit above which the reactor is susceptible to such instabilities is always quite far above actual or design operating levels. Nevertheless it is important to be able to anticipate such inherent instabilities in the design so that effective countermeasures can be taken (such as by modification of the original design or by the addition of stabilizing feedback control). Such considerations arise in reactor stability analysis. 3. L I N E A R S T A B I L I T Y A N A L Y S I S
The stability of a reactor with feedback can be investigated by examining the singularities of the closed-loop transfer function Z(s) L { s ) =
1\~P0PH{s)Z{s) „< w /
(6
^
"118)
in the complex s-plane. First notice that since Z (s) appears both in the numerator and denominator, its poles Sj [that is, the roots of the inhour equation = Z - 1 ( ^ ) = 0] "cancel." Hence the poles of L(s) are simply the zeros of \~P0H(s)Z(s)
= 0.
(6-119)
Now suppose that Eq. (6-119) were to have a simple root at s — s0 [i.e., a pole of L(s)]. Then when we invert the Laplace transform, this pole will contribute a term i n p ( t ) of the form exp(s 0 t). Hence if s0 is in the right half s-plane (RHP), t h e n p ( t ) will grow exponentially in time. This would imply an unstable response to an applied reactivity perturbation (within the linear approximation, of course). If the root s0 lies in the LHP, then terms of the form exp(5 0 /) will decay in time. Hence to study reactor stability, it is obviously important to determine if any of the poles of L(s) [i.e., zeros of Eq. (6-119)] lie in the RHP. Instabilities may arise for sufficiently large power levels P0 even with negative reactivity feedback, as the sequence of diagrams in Figure 6-8 indicate. For P 0 = 0, we can identify the poles of L(s) as just those of Z(s) [i.e., the inhour equation roots]. Then, for the case in which aP = H(0)<0, increasing P0 will provide more
*
NUCLEAR REACTOR KINETICS
/
267
negative feedback and hence shift the pole sY to the left and into the LHP. However the other poles will also shift. In particular, the pole s2 will shift to the right. For-some sufficiently large power level P 0 , the poles s{ and s2 will coalesce, forming a complex conjugate pair that moves off of the real axis and into the complex plane. For still larger P0, this pair moves back to the right, until for some critical power level P0 = Pcrit the poles move into the R H P and the reactor becomes unstable. When the poles cross the imaginary axis, we observe a "resonance" in the transfer function gain |L(/
P0 = 0
-x—x-
FIGURE 6-8. level Pn
P0 = P,> 0
-x—x-xs3
s2 S-1
P0-P2>
s—plane
s—plane
s—plane
s— plane
Pv
x4«x-
/
/
A
Pent/ 1I \
\
\
V
Shifting of poles of the closed-loop transfer function with increasing reactor power
\ Ther6 are numerous tricks for determining the signs of these poles. These include methods 1 2 such as those utilizing Nyquist diagrams, root-locus plots, and R o u t h Hurwitz criteria. However such subjects are more properly the concern of linear system analysis or control theory, so we will simply refer the interested reader to the references listed at the end of the chapter. 1 , 8 , 9 4. N O N L I N E A R POINT REACTOR KINETICS
Thus far our study of the point reactor kinetics equations with feedback has been restricted to situations in which the reactivity changes and corresponding power changes are sufficiently small that these equations can be linearized. In particular our study of the stability of the reactor has been restricted to the consideration of stability in the small—that is, to the study of perturbations and responses sufficiently small for a linear analysis. However for larger perturbations nonlinear effects must be taken into account. In these cases, the resulting conclusions about stability may be quite different. For example, we have seen that the linearized point reactor kinetics equation predicts that the reactor will be unstable if the power exceeds some critical value
268
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
P c r i t . However even though the reactor is linearly unstable, it may be stable in the nonlinear description. Hence it is of some interest to study the significance of linear stability theory within the more general framework of nonlinear stability theory. It should be remarked, however, that there are many different approaches to nonlinear point reactor kinetics—none of which is completely satisfactory. Furthermore such results as do exist provide only sufficient (as opposed to necessary) conditions for stability. Sufficiency conditions are usually m u c h too restrictive for practical application. Furthermore, the very limited validity of the models investigated by nonlinear stability methods usually discount their value for practical reactor design. A n d of course there is also the more pragmatic philosophy adopted in most nuclear reactor design favoring an ultraconservative design that guarantees linear stability under all conceivable operating conditions, hence obviating the need for a nonlinear stability analysis. 5. SOME F I N A L COMMENTS ON REACTOR STABILITY ANALYSIS
It might seem that an integral part of any reactor safety analysis would be an investigation of the stability of the reactor design. In particular such an analysis should consider the possibility that the reactor might become unstable under some feasible combination of operating conditions. In practice, however, such stability studies do not play near as significant a role in reactor design as one might expect. Reactor instabilities usually can occur only if one of the temperature feedback coefficients happens to be positive over some range of operating conditions. However it is usually quite easy to design a power reactor so that it will always be characterized by large, negative temperature coefficients (as we will see in Chapter 14). Indeed all power reactor designs tend to be ultraconservative in this respect (as they do with respect to all safety considerations). Hence there has been relatively little motivation to perform detailed stability analyses of reactor core designs. And in those instances in which methods of stability analysis have been applied to practical reactor designs, investigations have usually been limited to the linear domain.
IV. EXPERIMENTAL D E T E R M I N A T I O N OF KINETIC PARAMETERS AND REACTIVITY
REACTOR
F r o m our earlier development of the point reactor kinetics equations, we have found that the three most important parameters characterizing the kinetic behavior of a nuclear reactor are the reactivity p, the prompt generation time A, and the effective delayed neutron fraction A variety of experimental techniques have been developed to measure both these as well as dynamic (i.e., feedback) characteristics of the reactor. Such methods can be classified as either static or dynamic measurement techniques.
A. Static Techniques for Reactivity Determination 1. N E U T R O N MULTIPLICATION M E A S U R E M E N T (RECIPROCAL MULTIPLICATION METHOD)
Perhaps the most common measurement is the so-called critical loading experiment or reciprocal multiplication method, in which the steady-state neutron
NUCLEAR REACTOR KINETICS
/
269
flux resulting from a source in a subcritical assembly is measured as fuel is added to the assembly. If the reactor is characterized by a multiplication factor k, then one can crudely think of the amplification of the original source neutrons by the assembly as given by 9L
=
S q k S
0
k
'''
= (\-kylS0~MS0,
(6-120)
where S0 is the rate at which source neutrons are emitted in the reactor. The usual procedure for a safe approach to delayed critical in core loading consists of plotting M ~ x (or the reciprocal neutron counting rate) as a function of some parameter that controls reactivity (e.g., fuel mass) and then extrapolating this M ~1 plot to zero to determine the critical loading (as sketched in Figure 6-9). During a stepwise approach to delayed critical by the reciprocal multiplication method, the neutron level following each addition of reactivity must be allowed to stabilize in order to obtain an accurate indication of the asymptotic multiplication before proceeding with the next reactivity addition. As one approaches delayed critical, the buildup of the precursor concentrations can become bothersome.
\ \
Data points
\
X
\
X
\
x
\ \
/
\
\
Estimate of \ critical mass
Fuel mass
F I G U R E 6-9.
Critical loading measurements
2. FUEL SUBSTITUTION TECHNIQUES 4
One can determine reactivity by uniformly substituting a p .ison for a small fraction of the fuel. The poison is usually chosen such that it has the same scattering and absorption properties as the fuel (or as closely as possible). Then first-order perturbation theory yields a reactivity change due to the substituted poison of f
A p =
.
J
( 6 - 1 2 1 )
^V(r)2f(r)
This substitution method can be used for control-rod calibration by balancing the effect of a given rod movement by adding a proper amount of distributed poison.
270
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
B. Dynamic Techniques for Reactivity Measurements 1. A S Y M P T O T I C P E R I O D M E A S U R E M E N T S
Perhaps the simplest type of kinetic measurement one can perform is to make a small perturbation in the core composition of a critical reactor, and then to measure the stable or asymptotic period of the resultant reactor transient. Using the inhour equation, one can then infer the reactivity "worth" of the perturbation from a measurement of the asymptotic period. It should be noted that the period method for all practical purposes applies only to positive periods, since negative periods are dominated by the longest delayed neutron precursor decay and hence provide very low sensitivity to negative reactivity.
2. R O D D R O P M E T H O D
Let us consider a reactor operating at some equilibrium power level P0 when it is suddenly shut down by the introduction of a negative reactivity — 8p (the "rod drop"). From our earlier studies of the point reactor kinetics equations we know that after a few prompt neutron lifetimes, the reactor power level drops to a lower level Px, determined by the amount of reactivity insertion and remains at this "quasistatic" level until it is ultimately decreased by delayed neutron precursor decay. We can use our earlier result [Eq. (6-64)] from the prompt j u m p approximation to write H
P0
P+
(6-122) 8p'
Hence we can solve for the reactivity insertion in dollars in terms of the power levels P0 and Px as
- R R R
X
(6
-
-,23)
To determine Pv one need only extrapolate back the asymptotic behavior following the rod drop to t = 0. 3. S O U R C E J E R K M E T H O D
A very similar technique can be used to measure the multiplication of a subcritical assembly. Suppose we consider such a subcritical system maintained at a power level P0 by a neutron source of strength S0. One can express the equilibrium power level P0 using the point reactor kinetics equation as
0
(6-124) Sp+/?
where 8p is the degree of subcriticality of the system. Suppose the source is now jerked out of the core. Once again the prompt j u m p approximation can be used to express the lower "quasistatic" power level Px as
5p + j8
iO
(6-125)
NUCLEAR REACTOR KINETICS
/
271
Using Eqs. (6-124) and (6-125), along with the equilibrium forms of the pointreactor kinetics equation prior to the source jerk, one can find P0 Sp JT = ! +
(6-126)
Hence the dollars subcritical of the assembly is given very simply by
^
=
(6-127)
The source jerk method is very similar to the rod drop method in concept. However it is somewhat simpler to perform since it requires only the removal of a small mass of material (the source) in contrast to the rapid release of one or more control rods. 4. R O D OSCILLATOR M E T H O D
If one oscillates a control rod in a sinusoidal fashion in a critical reactor, there will be a corresponding oscillation in the reactor power. By measuring the gain and magnitude of the phase shift characterizing the power oscillations, one can measure the reactor transfer function, either at zero power or operating power: ^
= L(/co)Sp.
(6-128)
"o For low powers, L(/«)—»Z(/
can be used to'measure either the prompt generation time A or the reactivity worth of the oscillating rod, Sp = c o A ^ .
(6-130)
-M)
Such oscillator measurements can also be used to determine the stability characteristics of the reactor. 5. P U L S E D N E U T R O N M E T H O D S
By measuring the transient behavior of the neutron population following a burst of neutrons injected into an assembly, one can measure a variety of important parameters. Suppose we first consider a pulse of neutrons injected at t = 0 into a nonmultiplying assembly. Then according to one-speed diffusion theory, the flux satisfies =
(6-131)
272
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
subject to the initial condition applying to the pulsed source, <|>(r,0) =
(6-132)
where ^j(r) is the fundamental mode of the assembly geometry. Hence the asymptotic behavior of the flux is governed by the decay constant a
0
= v \ + vDB2.
(6-133)
If one measures a 0 for various assembly sizes, then plotting a 0 against B 2 will yield t>2a (the intercept at B* = 0) and vD (the slope). (See Figure 6-10.) Actually, there are higher order terms in Bg2 due to both transport and energy-dependent corrections a 0 = t ; 2 a + vDBj + CB% + • • •
(6-134)
which add a curvature to the a 0 ( B p l o t . In multiplying assemblies if one assumes that both the pulse injection and measurement are performed on a time scale short compared to delayed neutron lifetimes, then one can use i ^
= J DV 2 <|>-(2 a -^ f )<J>(r,0
(6-135)
to find decay constants of the form a 0 = u ( 2 a - ?2 f ) + vDB 2 .
experiment
(6-136)
NUCLEAR REACTOR KINETICS
/
273
In this case, the a 0 versus B 2 curve appears as that shown in Figure 6-10. One can use this technique to directly measure reactivity. For a rapid pulse, the time behavior of the flux or power will be determined by prompt neutron kinetics such as 1 dP
P~fi
,,
Hence a 0 can be used to infer p, provided P and A are known. If the core is maintained at delayed critical, then p = 0 and the pulsed neutron method measures a0=-A/(3.
(6-138)
C. Noise Analysis in Nuclear Reactors 1314 Thus far we have been analyzing the reactor as essentially a deterministic system. Yet we know that all of the dynamic variables describing the reactor (power level, flux, temperature, etc.) actually fluctuate in a statistical fashion about some mean value. W e are unable to predict with certainty the future values of these variables but rather can only specify the probability that they will assume a certain value. The statistical nature of the neutron diffusion process has been stressed repeatedly. Moreover the concept of a cross section, and, indeed, even the quantum mechanical description of the neutron interactions, must be interpreted in a probabilistic sense. There are numerous other sources of such statistical fluctuations or noise in a reactor, such as in fluid flow, coolant boiling, and mechanical vibration. In fact, statistics enters even into the measurement of the reactor state due to detector noise. At low power levels the statistical fluctuations associated with the fundamental nuclear processes occurring in the core will be dominant. However at higher power levels, the reactor noise will be predominantly due to disturbances of a nonnuclear nature (e.g., coolant flow). Regardless of its origin, reactor noise is important in reactor dynamics for at least two reasons. First, it interferes with the precision with which a desired quantity can be measured in a reactor; that is, one must extract the signal of interest out of the background noise. Second, however, since the noise originates from various processes occurring in the reactor, it can actually serve as a source of information about the system. The latter of these properties is of most interest to us here. W e will study how noise analysis can be used to measure the transfer function of the reactor. 1. CROSS CORRELATION METHODS
Let us begin by introducing two definitions. We will define the autocorrelation of a function by < ^ 0 0 = lim
fTx(t)x(t
T->oo 2.1 J —T
+ T)dt,
(6-139)
and the cross correlation of two functions, x{t) and y(t) by
lim - L (T x{t)y(t
T—>00 £ 1 J — T
+ r)dt.
(6-140)
274
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
If the functions a n d y ( f ) are periodic, then the limit process can be omitted provided T is chosen as the period. In general, however, x(t) and y{t) will not be periodic. In fact we will later consider them to be random variables in the sense that only their probability distribution can be specified. N o w we will set up the cross correlation between reactivity and power (using a notation in which the limit process T->cc is to be understood) r I
l
T
«/
P{t+r) «P„t(0—pT—
(6-141)
y
If we substitute in our expression f o r p { t ) / P 0 f r o m Eq. (6-107) (after extending the time domain to — oo < t' < t), we find
f
«Pex l t - r )
rdu8p^{t-u)l{u) o
J
dt
du - L f 00 = /
(6-142) l
U
( )
where we have identified the autocorrelation function of reactivity T
1
(6-143) 5
^ ) ^
As a final step, we take the Fourier transform of Eq. (6-142), defined as oo
/
(6-144) - rv-1
to find (6-145) but ^ {/(«)} is just the closed loop transfer function L(zco). Hence we find that we can write
L(io)) =
^{qy}
(6-146)
Here
NUCLEAR REACTOR KINETICS
/
275
then construct both the time-correlation functions
(6-147)
Then by taking the ratio of W{
(6-148)
Hence by merely measuring the cross correlation, one can determine both the amplitude and phase of the transfer function. This experiment serves, then, as an alternative to a reactor oscillator transferfunction measurement. Unlike the latter, it does not suffer from background noise (rather taking advantage of such random fluctuations), and hence does not require nearly so large an input signal. However both of these experiments suffer from the fact that one must perturb the reactor by introducing an externally controlled reactivity signal in order to perform the measurement. It is possible to bypass this difficulty and measure the amplitude of L(io>) directly from the inherent noise naturally present in the reactor. 2. AUTOCORRELATION MEASUREMENTS
Consider the autocorrelation of the fluctuations in the reactor power , ,
1
f
T
pO)p(t
I «/
If we substitute in Eq. (6-107) for p(t)/P0 time, we find
-P
+ r) Po
T
d
DT
•
(6-149)
and again take a Fourier transform in
%{
(6-150)
or (6-150 pp.
Hence the square of the magnitude (i.e., the gain) of the transfer function can be determined as the ratio of the power and reactivity spectral densities. Although we can easily measure the power spectral density, the reactivity spectral density characterizing inherent reactivity noise is not experimentally accessible. If one has reason to believe that such fluctuations are truly random in nature, then = constant, and the measurement of the power autocorrelation function by itself will yield the amplitude of the transfer function (although phase information will have been lost).
276
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
Such a measurement of |L(/
V. SPATIAL EFFECTS IN REACTOR KINETICS Thus far our analysis of time-dependent nuclear reactor behavior has been based on the point reactor kinetics model in which the neutron flux was assumed to be the product of a time-independent spatial shape factor ^ ( r ) and a timedependent amplitude factor P(t) [cf. Eq. (6-7)]. However this assumption will obviously be invalid in many cases of interest. For example, one of the most important safety questions concerning reactor analysis involves the reactor kinetic behavior following the postulated ejection of the control rod with highest reactivity worth. Such a strongly localized perturbation in the core composition would certainly cause a considerable deviation from the spatial shape factor ^ ( r ) and would invalidate the point reactor kinetics model. Furthermore most modern power reactor cores are quite large f r o m a neutronic point of view, being as much as 200 diffusion lengths in diameter. Hence the neutronic behavior in such cores tends to be quite loosely coupled from point to point. This means that a change in the flux or power density (or core multiplication) at one point in the core will not be felt at other points until after an appreciable time delay. For example, one frequently finds rather significant "tilts" in the spatial power distribution across the core due to nonuniform coolant temperature or fission product buildup. The phenomenon of such loose spatial coupling can be seen rather vividly by measuring the reactor transfer function, say, under zero power conditions, at several points in a large power reactor core. That is, one inserts a sinusoidal or pseudorandom reactivity variation at one point r in .the core (say, by moving a control rod in and out of the core), and then measures the amplitude and phase of the resulting power or flux oscillations by locating a neutron detector at various other points r' in the core. If the reactor were truly described by the point reactor kinetics equation, then the transfer function Z(/w; r,r') measured at various points r would be the same. And for low frequencies, one does indeed find that Z(/to;r, r') is essentially i n d e p e n d e n t . However for higher frequencies in large reactor cores, one finds that actual measurements will yield different results, depending on where the oscillator and detector are placed. Such measurements reveal that the transfer function is actually spatially dependent for higher frequencies, because of the time it takes to propagate a disturbance in the neutron flux f r o m one point to another in the core. This implies a breakdown in the point reactor kinetics model that characterizes every point in the reactor by the same time-dependence P(t). As a general rule, one finds that the point reactor kinetics equations are incapable of predicting the detailed behavior of reactor transients initiated by rapid local changes in reactivity. More precisely, if the neutron flux changes rapidly on a
NUCLEAR REACTOR KINETICS
/
300
time scale of the order of the effective neutron lifetime , the point reactor kinetics equation should be regarded as suspect. In these instances, one must take explicit account of the spatial dependence of the neutron flux. In fact one is frequently forced to perform a brute force numerical solution of the time-dependent neutron diffusion equation and precursor equations. Unfortunately in most cases in which such spatial effects are significant, one cannot rely on the one-speed approximation to provide an adequate description of the neutron energy-dependence. Hence a direct numerical study of nuclear reactor kinetics 15 usually involves the solution of the multigroup, time-dependent diffusion equations—at a considerable computational expense. For such finite difference solutions of the multigroup diffusion equations to be feasible, it is necessary to employ a variety of numerical techniques to accelerate the computation. These calculations are usually coupled with transient thermalhydraulic calculations and can become quite involved. In general they are only used as a last resort to describe situations in which detailed spatial behavior is of p a r a m o u n t importance in a rapid transient (such as the rod-ejection accident analysis). A variety of less direct methods exist. These can generally be classified as nodal, quasistatic, or modal techniques. The nodal approach 1 6 is very similar to that discussed in Section 5-II in that the core is divided into a number of regions or nodes. As i n ^ h e static case, the primary difficulty in this approach is determining the parameters^ that couple the flux at various nodal points. This is usually accomplished by a variety of approximate schemes adjusted by numerous empirical fits to either experimental or benchmark calculation data. So-called quasistatic methods 1 7 essentially work within the framework of the point reactor model, performing periodic static spatial calculations to obtain the shape functions necessary for evaluating the point kinetics parameters p, /?, and A. The practical utilization of this method has been rather limited to date. Perhaps the most popular alternative 18 to direct finite difference schemes involves expanding the flux in a finite series of known spatial functions or modes, \pj(r), and then obtaining a set of equations for the time-dependent expansion coefficients
(6-152)
Of course the ideal scheme would be to choose i^(r) as the spatial eigenfunctions of the perturbed reactor. Unfortunately one usually does not know these modes. Instead it is common to expand in an alternative finite set of spatial functions and then use either weighted residual or variational methods (recall Section 5-III-C) to determine the appropriate set of modal equations for the
278
/
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
REFERENCES 1. A. Z. Akcasu, G. S. Lellouche, and L. M. Shotkin, Mathematical Methods in Nuclear Reactor Dynamics, Academic, New York (1971). 2. R. Scalletar, in Proceedings of Conference on Neutron Dynamics and Control, D. L. Hetrick and L. E. Weaver Eds., USAEC CONF-650413 (1966). 3. W. M. Stacey, Jr., Space-Time Nuclear Reactor Kinetics, Academic, New York (1969). 4. G. R. Keepin, Physics of Nuclear Kinetics, Addison-Wesley, Reading, Mass. (1965). 5. A. Z. Akcasu, G. S. Lellouche, and L. M. Shotkin, Mathematical Methods in Nuclear Reactor Dynamics, Academic, New York (1971), p. 91. 6. R. Goldstein and L. M. Shotkin, NucL Sci. Eng. 38, 94 (1969). 7. P. H. Hammond, Feedback Theory and its Applications, English Universities Press, London (1958); J. L. Bower and P. M. Schultheiss, Introduction to the Design of Servomechanisms, Wiley, New York (1958). 8. D. L. Hetrick, Dynamics of Nuclear Reactors, University of Chicago Press (1971). 9. L. E. Weaver, Reactor Dynamics and Control, Elsevier, New York (1968). 10. F. W. Thalgott, et al., Proceedings of Second U. N. Conf. on Peaceful Uses of Atomic Energy, Vol. 12, 242 (1958). 11. J. A. Deshong, Jr., and W. C. Lipinski, USAEC Document ANL-5850 (1958). 12. A. Z. Akcasu, G. S. Lellouche, and L. M. Shotkin, Mathematical Methods in Nuclear Reactor Dynamics, Academic, New York (1971), Chapter 6. 13. J. A. Thie, Reactor Noise, Rowman and Littlefield, Totowa N.J. (1963). 14. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, Princeton, N. J. (1970), pp. 511-514. 15. D. R. Ferguson and K. F. Hansen, NucL Sci. Eng. 51, 189 (1973). 16. W. M. Stacey, Jr., Space-Time Nuclear Kinetics, Academic, New York (1969), Section 1.4. 17. K. O. Ott, NucL Sci. Eng. 26, 563 (1966). 18. J. B. Yasinsky and A. F. Henry, NucL Sci. Eng. 22, 171 (1965).
PROBLEMS At time t = 0, a decaying point source emitting S0e~Xt neutrons per second is placed at the center of a homogeneous bare spherical reactor which is being maintained in a subcritical state. Determine the time-dependence of the neutron flux that would be measured by a detector placed outside of the reactor. Use one-speed diffusion theory and ignore delayed neutrons. 6-2 Consider a spherical assembly operating at a critical steady-state power level at time t = 0. At this instance a neutron burst is suddenly inserted at the center of the reactor. Derive an expression for the length of time before the reactor flux will once again assume its fundamental mode shape to within 10%. Plot this time versus assembly radius R for a thermal assembly moderated with H z O. Ignore delayed neutrons. 6-3 (a) What is the maximum possible reactivity insertion capable in a 2 3 5 U fueled reactor? (Express your answer in $.) (b) What is the maximum possible reactivity of a 235 U-fueled reactor having a nonleakage probability of 0.6 for fission neutrons and 0.7 for delayed neutrons? 6-4 Estimate the prompt neutron lifetime in a large thermal reactor fueled with 2 3 5 U and moderated with: (a) H 2 0 , (b) graphite, and (c) D 2 0 . 6-5 Calculate the effective neutron lifetime for a thermal reactor fueled with either 233 U or 2 3 5 U and a fast reactor fueled with 239 Pu. 6-6 Give a physical discussion of the difference between the prompt neutron lifetime and mean neutron generation time A for both an infinite and finite reactor. (Refer to Keepin, 4 p. 166, for a more thorough discussion.) 6-1
NUCLEAR REACTOR KINETICS 6-7
6-8 6-9 6-10
6-11 6-12
6-13 6-14
/
279
In our development of an equation describing the delayed neutron precursor concentration, we assumed that these precursors did not migrate from the point of fission. This is a reasonable assumption in solid-fueled reactors. Consider, however, a reactor core in which the fuel is in gaseous form. Then the precursors may diffuse from the point of fission before decaying. If the diffusion coefficient characterizing a given precursor migration is Dp develop a generalization of the precursor concentration equations to account for this process. Derive an expression for the effective delayed neutron yield fractions $ characterizing a mixture of several fissile isotopes. Rederive the inhour equation for the case of a fast reactor fueled with 2 3 9 Pu and 2 3 8 U treating delayed neutrons from both isotopes explicitly. Repeat the solution of the point reactor kinetics equations with one effective delayed group for a constant reactivity insertion p 0 using Laplace transform methods (see Appendix G). Demonstrate that for small reactivity insertions and ( f t - p 0 + XA)2 » 4AAp0 the reactor power P ( t ) is given by the approximate form [Eq. (6-40)]. Estimate the reactor period induced by a positive reactivity insertion of 1 $ in an infinite 2 3 5 U-fueled thermal reactor moderated with water and a fast 2 3 9 Pu-fueled reactor. Proverlhat for six groups of delayed neutrons, the inhour equation has seven roots st of whifeh six have negative values. Consider a two-stage delayed neutron production process as sketched below: £ £ Fission product A -+B-* C + d e l a y e d neutron.
Assuming that B is the only delayed neutron precursor of interest, write the point reactor kinetics equations including both rate equations for A and B. By then deriving the corresponding inhour equation, demonstrate that such a two-stage decay process can actually yield a negative effective delayed neutron fraction. 6-15 Initially a reactor is operating at a steady-state power level P0. Using the point reactor kinetics equation with one equivalent group of delayed neutrons, determine the stable reactor period T for a positive step reactivity insertion of 1 $. Use the fact that A A / / ? < 1 to simplify your answer. Calculate a numerical value for T in the case in which A = 0.1 s e c " 1 and A = 0.001sec _ 1 . 6-16 According to the point kinetics equations with one equivalent group of delayed neutrons, how long should a steady source of S0 neutrons per second be left on in order to raise the steady state reactor power level from P{ to P21 Assume A A / / 3 < 1 . Roughly plot P(t) against t. 6-17 Using the point kinetics equations with one equivalent group of delayed neutrons, derive an expression for p(t) such that P(t)= P0+ at. 6-18 Use the one-speed, one equivalent delayed neutron group form of the point reactor kinetics equation to analyze the following situation: A reactor operator wishes to reduce the power level of a reactor from Px to P2. He therefore takes the reactor subcritical to a reactivity p 0 , where p 0 is a negative number, and after a time T, he restores the reactivity to zero, whereupon the reactor levels out to P2. In terms of Pu P2, p0, and the other constants of the system, how long must he choose the time T to be? (Assume AA/ ft \ for convenience.) 6-19 Repeat the derivation of the integrodifferential form [Eq. (6-51)] of the point reactor kinetics equation for the case in which P{t) and Cj(t) are specified at an initial time, chosen at f = 0. 6-20 Demonstrate that for a periodic power variation P(t)= P0+ Plsincot, one requires a periodic reactivity insertion of the form given by Eq. (6-54). Demonstrate that this reactivity insertion has a negative bias in the sense that its average over a period is negative.
280
/
6-21
Compute the time-dependence of the reactivity insertion for a positive power excursion of very short duration compared to delayed neutron lifetimes. Express this result in terms of the total energy released in the excursion. By considering the point reactor kinetics equation with one effective delayed group, determine the time-dependence of the reactor power for a ramp reactivity insertion of p(0~ Compare this with the result predicted by the prompt j u m p approximation, Eq. (6-68). Demonstrate that following a reactivity change from p, to p 2 , the power levels before and after the reactivity step are given by P2/P\ ~ ( f i — p\)/ ft — P2) in the prompt j u m p approximation. Provide the details omitted in the derivation of the alternative form of the pointreactor kinetics equation given by Eq. (6-72). In reactor kinetics problems in which reactivity is inserted at a very rapid rate, important changes occur in the neutron density while the delayed neutron emission changes only slightly. Assuming under this approximation that for short times after reactivity insertion Eq. (6-59) is applicable, determine the reactor power level P{t) if the time-dependence of reactivity is p(/) = p 0 e x p ( f / r ) . Explicitly perform the Laplace transform inversion of the zero power transfer function Z(s) to obtain the impulse response function %(t). Prove that a necessary and sufficient condition for stability of a linear system is that
6-22
6-23
6-24 6-25
6-26 6-27
THE ONE-SPEED DIFFUSION MODEL OF A NUCLEAR REACTOR
r 00
I I %(01 dt
NUCLEAR REACTOR KINETICS 6-33
/
281
Determine the total energy generated in the excursion modeled in Problem 6-32 in terms of the reactivity insertion, delayed neutron fraction, and reactivity feedback coefficient y. Also determine the peak power generated in the excursion. Discuss the implications of these results for reactor safety. 6-34 A homogeneous fuel-moderator mixture is assembled as an unreflected critical reactor. N o cooling is provided. Suddenly a control rod is removed, bringing k to 1.02. Estimate the maximum temperature that results. The thermal conductivity is high enough that the core can be treated isothermally. As data, use: a T = — 2.5 X 1 0 ~ 4 / ° C , 0 = 0.0075, and A = 1 0 " 5 sec. 6-35 A n interesting alternative to the pulsed neutron experiment is a modification of the diffusion length experiment in which an oscillating source of thermal neutrons is placed against one end of a long column of the material to be studied, and then the oscillating component of the resulting neutron flux in the column is measured at various positions. Determine expressions for the attenuation and phase shift (relative to the source) of the flux as functions of frequency co for a nonmultiplying column as described by one-speed diffusion theory. [This is known as the neutron wave experiment.] 6-36 Derive the expression obtained for the system gain in terms of the autocorrelation ftmctions in Eq. (6-151).
3 The Multigroup Diffusion Method
i
7 Multigroup Diffusion Theory
Thus far we have based our study of nuclear reactor theory on a particularly simple model of neutron transport, one-speed diffusion theory. This model certainly suffices to introduce most of the important concepts of reactor analysis as well as many of the computational methods used in modern reactor design. It can even be used on occasion to provide useful qualitative information such as in preliminary survey design studies. However for most of the problems encountered in practical nuclear reactor design the one-speed diffusion model is simply not adequate. Two very significant assumptions were made in deriving the one-speed diffusion model. We first assumed that the angular flux was only weakly dependent on angle (linearly anisotropic, in fact) so that the diffusion approximation was valid. Usually this assumption is reasonably well satisfied in large power reactors provided we take care to modify the analysis a bit in the vicinity of strong absorbers, interfaces, and boundaries to account for transport effects. The principal deficiency of the model is the assumption that all of the neutrons can be characterized by only a single speed or energy. As we have seen, the neutrons in a reactor have energies spanning the range from 10 MeV down to less than 0.01 eV—some nine orders of magnitude. Furthermore, we have noted that neutron-nuclear cross sections depend rather sensitively on the incident neutron energy. Hence it is not surprising that practical reactor calculations will require a more realistic treatment of the neutron energy dependence. (Inaeed it is surprising that the one-speed diffusion equation works at all. Its success depends on a very judicious choice of the one-speed cross sections that appear in the equation.) We will now allow the neutron flux to depend on energy, but rather than treat the neutron energy variable E as a continuous variable, we will immediately 285
286
/
THE MULTIGROUP DIFFUSION METHOD
discretize it into energy intervals or groups. That is, we will break the neutron energy range into G energy groups, as shown schematically below: Group g
I
e
G
1—« EG-I
1 Eg
Eg-1
— I E2
1
1
*
>
E0
Notice that we are using a backward indexing scheme, corresponding physically to the fact that the neutron usually loses energy during its lifetime (and mathematically to the fact that one always solves the discretized equations starting at high energies and working successively to lower energies). As in our earlier discrete ordinates approach, it would be possible to discretize
I. A HEURISTIC DERIVATION OF THE MULTIGROUP DIFFUSION EQUATIONS Perhaps the most straightforward manner in which to arrive at the form of the multigroup diffusion equations is to apply the concept of neutron balance to a given energy group by balancing the ways in which neutrons can enter or leave this
MULTIGROUP DIFFUSION THEORY
/
287
group. Consider then a typical energy group g:
After a bit of reflection, it should be apparent that such a balance would read as follows: Time rate of change of neutrons in group g
change due to leakage neutrons " scattering out of group g
absorption' + in groupg
—
r
T
source neutrons appearing in group g
neutrons ~ scattering into group g
(7-1)
It should be noted that we have taken explicit account of the fact that a scattering collision can change the neutron energy and hence either remove it from the group g, or if it is initially in another group g\ scatter it to an energy in the group g. We will characterize the probability for scattering a neutron from a group g' to the group g by something akin to the differential scattering cross section 2 S ( £ ) (a so-called group-transfer cross section), Note that the cross section characterizing the probability that a neutron will scatter out of the group g is then given by G =
2
(7-2)
sgg
S'-l
We will similarly define an absorption cross section characterizing the group g, 2 a g , and a source term S giving the rate at which source neutrons appear in group g. Finally we will define a diffusion coefficient Dg so that the leakage from group g can be written within the diffusion approximation as VDgV(j>g. If we combine all of these terms, we find a mathematical representation of the balance relations [Eq. (7-1)] 1 3
v.
3;
=
1 2sg,A„ g'= 1
g= 1,2,...,G.
(7-3)
If we separate out that component of the source due to fissions, then we can write Sg=xg
2
(7-4)
where Xg is the probability that a fission neutron will be born with an energy in
288
/
THE MULTIGROUP DIFFUSION METHOD
group g, while 2 f g , is the fission cross section characterizing a group g' and v . is the average number of fission neutrons released in a fission reaction iduced by a neutron in group g'. Hence we now have a set of G coupled diffusion equations for the G unknown group fluxes
2
^
2
2
s*'g> Xg> 2 f g > V
( 7 -15)
We have only given a very vague definition of these constants in our heuristic derivation of the multigroup equations. Hence before we can concern ourselves with just how these equations are to be solved, we must go back and give a more careful derivation of these equations in an effort to obtain a more explicit and useful definition of the group constants.
II. DERIVATION OF THE MULTIGROUP EQUATIONS FROM ENERGY-DEPENDENT DIFFUSION THEORY Perhaps the most satisfying manner in which to derive the multigroup diffusion equations characterizing the average behavior of neutrons in each energy group is to integrate (i.e., average) the equation for the energy-dependent neutron flux, <j>(r,E,t), over a given group, E <E<Eg_v We will assume that this flux can be adequately described by the energy-dependent diffusion equation: - ^-V-DV<j> V Ot
+ 2t(r,E,t) =
rdE'UE'->EWt,E'9t)
Jo
/•OO +
X
{E)j
+ Sext(r,E,t).
dE'v{E')^{E')${x,E',t) (7-6)
Notice that we have inserted the explicit form for the fission source developed earlier in Eq. (4-50). We might mention that other energy-dependent equations could be used as a starting point for the development of the multigroup diffusion equations. 1,2 For example, we could have first developed the multigroup form of the transport equation and then introduced the diffusion approximation for each group. We will consider an alternative approach based upon the P} equations in Chapter 8. However all of these approaches yield very similar forms for the multigroup diffusion equations, with only some minor variations in the expressions for the group-averaged cross sections. We will begin by eliminating the energy variable in the energy-dependent diffusion equation by integrating Eq. (7-6) over the gth energy group characterized
MULTIGROUP DIFFUSION THEORY
/
289
by energies Eg < E < Eg_ x:
r~
d_ Eg I dt
r~
1
l
Eg
x
d E - ^ - V - |
r~ Eg
dEDV$
+ /
l
dEZt4>
= f ' *dE f"dE'2t(E'^>E)4>(T9E'9t)+ J
f
'
dES.
(7-7)
J
J
E0
Ee
0
We will proceed further by making some formal definitions. First define the neutron flux in group g as 4>g(r,t)= J
(7-8)
dE
Next define the total cross section for group g as E.
^
=
'
'dE2t(E)4>(T,Ettl
(7-9)
the diffusion coefficient for group g as fEg-1dED(E)VMriE,t)
J r
f.
(7-10) "1 dE Vj<j>(r, E, t)
and the neutron speed characterizing group g as Eg_,
f
v g
%
(7-11)
dE±
Je,
The scattering term requires a bit more work. If we break up the integral over E' to write fEg~ldE JEg
(°°dE'2t(E'^>E)4>(T9E,9t) Jo G
= 2
f r^ d E j ^ d E ' Z J t E ' ^ E M T t E ' t t ) , Eg g'-1 *
(7-12)
then it becomes evident that we want to define the group-transfer cross section as d E
= g
f
8 £
ldE
"2t(E'^>E)4>(T9Ef,t).
(7-13)
^g
A very similar procedure is followed for the fission term by writing Eg_,
E
g-i
f
dESt(r9E9t) Eg
=
C I *E0
G
dEx(E)
Eg._x
2 / g'= i E*'
dE'v(E')2f(E%(r9E'9t) ,
(7-14)
290
/
THE MULTIGROUP DIFFUSION METHOD
and then defining the fission cross section for group g' as
» V 2 v = T " J/ 9g'
Eg'
l
'
dE'r(E')2AE')*(r,E',t)9
(7-15)
while defining E
g-1
Xg=f
dE x(E).
(7-16)
If we now use these purely formal definitions to rewrite Eq. (7-7), we arrive directly a t t h e multigroup
diffusion
equations:
1
v
s
G
01
G
g'=l
g'= I
g=l,2,...,G.
(7-17)
Several comments concerning these equations are necessary. The multigroup diffusion equations (7-17) are still quite exact (within the diffusion approximation, that is), but they are also quite formal in the sense that the group constants are as yet undetermined. While it is true that our derivation has yielded explicit expressions for these group constants, it is apparent that in order to calculate them we would need to know the flux
(7-18)
in which case they reduce to group averages over the neutron flux energy spectrum
(7-19)
in our calculation of the group constants, e.g.,
J^-'dEZt(E)^ (r,Ej) pprox
2tg a —
(7-20) fEg~'dE
(r,E,t)
as averages over these approximate intragroup fluxes. In the next two chapters we will develop a number of schemes for approximating
MULTIGROUP DIFFUSION THEORY
/
291
the flux §{r,E,t) within a group—usually by first neglecting its spatial and time dependence. The accuracy required of this flux estimate is primarily dictated by the group structure itself. Of course for an extremely fine group structure, the cross sections and hence the flux would tend to be smoothly varying over a given group, and hence even a very crude approximation to the flux would be sufficient. For example, in Chapter 8 we will demonstrate that in a thermal reactor the flux behaves very roughly as §{E)~ \ / E for energies between l e V and 10 5 eV. Hence this functional form could be inserted into our definitions to calculate the group constants characterizing groups in this energy range. However when we remember the very complicated dependence of the cross sections on energy (in particular, their resonance structure), it becomes apparent that the groups would have to be very finely divided indeed for such a crude approximation to the intragroup fluxes to yield meaningful results. In actual practice, one usually works with from two to 20 groups in reactor calculations. Such few group calculations can only be effective with reasonably accurate estimates of the group constants (and hence the intragroup fluxes). The most common approach is to actually perform two multigroup calculations. In the first of these calculations, the spatial and time dependence is ignored (or very crudely approximated), and a very finely structured multigroup calculation is performed to calculate the intragroup fluxes (usually relying on various models of neutron slowing down and thermalization). The group constants for this fine spectrum calculation are frequently taken to be just the tabulated cross section data averaged (with, for example, a 1 / E weighting) over each of the fine groups. These intragroup fluxes are then used to calculate the group constants for a coarse group calculation
(including spatial dependence):
This scheme of first calculating a neutron flux spectrum and then collapsing the cross section data over this spectrum to generate few group constants is the most common method in use today. It should be noted that in a calculation of this type, the spectrum calculation (and hence the few-group constants) will depend sensitively on the particular reactor being analyzed and its operating conditions (e.g., fuel loading, isotopic composition, temperature, and coolant conditions). In fact such group constants will have to be calculated in each region of the reactor in which the core composition varies appreciably, for example, because of variations in fuel enrichment or moderator density. Furthermore these group constants will have to be recalculated whenever these regionwise core properties change, such as during fuel burnup or reactivity adjustment via the movement of control rods. Hence spectrum calculations and the generation of few-group constants must be performed repeatedly in reactor design. It is usually necessary to take some account of the spatial dependence of the flux in the generation of group constants. This is frequently done by performing the calculation of the'neutron spectrum for a typical cell of the reactor core lattice, using a variety of approximate techniques to account for flux variation throughout
292
/
THE MULTIGROUP DIFFUSION METHOD
the cell. The flux spectrum c p ( E ) that one then uses to generate group constants will be a spatial average of the true flux over the cell. Such cell calculations are essential for an adequate description of thermal reactors in which the neutron mfp is relatively short, and we will devote considerable attention to them in Chapter 10. The above discussion illustrates a very important feature of nuclear reactor analysis: the separation of the treatment of energy- and space-dependence in multigroup diffusion theory. That is, one first utilizes a rather crude description of the flux spatial dependence (say, fon only a given cell of the reactor lattice) to generate a detailed representation of the energy spectrum
for g' > g.
(7-21)
Since most few-group diffusion calculations utilize only one thermal group to describe the neutrons with E < 1 eV (assuming that neutrons cannot scatter up out of the thermal group), we can generally simplify the scattering term to write 2
2 1
+
(7-22)
g'= i
Here we have taken care to separate out the in-group scattering term which characterizes the probability that a neutron can suffer a scattering collision and lose sufficiently little energy that it will still remain within the group. It is customary to transfer this term to the left-hand side of the multigroup equation (7-17) a n d t o d e f i n e a removal
cross
^
section,
=
(7-23)
MULTIGROUP DIFFUSION THEORY
/
293
which characterizes the probability that a neutron will be removed from the group g by a collision. Note that the removal cross section is sometimes defined such that it does not contain absorption 2 a g . We will use the above definition in our development, however. We will see later that the neglect of upscattering (that is, the assumption that the neutron can never gain or scatter up in energy in a collision) greatly simplifies the solution of the multigroup diffusion equations. One frequently achieves an additional simplification of the multigroup equations by choosing the group spacing such that neutrons will only scatter to the next lowest group—that is, such that
2
1,A-1+
=
(7"24)
In this case, one refers to the multigroup equations as being directly coupled. If we recall from Section 2-II-D that a neutron of energy E cannot scatter to an energy below aE in a single elastic scattering collision, then it is apparent that to achieve direct coupling we should choose our group spacing such that Eg_x/Eg> 1 / a . For heavier moderators (e.g., 12C), this is easy to do. Unfortunately in hydrogeneous moderators a H = 0, and hence direct coupling cannot be strictly achieved. However, if one chooses Eg_x/Eg>\50, then the probability of the neutron "skipping" the next lowest group in a scattering collision with hydrogen is less than 1%, and hence direct coupling is effectively achieved. We might mention as well that one typically chooses a group structure such that the ratio, Eg_ x/ Eg, is kept constant from group to group. The motivation for this choice will become apparent in Chapter 8. We will most frequently be concerned with situations in which both the timedependence and the presence of an external source can be ignored (e.g., criticality
'g- 3
1
E
g—1
*\
£g
g
g
—2
- 2
E
g-1
-1
u
g -3
I—*,
\* \ *
I I ^ T
*
No upscattering F I G U R E 7-1.
+1
Vi
E g+2
i— E g
Directly coupled
+ 2
Alternative types of multigroup coupling
g + 1
Nondirectly coupled
g+ 2
294
/
THE MULTIGROUP DIFFUSION METHOD
calculations). In this case the multigroup equations can be written as
+
21
2 K
g'=l
(7-25)
8'= 1
The structure of these equations can be seen more clearly in matrix form -VvDjV+2, —2,
0 V-Z)2V+2R2
0
/<M
0
02 3
'S23
: <
V
M /"iXA,
f2XiSf2
l
\
:
:
(7-26)
1 :
where we have inserted the usual criticality eigenvalue k. Notice in particular that the neglect of upscattering has led to a lower triangular form for the "diffusion" matrix M . The fission matrix F is full, however, since fission neutrons induced by a neutron absorption in a lower group will appear distributed among the higher energy groups. In the case of directly coupled groups, M becomes a simple bidiagonal matrix of the form
By way of contrast, if one chooses to assign several groups to the thermal energy range in which appreciable upscattering occurs, there will be a full submatrix within M corresponding to 2 , for g' or g in the thermal range:
MULTIGROUP DIFFUSION THEORY
/
295
We will return in a later section to discuss a general strategy for solving such systems of diffusion equations. Before doing so, however, it is useful to consider several simple applications of the multigroup diffusion equations.
III. SIMPLE APPLICATIONS OF THE MULTIGROUP DIFFUSION MODEL A. One-Group Diffusion Theory Eq—co
First suppose we set up the "one-group" diffusion equation by defining and is^ = 0. Then if we note that
J
r.
oo
o dEX(E)
=h
(7-29) E0= oo
and
1 group Ex= 0
f ^ d E ' ^ E ^ E ' ) = 2S(£),
(7-30)
we find that the multigroup equations yield an old friend, the one-speed diffusion equation
i
*
+
(7-31)
Of course it should be stressed that this equation is still of only formal significance until we provide some prescription for calculating the group constants [that is, the intragroup flux—which, in this case, is
B. Two-Group Diffusion Theory A more enlightening application involves the case of two energy groups, one chosen to characterize fast neutrons and a second for thermal neutrons. The cutoff energy for the thermal group is chosen sufficiently high such that upscattering out of the thermal group can be ignored. This corresponds to an energy of 0.5-1.OeV in water moderated reactors, but may range as high as 3eV in high-temperature gas-cooled reactors. If we choose the group structure as shown below:
r
Thermal group
"
v Ey ~ 1 eV
Fast group
*— Eq = 10 MeV
296
/
THE MULTIGROUP DIFFUSION METHOD
then we can identify: E0 = f dE<j>(r,E,t) j eEx x
$x(r,t)
cE'
02(r,/)= I
Jr.
= f&st flux,
'
dE$(r,Ej)=
(7-32)
thermal flux.
(7-33)
We can simplify the group constants for this model somewhat. Consider first the fission spectrum. Since essentially all fission neutrons are born in the fast group (recall the fission spectrum x(E) illustrated in Figure 2-21), we can write
Xl=
,E E0 dEX(E)= f °dE Jr. 'Ex
Et E 1, 1,
xx22== f/ XdEX{E) JJEr-2
= 0.
(7-34)
Hence the fission source will only appear in the fast group equation: S fi = v^fi
(fast),
(7-35)
S fz = 0
(thermal).
(7-36)
We can proceed to calculate the scattering and removal cross section. First since there is no slowing down out of the thermal group, , E i — 1 eV
I
J eC = ft o
E2<E,<El.
d E \ { E ' ^ E ) = ^s{E%
(7-37)
2
Hence we find
2 E J
2
JE2
J
2
2
(7-38) Thus the removal cross section for the thermal group is just 2
R2 = 2 t 2 - 2 S 2 2 = 2 t 2 - 2 S 2 = 2a2,
(7-39)
as we might have expected. The remainder of the group constants are defined as before in the previous section. In practice they would be calculated by first performing a fine spectrum calculation for the group of interest, and then averaging the appropriate cross section data over this spectrum to obtain the group constants. For example, a fast spectrum calculation would be performed to calculate the fast group constants vu 2 f , 2 R , 2 , a n d Dx (as d e s c r i b e d in C h a p t e r 8) w h i l e a thermal spectrum calculation w o u l d b e p e r f o r m e d to c a l c u l a t e t h e thermal group an constants v2> d 2 a 2 (as described in Chapter 9). We will consider the application of two-group diffusion theory to a reactor criticality calculation. 7 Then we can set both the time derivatives and the external
MULTIGROUP DIFFUSION THEORY
/
297
source terms equal to zero to write the two-group diffusion equation as
- VD V
=
x
+
"2^2],
~V-D2V^2^)t>2=\
(7-40)
Notice that we have inserted a multiplication factor (1 / k ) in front of the fission source term since we are eventually going to be performing a criticality search. Also notice that while the source terms in the fast group correspond to fission neutrons, the source term in the thermal group is due only to slowing down from the fast group. As a specific illustration, we will apply the two-group diffusion equations, Eq. (7-40), to analyze the criticality of a bare, uniform reactor assuming that both fast and thermal fluxes can be characterized by the same spatial shape ^(r): V ^ + 5 V ( r ) = 0,
^(f s ) = 0.
(7-41)
W e have omitted the subscript g from the geometric buckling Bg= B2 so as not to confuse it with the group index g. Then if we substitute ,(!•) = ^ ( r ) ,
(7"42)
00
into Eq. (7-40), we find the algebraic equations (DXB2
+2
R
- k - ' v ^ ) ^ - k~lv2Zf2
-2Si2<#>I + (Z) 2 5 2 + 2 a > 2 = 0.
(7-43)
However this algebraic system has a solution if and only if
+
-
J ^ '
+S j
^
= 0.
(7-44)
We can now solve for the value of the multiplication factor k, which will yield a nontrivial solution of the two-group equations:
k
——7 + 7
i^+D.B2)
T7
^ - ^ r .
(2a:+D2B2)
(7-45)
It is of interest for us to see if we can relate this expression to our earlier expressions for k—notably the six-factor formula. First notice that the first term in Eq. (7-45) represents neutron multiplication due to fissions occurring in the fast group, whereas the second term represents multiplication due to thermal fission. Since we expect the thermal fission contribution to be dominant in those situations
298
/
THE MULTIGROUP DIFFUSION METHOD
in which such a two-group analysis makes sense, let us first examine 2
*2 =
"22f,
s„
(2Ri + J>i*2)
(1 + L2lB2)
{\+D2B2)
(7-46)
L2B2)
(1 +
From our earlier discussion in Section 5-III-D, it is evident that P N L i - ( 1 + L?2?2)"\
/>NL2 = (1 + L 2 2 S 2 ) _ 1
(7-47)
are just the fast and thermal nonleakage probabilities. Notice that the diffusion length Lx characterizing the fast group is defined somewhat differently as ,
D
D
i K
i
a
1
s
l
12
but this is consistent with our earlier definition of the diffusion length, since both 2 a and 2 Sj2 act to remove neutrons from the fast group. The only unidentified term is the ratio 2 / 2 R . However for a homogeneous reactor we know that this ratio is just: Rate at which neutrons slow down to thermal group Rate at which neutrons are removed from fast group
=
r^ J r J
^
. •^ s,2
=
2 Siz
= p9
(7-49)
. /r\ Ri^iv )
which we can identify as the resonance escape probability p characterizing slowing down from group 1 to group 2. Hence
In a very similar manner we can identify the fast multiplication factor as "i^/SR,
where ril = v l 2 f i / 2 ^ and we have defined a "fast utilization factor" / 1 = 2 a | / 2 R i in analogy to the thermal utilization f 2 . To complete our identification with the usual six-factor formula, we evidently must identify the fast fission factor e as just
MULTIGROUP DIFFUSION THEORY
/
299
Then we find k=kx
+ k2 = tk2=
Ifc-A/^NL^NL,
(7"53)
= Wth/K^FNL^TNL*
which is the usual six-factor formula. Of course, this definition of c is somewhat awkward, since it depends upon quantities such as the thermal utilization, the thermal nonleakage probability, and the resonance escape probability. The identification of the various components of the six-factor formula in terms of two group-constants is not unique, and alternative schemes can be found in the literature. 1,7 Such arbitrariness is indicative of the limited usefulness of the sixfactor formula for more realistic reactor analysis. In fact the two-group expression for k [Eq. (7-45)] is far more appropriate for the description of most thermal power reactor types which are characterized by a somewhat harder neutron energy spectrum than the natural uranium, graphite-moderated reactors which motivated the development of the six-factor formula. The two-group diffusion model can be used to demonstrate a number of the various applications of the multigroup formalism. For example, one frequently wishes to generate the group constants for a few-group calculation using the neutron spectrum generated by a many-group calculation. Such a procedure is known as group collapsing, since it expresses few-group constants in terms of many-group constants. To illustrate this, we can derive expressions for the one-group constants in terms of two-group constants. For example, E
E
f " d E S j E M S ) Jr.
E
f "dEZJtE)
f
l
dE^(E)
Jr.
2a = f °dE
f
JE r .2
°dE
Jr.
f
'dE<j>(E)
Jr.
(7-54) 01+02
or using Eq. (7-43) to eliminate <j>2 in terms of
a
~
D2zB2
+ 2 aa + 2 2
st2
'
The remaining one-group constants can be given as
j»2 f =
=
.
(7-57)
Z>i 2 5 + 2 a 2 + 2s 12
(These relations can be generalized to the case of collapsing from G groups, but we will leave this development to the problem set at the end of the chapter.)
300
/
THE MULTIGROUP DIFFUSION METHOD
C. Modified One-Group Diffusion Theory In the analysis of large thermal reactors it is sometimes possible to simplify the two-group diffusion equations even further by ignoring thermal leakage. If we note that in a large L W R , D2~0.5 cm, B2^ 10" 4 c m ' 2 , while 2 ^ 0 . 1 c m " 1 , then we find -D2V2
D2B2
S < a2 #>2
2
a2
Hence the neglect of - D2V2
fcOO^^r)
(7-59) a
2
and substitute this into the fast-group equation, we find the modified one-group diffusion model - V - D . V ^ + 2R1*, = v & f r + r 2 S f 2 ( 2 S i 2 / 2 a > „
(7-60)
or rearranging for the case of a homogeneous reactor
+
=
(7-61)
where p,2f
v^f
2
= —-L + — - i - J i = * ^R,
a2
+ *
,,
(7-62)
R]
where we recall L2=DX/2R. This model is sometimes referred to as one- and one-half-group diffusion theory, since it is midway between the one- and two-group schemes. A very useful (and common) modification of this scheme which takes some account of both fast and thermal leakage within a one-group treatment is obtained by replacing L2 in Eq. (7-61) by the so-called migration area M2 defined as M 2 = L 2 + L 2 2 = Z ) 1 / 2 R I + Z) 2 /2 R 2 .
(7-63)
We will show in Chapter 9 that M2 is essentially just 1 / 6 the mean square distance traveled by a neutron from its birth in fission to its eventual demise via thermal absorption. We have developed these very simple few-group models to demonstrate how
MULTIGROUP DIFFUSION THEORY
/
301
multigroup techniques can be used to evaluate many of the quantities of interest in reactor analysis. (Still further examples are included as exercises at the end of this chapter.) We could continue our discussion by considering more elaborate multigroup diffusion models, such as the four-group model customarily used in LWR analysis, or the ultrafine group structure required in fast reactor studies. However we will defer such topics until we have studied schemes for the generation of multigroup constants. Thus far our illustrations of multigroup diffusion calculations have been extremely simple. They usually ignored spatial dependence (or at best, assumed only a fundamental mode dependence). In any realistic calculation, one must take into account the inhomogeneous nature of the reactor core by actually solving the multigroup diffusion equations in detail. Hence we now turn our attention to the strategy for solving these more general equations.
IV. NUMERICAL SOLUTION OF THE MULTIGROUP DIFFUSION EQUATIONS A. Successive Solution of the Multigroup Equations We now consider a strategy for solving the multigroup diffusion equations on digital computers. Suppose we begin by writing these equations (7-26) out in detail as
-V D V
2
R
2
= ^XiS +
- VD V<j> 3 + 2r3<#>3 = 3
I x 3 S + 2 Bis *, + 2 ^
(7-64)
Notice that here we have assumed that there is no upscattering and also defined the fission source as
S(r)=
2
(" ) 7
g' = i
65
It is very important to note that the spatial dependence of the fission source is identical in each group diffusion equation. Now the essential scheme is just as before. We begin by guessing a fission source, S(r) and a multiplication eigenvalue k: S(r)-J(0,(r),
k—k®\
(7-66)
302
/
THE MULTIGROUP DIFFUSION METHOD
Next, we calculate the flux in the first group: - V - Z J ^ +S ^
1
^
(7-67)
Having obtained this flux, we can then proceed to the diffusion equation characterizing the next lowest energy group - V - Z J ^ +Z ^ -
- L x ^ t o +Z.^"
(7-68)
and solve this for ^ V ) since the right-hand side is now known. We can continue on in this fashion to determine all of the group fluxes:
*<»(r), *<'>(r), <^(r), < | > < ; > < » .
(7-69)
Having done so, we can then calculate a new fission source s°>(r)= i
(7-70)
«'=i
and a new value of k f d3rS(1)(
*<•>=-=
r)
.
(7-71)
We can then proceed to perform each source iteration by solving down the multigroup equations toward increasingly lower energies. This scheme of solving successively the equations in the direction of lower energies is enabled by the assumption that there is no upscattering. This implies that the flux in the higher energy groups always determines the source term in the lower energy groups. In effect, we are merely inverting a lower triangular matrix (as the matrix formulation in the previous section made apparent). If one chooses a multigroup structure in which more than one group is assigned to the thermal energy range in which appreciable upscattering can occur, then such a successive groupwise solution of the multigroup diffusion equations is no longer possible. One must solve the equations characterizing the thermal group simultaneously. If the number of such fully coupled groups in which both upscattering as well as downscattering occurs is small (e.g., in H T G R calculations, 4 it is common to use three to four thermal groups), a direct simultaneous solution (i.e., matrix inversion) can be accomplished. However if the number of thermal groups is large, as it may be in thermal spectrum calculations, then iterative solution schemes will be necessary (similar to the inner iterations used in multidimensional diffusion calculations). A great deal is known about the mathematical nature of such multigroup diffusion eigenvalue problems. 8,9 Under rather weak restrictions on the group fluxes and their boundary conditions, one can show that there will always exist a
MULTIGROUP DIFFUSION THEORY
/
303
maximum eigenvalue keff that is real and positive. The corresponding eigenfunction is unique and nonnegative everywhere within the reactor. These features are reassuring, because we would anticipate that the largest eigenvalue will characterize the multiplication of the system, and the corresponding eigenfunction will describe the flux distribution within the core (which cannot be negative). One can also demonstrate that the above source iteration will converge to this "positive dominant" eigenvalue keff and the corresponding eigenfunction. Such formal considerations are interesting in their own right, as well as being useful in the investigation of algorithms devised for the solution of the multigroup diffusion equations. In actual practice, however, one must also discretize the spatial dependence in order to solve the group-diffusion equations. That is, one chooses a spatial mesh and finite difference scheme, just as we did in Chapter 5, and then discretizes the diffusion equations for each group. We now turn to a brief discussion of several more practical aspects of the solution of such equations.
B. Strategies for Solving the Finite-Differenced Multigroup Diffusion Equations If we recall the general discussion of finite difference representations of the neutron diffusion equations given in Section 5-II-B, it is apparent that the general structure of the finite-differenced multigroup diffusion equations takes the form: 11
(Note here that we will occasionally write the group index g as a superscript in order to avoid confusing it with the spatial mesh indices i and j.) Notice that in addition to the coupling to different energy group fluxes at a given mesh point due to the fission source and scattering, the finite difference equation is coupled as well to the flux at adjacent spatial mesh points because of the effect of spatial diffusion. If we denote the number of spatial mesh points by N and the number of groups by G, then Eq. (7-72) represents a set of G X N simultaneous linear algebraic equations. One usually normalizes the flux at one energy group-space mesh point (since the overall normalization of the flux is arbitrary in a criticality calculation). Hence we have GxN equations available to determine the G x i V - 1 fluxes and the multiplication eigenvalue &eff. As. before, it is convenient to rewrite this set of equations as a matrix eigenvalue problem M ± = t £ ± -
(7-26)
EXAMPLE: Consider four energy groups with a five by four two-dimensional spatial mesh. Then each matrix has (4 X 5 X 4)2 = 6400 elements and the flux vector <j> has 80 elements. One typically allows fission neutrons to appear only in the = uppermost energy group, X i ~ h g > 1. Furthermore, we will assume directly coupled groups such that 2 sg , g = 0 if g'¥=g — 1. We can explicitly exhibit the structure of the matrix eigenvalue problem in this case as shown in Figure (7-2).11
MULTIGROUP DIFFUSION THEORY
/
305
Let us now review the general iterative strategy for solving this eigenvalue problem. (T) One first makes an initial guess of the source vector S<0) and the multiplication eigenvalue (2) At this point one proceeds to solve the inhomogeneous matrix equation M$(n+l)=-^S(n)
(7-73)
for the next flux iterate, (w+1). This solution involves a number of substeps: 2-1 One solves the inhomogeneous diffusion equation characterizing each of the energy groups g ( « + ! ) _
k
1 M
qOO
i
d
_
^
(«)
by solving first for the highest energy group, g = 1 (noting ^o 0 — and then using to solve for
Of course, solving even the inhomogeneous diffusion equation [Eq. (7-74)] for a single group is no trivial matter. For multidimensional problems, iterative techniques will be necessary such as those discussed in Chapter 5 (e.g., SOR). Such inner iterations usually take the previous flux estimate g_I) as their first guess in solving Eq. (7-74). It should be mentioned that a variety of schemes have been proposed (and utilized) for coupling such inner iterations to the outer (source) iterations to accelerate convergence. Q) Having obtained the flux estimate
(7-76)
to find k(n+1) = k
M
(F
(7-77)
I if*=l
\
I if*>l
MULTIGROUP DIFFUSION THEORY @
/
307
A t this p o i n t o n e tests t h e s o u r c e i t e r a t i o n f o r c o n v e r g e n c e , s u c h as comparing
by
9 jc(n+
(7-78)
o
o r a p o i n t w i s e criterion
max
$ ( « + D _ $ («) Si Si c
®
?
(7-79)
(n+ 1)
(or b o t h ) . If t h e c h a n g e s in k(n) or t h e e l e m e n t s of S ( n ) o r <j!>(n) a r e s u f f i c i e n t l y small, o n e a s s u m e s t h a t c o n v e r g e n c e h a s b e e n a c h i e v e d , a n d t h e iterative p r o c e d u r e is e n d e d . If n o t , t h e n a n e w fission s o u r c e is c a l c u l a t e d a n d t h e iteration continues. U s u a l l y t h e fission s o u r c e 5 ( n ) u s e d in t h e n e x t i t e r a t i o n is c h o s e n via a n e x t r a p o l a t i o n s c h e m e (cf. S e c t i o n 5 - I V - C ) t o a c c e l e r a t e c o n v e r g e n c e of t h e source iterations, for example,
s
~
( n )
=
k
in)
\
(n) +
F =
~
a
k( n)=J-
(7-80) —
T h e positive n a t u r e of t h e m u l t i g r o u p d i f f u s i o n o p e r a t o r s implies similar p r o p e r ties f o r t h e m a t r i c e s resulting f r o m f i n i t e - d i f f e r e n c i n g these e q u a t i o n s . H e n c e m a n y of t h e f o r m a l c o n c l u s i o n s c o n c e r n i n g c o n v e r g e n c e t o t h e positive d o m i n a n t eig e n v a l u e a n d e i g e n f u n c t i o n c a n a l s o b e s h o w n t o h o l d f o r the f i n i t e - d i f f e r e n c e d m u l t i g r o u p d i f f u s i o n e q u a t i o n s . T h e t h e o r y of s u c h n u m e r i c a l p r o c e d u r e s h a s b e e n placed o n firm g r o u n d by Varga10 a n d Wachpress,8 a n d the interested reader is r e f e r r e d t o their treatises f o r m o r e detail. T h u s f a r w e h a v e o n l y c o n c e r n e d ourselves w i t h t h e u s u a l criticality c a l c u l a t i o n f o r kefV A c t u a l l y a variety of d i f f e r e n t t y p e s of criticality s e a r c h e s a r e typically c o n d u c t e d in a n e f f e c t to a c h i e v e a d e s i r e d c o r e m u l t i p l i c a t i o n (which is f r e q u e n t l y n o t e q u a l t o 1.0). F o r e x a m p l e , t w o s u c h s c h e m e s a r e : (a) c o r e size s e a r c h e s (usually t h e size of a given c o r e r e g i o n is v a r i e d , say b y v a r y i n g t h e p o s i t i o n of o n e b o u n d a r y , u n t i l t h e d e s i r e d v a l u e of & eff is a c h i e v e d ) a n d (b) c o m p o s i t i o n s e a r c h (the a t o m i c d e n s i t y of a specific f u e l isotope, o r several s u c h isotopes, is varied, p e r h a p s o n l y i n a given s u b r e g i o n of t h e core). T h e m u l t i g r o u p d i f f u s i o n e q u a t i o n s c a n also b e u s e d f o r a v a r i e t y of o t h e r a p p l i c a t i o n s . F o r e x a m p l e , o n e c a n u s e these e q u a t i o n s t o d e t e r m i n e t h e n e u t r o n f l u x m a i n t a i n e d in a subcritical a s s e m b l y b y a s t e a d y - s t a t e s o u r c e ( b y r e t a i n i n g t h e s o u r c e t e r m in t h e m u l t i g r o u p d i f f u s i o n e q u a t i o n s ) . T h e y c a n also b e u s e d t o c a l c u l a t e t h e t i m e eigenvalues a o c c u r r i n g in s o l u t i o n s of t h e f o r m eat b y inserting a n e f f e c t i v e a b s o r p t i o n t e r m (a/vg)
308
/
THE MULTIGROUP DIFFUSION METHOD
V. MULTIGROUP PERTURBATION THEORY Before we conclude this general discussion of the multigroup diffusion equations, it is useful to redevelop the procedure of perturbation theory as it applies to the multigroup equations as written in matrix form: (7-81)
In order to determine the change in reactivity induced in the core by a small perturbation, we must generalize somewhat our concept of inner products and adjointness to account for the matrix nature of the multigroup eigenvalue problem [Eq. (7-81)]. To this end, define the inner product between two G-dimensional vectors /(r) and g(r) as ( / , g ) = [^/•[/t(r)g1(r)+/*(r)g2(r)+...]. —
(7-82)
J y
We can now use this inner product to construct the adjoint of the operators M and F:
/ > £ ) = (/> M S ) -
(7-83)
Since the adjoint of a matrix is obtained by first taking the transpose of the matrix and then complex-conjugating each of its elements, it is evident that ' - V - ^ V + Sr, M M
-2Sl2
0
- V D 2 V + ZK2
I
(7-84)
and similarly '"ixA,
*'iX2 2 f I
*
£+ = [ *2XI2,2
f 2 X 2 2f 2
• • • ).
(7-85)
Note in particular that and F ^ ^ F — t h a t is, the multigroup criticality problem is not self-adjoint. Hence we find <j>*^=
M + 8M9
(7-86)
F+
(7-87)
and 8F.
MULTIGROUP DIFFUSION THEORY
/
309
For example, one could imagine a perturbation in the absorption cross section characterizing the second group as being represented by r
m = \
0
0
0
0
S2 a
0
2
o
o
0
I-
<7-88)
•
(7-89)
The corresponding reactivity change is then given by (
8F~
*P=
8M1 ~
To make these ideas more precise, let us consider the particularly simple example of two-group diffusion theory as described by Eq. (7-40). In matrix form, these equations become
/-v-D.v+z^ \
o
-SSi2
-V-D2V
W<M = i("i 2 f, +2J
U2/
M
^v
"2sf2\ /<M
0
0 /U 1 | F—
±
2
/.
(7-90)
±
k
—
The adjoint equations are
\
0
-V-D2V+2j
\)
M+
i
(7-91)
Ik F
+
=
f
±f
Note that M1~¥= M and F 1 "^ F, hence Suppose we consider the reactivity change induced by perturbing the thermal absorption cross section by an amount Then 8F = 0,
m=(°o
< 7 " 92 )
J -
Hence we can compute (if, l i ) =
Jyd3r
(o
S2 a
=
(7-93)
to find d3r
f A p =
~
V
< t
P
x
=
~ 7 1
(7-94)
310
/
THE MULTIGROUP DIFFUSION METHOD
In analogy with our earlier one-speed calculation, suppose we set S2aj=«S(r-r0).
(7-95)
Ap=-^(r0)*2(r0).
(7-96)
Then
Thus we find
4>l(r0) =
cAp fractional change in —— = reactivity per neutron 2 absorbed per unit time.
(7-97)
In particular, if | is large at r0, then the change in p introduced by a thermal absorber at r0 will be large. That is, |(r) measures the "importance" of the point r with respect to reactivity changes induced by perturbing the thermal absorption at that point. For the more general multigroup problem, ^J(r) can be identified as the neutron importance function for group g, since J(r) is proportional to the gain or loss in reactivity of a reactor due to the insertion or removal of one neutron per second in the group g at point r. In general, one finds that the multigroup neutron importance (or adjoint fluxes) differs substantially f r o m the multigroup fluxes. This is shown for a two-group calculation for a reflected slab geometry core in Figure 7-4.
FIGURE 7-4.
The two-group fluxes and adjoint fluxes for a reflected slab reactor
MULTIGROUP DIFFUSION THEORY
/
311
VI. SOME CONCLUDING REMARKS W e have n o w developed the multigroup diffusion equations a n d outlined a s t r a t e g y f o r solving t h e m . T h e m a j o r u n c e r t a i n t y at this p o i n t c o n c e r n s j u s t h o w o n e d e t e r m i n e s t h e g r o u p c o n s t a n t s . T h e s e latter c a l c u l a t i o n s rely o n clever guesses o r a p p r o x i m a t i o n s f o r t h e i n t r a g r o u p fluxes, a n d m o r e specifically o n o n e ' s ability to d e t e r m i n e t h e n e u t r o n e n e r g y s p e c t r u m c h a r a c t e r i z i n g f a s t a n d t h e r m a l n e u t r o n s . T h i s is t h e s u b j e c t t o w h i c h w e will n e x t t u r n o u r a t t e n t i o n as w e s t u d y t h e slowing d o w n a n d t h e r m a l i z a t i o n of n e u t r o n s .
REFERENCES 1. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press, (1958), p. 509. 2. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, Princeton, N. J. (1970), p. 181. 3. R. L. Hellens, The Physics of P W R Reactors, in New Developments in Reactor Physics and Shielding, USAEC CONF-720901 (1972). 4. M. H. Merrill, Nuclear Design Methods and Experimental Data in Use at Gulf General Atomic, GULF-GA-A12652 (GA-LTR-2), 1973. 5. R. Avery, Review of F B R Physics, in New Developments in Reactor Physics and Shielding, U S A E C CONF-720901 (1972). 6. B. J. Toppel, A. L. Rago, and D. M. O'Shea, MC 2 , A Code to Calculate Multigroup Cross Sections, ANL-7318, Argonne National Laboratory, 1967. 7. P. F. Zweifel, Reactor Physics, McGraw-Hill, New York (1973). 8. E. L. Wachspress, Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equation of Reactor Physics, Prentice-Hall, Englewood Cliffs, N. J. (1966). 9. G. J. Habetler and M. A. Martino, Proceedings of Symposium on Applied Mathematics, Vol. XI, American Mathematical Society (1961), p. 127. 10. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N. J. (1962). 11. R. G. Steinke, A Review of Direct and Iterative Strategies for Solving MultiDimensional Finite Difference Problems, University of Michigan Nuclear Engineering Report (1971). 12. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, Mass. (1966).
PROBLEMS 7-1
7-2 7-3 7-4
Estimate the fast-group constants characterizing H 2 0 if the fast group is taken from El = 1 eV to E0 = lOMeV and the neutron energy spectrum over this group is taken as
312
/
7-5
Repeat Problem 7-4 for the case in which the lowest two groups correspond to thermal neutrons (which will be characterized by appreciable upscattering). Calculate the critical size and mass of a bare sphere of pure 2 3 5 U metal using the group constants characterizing groups 1 and 2 in Table 7-1.
7-6
THE MULTIGROUP DIFFUSION METHOD
TABLE 7-1
8
ANL Four-Group Microscopic Cross Sections (in barns)*
Lower Energy
u [ln£0/£L]
EL MeV keV eV
1
1.353
2
9.12
3
0.4
4
0
8
V
1 4.7 2 7.0 3 51.0 4 .597.0
H2O
Fission Spectrum
2
->* + I
<*tr
0.575
3.08
0
2. 81
0.425
10.52
0
4.04
17.03
0
16.55
0.035
4.14
—
0
68.6
0.57
0
7
U 235 OF o
u238 y
a
CT
tr
inR
2.65 1.3 0.1 1.4 1.4 0.3 0 2.55 23.0 18.0 0 2.5 0 2.5 490 97
0 0 0.01 0
4.7
7.0 11.0
13.0
V
2.65 ----
ff
0.53 0 0 0
0.04 0.18 0.8 2.4
inR
a
2.1 0 0 0
0 0 0.01 0
eR
t
From Reactor Physics Constants, ANL-5800 (1963). [Here, a i n R and a e R refer to the inelastic and elastic scattering removal cross sections for the group similar to the definition given in Eq. (7-23).]
7-7
7-8
7-9
7-10
Compare the critical radius of a 2 3 5 U sphere as given by one-group, two-group, and modified one-group (1-1 / 2 group) diffusion models. Again use the data from groups 1 and 2 of Table 7-1. Determine the thermal flux due to an isotropic point source emitting S0 fast neutrons per second in an infinite moderating medium. Use two-group diffusion theory. In particular, discuss the solution to this problem for the case in which Lx> L2 and L,
TABLE 7-2
Few-Group Diffusion Theory Constants for a Typical PWR Reactor Core
Four-Group
Two-Group Group Constant
1 of 2
2 of 2
1 of 4
3 of 4
3 of 4
4 of 4
D 2R
.008476 .003320 .01207 1.2627 .02619
.18514 .07537 .1210 .3543 .1210
.009572 .003378 .004946 2.1623 .08795
.001193 .0004850 .002840 1.0867 .06124
.01768 .006970 .03053 .6318 .09506
.18514 .07527 .1210 .3543 .1210
MULTIGROUP DIFFUSION THEORY
/
313
7-11
Using the group constants of Table 7-2, calculate each of the terms in the six-factor formula for the core described in Problem 7-10. 7-12 The neutron "age" r is defined as 1 / 6 of the mean-square distance a fast neutron will travel before it slows down or is absorbed. Derive an expression for the age r in terms of two-group constants. In particular, compute the age for the group constants characterizing a L W R in Table 7-2. 7-13 Recall that we defined the conversion ratio C R characterizing a reactor as the ratio of the production rate of fissile nuclei to the destruction rate of fissile nuclei. Derive an expression for the conversion ratio in a slightly enriched uranium-fueled reactor (such as a LWR) at the beginning of core life (i.e., such that the plutonium density is zero). 7-14 One important method of controlling reactivity in a P W R is to dissolve a poison such as boron in the coolant. If we assume that boron only affects the thermal absorption, 2 a z , derive an estimate of the critical boron concentration that will render k 00 = 1. Use this expression to estimate the boron concentration necessary to render the reactor described in Problem 7-10 critical. Take o f s O , a® =2207 b. 7-15 Calculate the one-group constants corresponding to collapsing the two-group set given in Table 7-2. 7-16 Derive a general group collapsing expression for TV-group constants in terms of N X M-group constants (e.g., 2-group in terms of 4-group). 7-17 Calculate the two-group constants corresponding to collapsing the four-group constants in Table 7-2 by assigning groups 1, 2, and 3 to group 1 and group 4 to group 4, respectively. Compare these with the two-group constants listed in Table 7-2. 7-18 Using two-group theory, determine the critical core width of a slab reactor with core composition similar to that of a P W R and surrounded by an infinite water reflector. Use the two-group constants of Table 7-2 supplemented with group constants characterizing the water reflector: D j = 1.13 cm, D 2 = 0.16 cm, 2 R =0.0494 c m - 1 2 a 2 = 0.0197 c m " 1 , 2 a i = .0004 cm" 1 . 7-19 A beam of Jin fast n e u t r o n s / c m 2 / s e c is incident on the plane face of an infinite homogeneous nonmultiplying half-space. Use two-group diffusion theory and partial current boundary conditions to calculate 7 o u t , the partial current of thermal neutrons going back toward the fast source. 7-20 Consider a thin slab of fuel surrounded by an infinite moderator. Suppose that the slab is thin enough that it is essentially transparent to nonthermal neutrons. However because of its large thermal absorption cross section, all thermal neutrons striking the surface of the slab are absorbed. Suppose further that every thermal neutron absorbed in the fuel results in the net production of t j fission neutrons. Using the two-group diffusion theory model, develop an equation for the effective multiplication factor for this system. 7-21 Assuming that fwo-group diffusion theory with one group of delayed fission neutrons represents a valid description of the reactor, write the three coupled differential equations for the time and spatial dependence of the neutron precursor fragment concentration C(r,f) and the fast and slow group fluxes ^ ( r , t) and <J>2(r, 0 respectively. The average speed, diffusion constants, and removal (by all types of collision) cross sections in the fast and slow groups are u,, Dh and v2, D2, and respectively. Fission can be produced only by slow group neutrons, and the cross section is 2 f 2 . There are v neutrons emitted per fission, all into the fast group, with a fraction /? delayed. The decay constant of the neutron precursor is X, and the cross section for transfer of neutrons from the fast group to the slow group is 2 Si2 . By assuming that the spatial shapes of <j>l9
Write out the explicit form of the matrix equations A/j>= k ~ XF<$> and Sg + R g - f o r a four-group, 4 x 4 x 3 three-dimensional spatial mesh problem in which the lowest two groups are both taken in the thermal range in which significant upscatter can occur.
=
314 7-23
7-24
7-25 7-26 7-27
/
THE MULTIGROUP DIFFUSION METHOD Write a simple two-group, one-dimensional diffusion code (similar to the one-group code written in Problem 5-27). Then treating the L W R described in Problem 7-10 as a slab geometry, calculate the width necessary for reactor criticality. Using ANL-7411, list and contrast various multigroup diffusion codes. In particular, compare their group structure, inner-outer iteration strategy, source-extrapolation methods, criticality search options, and estimated running times. Calculate i,02 and the adjoint fluxes, <£[,<J>2 for a bare, spherical reactor of radius R. Compute and sketch both the two-group fluxes and adjoint fluxes for the reflected slab reactor described in Problem 7-18. A critical system consisting of a slab surrounded on both sides by infinite reflectors is to be described by two-group diffusion theory. The center third of the core suddenly has its thermal group absorption cross section increased slightly. Find the change in core multiplication by perturbation theory. Then to demonstrate that the work saved over directly calculating the multiplication factor for the perturbed system is very great, carry out the latter calculation until as many pages are filled as were required to perform the perturbation calculation.
8 Fast Spectrum Calculations and Fast Group Constants
The principal tool of nuclear reactor analysis is multigroup diffusion theory. In Chapter 7 we developed the general form of the multigroup diffusion equations and prescribed a strategy for their solution. However these equations contained various parameters known as group constants formally defined as averages over the energy-dependent intragroup fluxes (r,£) which must be determined before these equations assume more than a formal significance. The determination of suitable approximations for the intragroup fluxes, that is, the neutron energy spectrum, is the key to the generation of group constants that will yield an accurate few-group description of nuclear reactor behavior. Hence our goal in this chapter and Chapter 9 is to develop methods for generating few-group constants by averaging fundamental microscopic cross section data over suitable approximations to the neutron energy spectrum. Of course there are two aspects to this problem, since both the energy and the spatial dependence of the intragroup fluxes must be estimated. Because of the extremely complicated dependence of microscopic neutron cross sections on energy, it is necessary to provide a rather careful treatment of the energy dependence of the intragroup flux. Of course if the number of groups is very large ( > 5 0 ) , one can frequently get by with rather crude estimates of this energy dependence. For example, for energies above 1 MeV, one might approximate <$>(E) by the fission spectrum x(E)\ for intermediate energies, 1 e V < E< 1 MeV, we will see that
316
/
THE MULTIGROUP DIFFUSION METHOD
usually required to perform a detailed multigroup calculation of the neutron energy spectrum using a fine-group structure (with a crude treatment of spatial dependence) which can then be used to average or "collapse" fundamental cross section data to generate few-group constants. The methods used to generate the neutron energy spectrum vary, depending on the range of neutron energies of interest. For example, at high energies the dominant process is neutron slowing down via both elastic and inelastic scattering. At intermediate energies, resonance absorption becomes quite important. At low energies, upscattering becomes appreciable as the neutrons tend to approach thermal equilibrium with the nuclei comprising the reactor core. Hence it is customary to divide the range of neutron energies into three different regions, each characterized by these different types of interaction, as indicated below: Upscattering Chemical binding
Elastic scattering from stationary, free nuclei (isotropic in CM, s-wave)
Elastic scattering (anisotropic in CM, p-wave)
N o upscattering resonance absorption (resolved resonances)
N o upscattering resonance absorption (unresolved resonances) fission sources
Inelastic scattering
Diffraction
0
105 eV
1 eV
V
^
J
Neutron Thermalization
V
^
J
Neutron moderation or slowing down
107 eV
\
^
J
Fast fission
In this chapter we will be concerned with the calculation of the neutron energy spectrum characterizing fast neutrons. Hence our study of the calculation of fast neutron
spectra
a n d the g e n e r a t i o n of fast
group
constants
will b e d o m i n a t e d b y a
development of the theory of neutron slowing down and resonance absorption. In C h a p t e r 9 w e t h e n c o n s i d e r t h e c a l c u l a t i o n of thermal
neutron
spectra
and
thermal
group constants involving the development of the theory of neutron thermalization. One must also account for the spatial dependence of the intragroup fluxes, as well as the spatial variation of the reactor core composition. Of course a detailed treatment of this spatial dependence in the generation of multigroup constants would be impractical—indeed the primary motivation for the generation of fewgroup constants is to yield a sufficiently coarse treatment of neutron energy dependence to allow such a detailed spatial calculation. Instead we must introduce a very approximate treatment of the spatial dependence of the intragroup fluxes. Of course the most drastic such approximation is simply to ignore the spatial dependence altogether in the calculation of the neutron spectrum. That is, one effectively assumes that the intragroup fluxes can be calculated by assuming that neutron slowing down and thermalization occur in an infinite medium in which there is no spatial dependence. We will begin our study of both fast and thermal spectra for this situation because of its simplicity.
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
317
In most cases, however, one cannot so cavalierly ignore the spatial dependence of the intragroup fluxes. Hence we will modify our study of infinite medium spectrum calculations to include a very simple treatment in which the intragroup flux spatial and energy dependence are assumed to be separable, and this spatial dependence is then characterized by a single parameter (an effective value of the geometric buckling B g ). However even this extended treatment is inadequate for many situations in which the detailed geometry of the reactor lattice must be taken into account. This is particularly true for an adequate treatment of both resonance and thermal neutron absorption. The few-group constants must be modified to account for the variation of both the neutron flux and the material composition in a small subregion or cell of the reactor core. In effect one must spatially average the few-group constants over a unit cell of the core lattice (much as the microscopic group constants have been averaged over energy to generate the few-group constants) before they can be used in a few-group diffusion calculation. We will consider such cell-averaging techniques in Chapter 10. Therefore the generation of appropriate few-group constants usually involves a sequence of both energy and spatial averages of the microscopic cross section data over approximate estimates of the neutron energy spectrum and spatial dependence characterizing the neutron flux in a given subregion of the reactor of interest. As such, these group constants are dependent on the reactor design of interest, and also the region of the reactor core under investigation, its present operating conditions (e.g., coolant densities, temperatures), as well as its past history (e.g., fuel depletion, fission product buildup). The generation of multigroup constants is an extremely important (and usually time consuming) facet of nuclear design since it is the key to the successful implementation of the multigroup diffusion technique.
I. NEUTRON SLOWING DOWN IN AN INFINITE MEDIUM A. Introduction Our initial goal is the calculation of the energy dependence of the neutron flux <£>(r,£) in the fast energy range, E > 1 eV, in which upscattering can be ignored. To simplify this calculation, we will first consider neutron slowing down from sources uniformly distributed throughout an infinite, uniform medium. 1 ' 2 ' 4 In this case, all spatial dependence disappears, and the neutron continuity equation, Eq. (4-79), reduces to an equation for the neutron energy spectrum <{>(r, is )-»<£(£) of the form (8-1) That is, the neutron transport equation simplifies rather dramatically to an integral equation in the single variable Ey which we will refer to as the infinite-medium spectrum
equation.
Of course the neglect of all spatial dependence may seem very drastic, and we will later study schemes for reincluding it in our analysis. In most large reactors, however, leakage is a relatively minor effect in comparison to neutron energy variation. Furthermore even spatial effects due to core heterogeneities are usually
318
/
THE MULTIGROUP DIFFUSION METHOD
of secondary importance, particularly in the fast neutron energy range. Hence we are certainly justified in focusing our initial attention on the energy dependence of the neutron flux. Before we can proceed further, we must provide more information concerning the energy dependence of the neutron cross sections 2 t (is) and 2 S ( £ " — C o n s i s tent with our present concern with fast neutron spectra, we will assume that the scattering nuclei are initially at rest and free to recoil. That is, we will consider only fast neutron energies much greater than the thermal energy of the nuclei E ^ > k T so that upscattering in collisions can be ignored. We will further restrict our initial investigation by considering only the process of elastic scattering that is isotropic in the center of mass system. In this case, our earlier study of two-body collision kinematics has indicated that the differential scattering cross section is of the form E
(1 - a ) E " 0.
a
< E
'
< E
(8-2)
otherwise.
where a = (A — \/A +1) 2 . This form is sufficiently simple to enable us to obtain a n a l y t i c a l s o l u t i o n s t o t h e neutron
slowing down r
+
equation
00 dE"29(E'^>E)t(E')
+ S(E)
(8-3)
J
E
in many cases of interest, even without specifying the detailed forms of the cross sections 2 s (is) and 2 a (is). Of course it will be necessary to use more general numerical methods in solving Eq. (8-3) for most practical situations. However those analytical results we will develop are of considerable use in formulating efficient numerical schemes, as well as in checking their accuracy. By restricting our attention to s-wave elastic scattering, we have limited the applicability of our initial analysis to neutron energies below 1 MeV in light moderators, and below 0.1 MeV in heavier materials, 3 since the average scattering angle cosine Ji0 in the CM system is given roughly by /Z0 = 0.07 A 2 / 3 £[MeV]. More specifically, in hydrogen, s-wave scattering is dominant below several MeV. By way of contrast, for heavier nuclei such as 1 6 0, p-wave scattering becomes significant above roughly 50 keV. We will justify our restricted study of only s-wave scattering by noting that in thermal reactors, most neutron moderation is due to light elements such as jH, 2 D, or X\C. For neutron energies above 0.1 MeV, inelastic scattering from heavy nuclei becomes important and cannot be ignored. Unfortunately the complexities of inelastic processes severely limits the extent to which one can use analytical methods to investigate neutron slowing down. Hence our consideration of inelastic scattering will be deferred until our discussion of the numerical solution of the neutron slowing down equations. We will begin our study of Eq. (8-3) for the particularly simple case of neutron slowing down in an infinite medium of hydrogen. This study is of particular importance since hydrogen is a very common moderator, and moreover this analysis will serve to introduce a number of concepts which prove useful in the study of more general neutron slowing down problems.
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
319
W e will t h e n t u r n to the s t u d y of n e u t r o n slowing d o w n via elastic s c a t t e r i n g f r o m n o n h y d r o g e n e o u s m o d e r a t o r s . T h e r e a d e r s h o u l d b e c a u t i o n e d t h a t while the d e t a i l e d a n a l y s i s of n e u t r o n slowing d o w n in this section m a y look l e n g t h y a n d c o m p l i c a t e d , t h e a c t u a l results of this a n a l y s i s a r e e x t r e m e l y simple. W e will f i n d t h a t t h e n e u t r o n f l u x will usually b e h a v e essentially as (£)— l/1,s(E)E, provided o n e is n o t n e a r energies c h a r a c t e r i z i n g a n e u t r o n s o u r c e or a s t r o n g cross section r e s o n a n c e . H o w e v e r , as is, c h a r a c t e r i s t i c of so m u c h of physics, the analysis n e c e s s a r y to d e r i v e a n d j u s t i f y this result is c o n s i d e r a b l y m o r e c o m p l e x t h a n the result itself.
B. Neutron Moderation in Hydrogen 1. S L O W I N G D O W N IN T H E ABSENCE O F ABSORPTION W e will first e x a m i n e t h e very simple c a s e in w h i c h all a b s o r p t i o n is i g n o r e d . A c t u a l l y s u c h a n a s s u m p t i o n is n o t t o o d r a s t i c in h y d r o g e n , since a " / a " ~ . 0 1 4 . H o w e v e r w e a r e m o r e generally c o n c e r n e d with n e u t r o n slowing d o w n in a h y d r o g e n e o u s m o d e r a t o r c o n t a i n i n g in a d d i t i o n a h e a v y m a s s a b s o r b e r ( s u c h as 238 U ) . E v e n in this case, m o s t n e u t r o n a b s o r p t i o n o c c u r s in fairly n a r r o w energy r a n g e s c o r r e s p o n d i n g t o a b s o r p t i o n r e s o n a n c e s , a n d f o r energies f a r f r o m these r e s o n a n c e s , t h e neglect of a b s o r p t i o n will yield t h e correct q u a n t i t a t i v e b e h a v i o r of the n e u t r o n s p e c t r u m ^(-E1). F o r h y d r o g e n w e set A = 1 a n d h e n c e a H = 0. T h e n in the a b s e n c e of a b s o r p t i o n , E q . (8-3) b e c o m e s f00 I dE'
2 a ( i ? )*(£)=
2S(£>(£') +S{E).
(8-4)
Je
W e will b e g i n b y d e t e r m i n i n g t h e n e u t r o n e n e r g y s p e c t r u m
+ S ^ E - E t ) .
(8-5)
H e r e w e h a v e set the u p p e r limit of i n t e g r a t i o n at E0 since n o n e u t r o n s c a n achieve energies h i g h e r t h a n the s o u r c e e n e r g y if o n l y d o w n s c a t t e r i n g is possible. W e c o u l d solve this p r o b l e m as w e d i d the p l a n e s o u r c e d i f f u s i o n p r o b l e m of C h a p t e r 5, b y restricting o u r a t t e n t i o n to energies E < E 0 a n d using the s o u r c e as a b o u n d a r y c o n d i t i o n as E ^ E 0 . H o w e v e r it is m o r e c o n v e n i e n t in this case to solve Eq. (8-5) directly. F i r s t n o t e t h a t b e c a u s e of t h e singular source, t h e s o l u t i o n F(E) m u s t c o n t a i n a t e r m p r o p o r t i o n a l to 8(E— E0). H e n c e w e will seek a s o l u t i o n of the form F(E)
= Fc(E)
+
C8(E-E0),
(8-6)
320
/
THE MULTIGROUP DIFFUSION METHOD
where C is an undetermined constant and FQ(E) is the nonsingular component of F{E). Substituting this form into Eq. (8-5), one finds rE° F(Ef) c = J d E ' - ^ j - + ±- + JF 0
Fc(E)+C8(E-E0)
SQ8(E-Ej.
(8-7)
However the singular contributions must be equal. Hence we demand C = S0. Then we find that FC(E) satisfies the remaining nonsingular equation
F(E)= -
J
I rE
+
E'
E<E0.
E
(8-8)
Notice something rather interesting here. FC{E) satisfies an equation in which the source term is just that corresponding to source neutrons making a single collision at the source energy E0—that is, the effective source presented by "once-collided" neutrons S(E) = S 0 P ( E 0 ^ E ) = ^ .
(8-9)
Thus FC{E) can be interpreted as the collision density due to neutrons that have suffered at least one collision (hence the subscript "c"). In a similar sense, we now identify the singular component of F{E), S08(E— E0), as simply the contribution due to uncollided source neutrons (present only at the source energy E0). We now must solve the remaining equation (8-8) for FC(E). We can solve this integral equation by first differentiating it to convert it into an ordinary differential equation (refer to Appendix B): dpc dE
E \ rfi-o ° ( dE' dE J w d
w
)
E'
_d_l So dE\E0
(8-10)
or dFc d E
i = - ^ H E ) .
(8-11)
The general solution to Eq. (8-11) is of the form FC{E)= C/E. To determine the constant C, note that we can infer an initial condition at E = E 0 directly from the integral equation (8-8): (8-12) Hence we must choose C = 5"0 to find the total solution F ( E ) = ^ +
S08(E-E0),
(8-13)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
321
or S0 +
jirE
^r>
H E
-
E
°
y
<8
-' 4 )
It s h o u l d b e o b s e r v e d t h a t if t h e s c a t t e r i n g cross section is o n l y w e a k l y d e p e n d e n t o n e n e r g y ( w h i c h it usually is), t h e n b e l o w the s o u r c e e n e r g y E0 the flux a s s u m e s a \ / E f o r m . T h i s f u n c t i o n a l f o r m c o u l d in f a c t b e u s e d as a c r u d e e s t i m a t e of t h e f l u x in g e n e r a t i n g g r o u p c o n s t a n t s c h a r a c t e r i z i n g n e u t r o n slowing d o w n , b u t w e will p r o c e e d to d e v e l o p m o r e s o p h i s t i c a t e d m o d e l s . A very similar analysis c a n b e u s e d to c o n s t r u c t t h e s o l u t i o n c o r r e s p o n d i n g to a d i s t r i b u t e d s o u r c e S(E). I n this case, o n e f i n d s (see P r o b l e m 8-2) t h a t 1 '
2
b
/-00 \ dE'S
(E)EJ
S(E) —
(£")+
(
8
-
1
5
)
2b(E)
e
O n c e a g a i n w e n o t e t h a t b e l o w t h e s o u r c e energies, t h e flux b e h a v e s essentially as \/E. 2. SOME U S E F U L D E F I N I T I O N S (a) THE SLOWING
DOWN
DENSITY
It is u s e f u l t o i n t r o d u c e several n e w d e f i n i t i o n s at this p o i n t in o r d e r to facilitate o u r later analysis. W e first w a n t to d e f i n e t h e n e u t r o n slowing down density q(E) d e s c r i b i n g t h e r a t e at w h i c h n e u t r o n s slow d o w n p a s t a given e n e r g y E. T o b e m o r e general, we define q(r E)d3r=
n u m
^
e r
n e u t r o n s
slowing d o w n p a s t e n e r g y
^g
^
E p e r sec in d3r a b o u t r. If w e recall t h a t t h e d i f f e r e n t i a l s c a t t e r i n g cross section d e s c r i b e s t h e p r o b a b i l i t y t h a t a n e u t r o n will s c a t t e r f r o m a n initial e n e r g y E' to a f i n a l e n e r g y E" in dE", t h e n w e c a n write R a t e at which neutrons t h a t s u f f e r collision a t e n e r g y E' in dE' slow d o w n p a s t E
f\(E'^E")<j>(r,E')dE" Jc\
dE'.
(8-17)
H e n c e t h e t o t a l slowing d o w n d e n s i t y resulting f r o m all initial energies E'>E given b y r
q(r,E)=
co
rE
dE' J e
is
dE"2s(E'^>E")
(8-18)
^o
W e c a n a p p l y this d e f i n i t i o n t o c a l c u l a t e t h e slowing d o w n d e n s i t y in o u r infinite,
322
/
THE MULTIGROUP DIFFUSION METHOD
nonabsorbing, hydrogeneous moderator as 00
q(E)
dE
=
dE
I i. o
(8-19)
Hence for this simple problem, the slowing density is constant and equal to the source. This is of course understandable, since in the absence of absorption and leakage, all source neutrons must eventually slow down below the energy E. (b) THE NEUTRON
LETHARGY
The energy range spanned by neutron slowing down is extremely large, ranging from 107 eV down to 1 eV. Furthermore we have found that in elastic scattering the neutron tends to lose a fraction of its incident energy rather than a fixed amount of energy. These considerations suggest that it would be more convenient to use as an independent variable, the logarithm of the neutron energy E. T o this e n d , w e d e f i n e t h e neutron
lethargy
(8-20) Here the energy E0 is chosen to be the maximum energy that neutrons can achieve in the problem. It is usually set either at the source energy for a monoenergetic source problem, or chosen as 10 MeV for a reactor calculation. Notice that as a neutron's energy E decreases, its lethargy u increases. That is, as a fast neutron loses energy via scattering collisions, it moves more slowly, becoming more "lethargic." The neutron lethargy is such a convenient variable that it is customary to perform fast spectrum calculations in terms of u rather than E. Hence we must convert all of our earlier equations over to this new independent variable. To accomplish this, we first compute the relationships between differentials
(8-21)
Then, for example, we can calculate the collision density in terms of lethargy by writing
(8-22)
F(u)du=-F(E)dE
or F(u)
=
EF(E).
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
323
( T h e m i n u s sign a p p e a r s b e c a u s e l e t h a r g y increases as e n e r g y decreases.) If we a p p l y this t o o u r earlier e x a m p l e of slowing d o w n in a n infinite, n o n a b s o r b i n g m e d i u m of h y d r o g e n , w e f i n d f o r this case, the collision rate d e n s i t y is a c o n s t a n t in the lethargy variable F(u)
= EF(E)=S0.
(8-23)
I n m o r e g e n e r a l cases, w e will f i n d t h a t t h e collision d e n s i t y F is usually a m u c h m o r e slowly v a r y i n g f u n c t i o n of u t h a n E a n d h e n c e is easier to a p p r o x i m a t e in the lethargy variable. A s a s e c o n d e x a m p l e , w e c a n c o m p u t e t h e elastic s c a t t e r i n g p r o b a b i l i t y f u n c t i o n P ( E ' ^ E ) in l e t h a r g y b y n o t i n g P(u'^u)du=
- P(E'^E)dE.
(8-24)
= -
P ( E ' ^ E ) = EP
Thus P (u'^u)
du 1
(E'-*E)
eu'~u
/ E \
(1 - a ) \ E ' J
(8-25)
(1 — a) *
Also E<E'<E/a=>u-ln(\/a)<
u'<
u.
(8-26)
Hence we find
(1 — a ) '
w — l n ( l / a ) < w' < u
0,
(8-27)
otherwise.
W e c a n n o w rewrite o u r slowing d o w n e q u a t i o n f o r h y d r o g e n in t e r m s of the lethargy variable as 2s(w)<J>(w)= fUduf eu'-u^{u%{uf)
+
S(u).
(8-28)
N o w n o t i c e s o m e t h i n g r a t h e r interesting; f r o m E q s . (8-18) a n d (8-27), it is a p p a r e n t t h e slowing d o w n d e n s i t y in t h e l e t h a r g y v a r i a b l e is given b y q(u)=
fVjiwW)^'"-
( 8 " 29 )
Jo N o w s u p p o s e w e d i f f e r e n t i a t e this e x p r e s s i o n w i t h respect t o u dq du
= -
fUdu'2,(u')
+ 2M(u)
(8-30)
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/
THE MULTIGROUP DIFFUSION METHOD
Hence we can reidentify q{u) from Eq. (8-29) to write ^ + q ( u ) = 2s(u)
(8-31)
These equations relating the slowing down density to the flux are peculiar to slowing down in hydrogen, but we will later find that they also hold when absorption and spatial dependence are included. They will prove to be of considerable use in numerical studies of neutron slowing down. As a final exercise involving the lethargy variable, let us compute the average lethargy gain (corresponding to the average logarithmic energy loss) of a neutron in a collision with a nucleus of arbitrary mass number:
1
(1 —
J
<*E,
a)Ej
dEf
(8-32)
or
(8-33)
+
By way of example, the mean lethargy gain per collision, moderators of interest in Table 8-1. TABLE 8-1
Moderator H D H2O D2O He Be C 238 u
is tabulated for several
Slowing Down Parameters of Typical Moderators
A 1 2 —
a
£
0 .111
1 .725 .920 .509 .425 .209 .158 .008
—
—
—
4 9 12 238
.360 .640 .716 .983
p[g/cm3] gas gas 1.0 1.1 gas 1.85 1.60 19.1
Number of collisions from 2 MeV to 1 eV 14 20 16 29 43 69 91 1730
^[cm-1] — —
1.35 0.176 1.6 X 10" 5 0.158 0.060 0.003
£2s/2a — —
71 5670 83 143 192 .0
An interesting application of this quantity is to compute the average number of collisions necessary to thermalize a fission neutron, that is, to slow it down from 2 MeV to 1.0 eV. This is given by l
2 x 10 6
< # > = ^ -
= 4 ^
(8-34)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
325
and is compared for several moderators of interest in Table 8-1. We can see rather dramatically how much more effective low mass-number nuclei are at moderating fast neutrons. (c) MODERATING POWER AND MODERATING
RATIO
The above analysis has indicated that the number of scattering collisions necessary to slow a neutron to thermal energies is inversely proportional to Certainly the better moderators will be characterized by large values of However they must also be characterized by large scattering cross sections 2 S since otherwise the probability of a scattering collision occurring will be too small. Hence a more a p p r o p r i a t e m e a s u r e of t h e moderating
o r slowing
down power
of a m a t e r i a l is the
product ££ s . To this end, one defines: Moderating p o w e r = | 2 S .
(8-35)
However, even this parameter is not sufficient in itself to describe the effectiveness of a material for neutron moderation, because obviously one also wishes the moderator to be a weak absorber of neutrons. Thus it is customary to choose as a figure of merit the moderating ratio, defined as Moderating r a t i o = - = -
(8-36)
The moderating power and moderating ratio are given for several materials of interest in Table 8-1. From this comparison it is evident that D 2 0 is the superior moderator. Indeed the moderating ratio of D 2 0 is sufficiently large that it can be used to construct a reactor fueled with natural uranium. Unfortunately it is also very expensive. Hence most thermal reactor designs choose to use somewhat poorer, but cheaper, moderators such as light water or graphite, although this usually requires the use of enriched uranium fuels. 3. S L O W I N G D O W N W I T H ABSORPTION
We will now include an absorption term in our infinite medium slowing down equation. Actually the absorption cross section of hydrogen is negligible. The physical situation we want to describe is that of a strongly absorbing isotope mixed in hydrogen (such as 238 U). However this isotope can also scatter, and since its mass number is not unity, it would invalidate our analysis. For simplicity, therefore, we will assume the absorber to be "infinitely massive" so that it does not slow down neutrons—it only absorbs them. We will also ignore inelastic scattering. Then the appropriate slowing down equation (again with a monoenergetic source at energy E0) is just
+
)]*(£)=
/
dE>
^
;
+S08(E~E0).
(8-37)
J e
As in our earlier analysis, we seek the solution for the total collision density F(E) = '2t(E)4>(E) as the sum of a collided and uncollided contribution, similar to
326
/
THE MULTIGROUP DIFFUSION METHOD
Eq. (8-6). Then we find that the collided contribution F (E) must satisfy
FXE)
-I
dE'2s(E')Fc(E>)
2S(£0) S0
£'2t(£')
(8-38)
Note that again the effective source term in this equation corresponds to oncecollided source neutrons, suitably scaled down to account for neutron absorption at the source energy E0. We again solve this equation by first differentiating to find U E ) dE
(8-39)
FXE).
E2t(E)
We can easily integrate this ordinary differential equation using the initial condition at E= E0 implied by Eq. (8-38) to find rEodE'UF')
2s(£0)S0 J
(8-40) HE')E'
Notice in particular that if we were to set 2 a = 0 we would return to our earlier solution FC(E)= S0/E. We can compute the slowing down density as before to find
q(E)
= EFc(E)
(8-41)
= S0ex p I
St(E')E'
If we now note that So = rate at which source neutrons are emitted at energy E0 q(E) = rate at which neutrons slow down past E, then we can identify
So
_ probability that a source neutron is not absorbed while slowing down from E0 to E.
(8-42)
Since most absorption in the slowing down energy range is due to resonances in the absorption cross section, it is natural to identify the resonance escape probability to energy E in hydrogen as just
p(E)=——
=exp
I
E'UE')
(8-43)
To proceed further we need explicit forms for 2 a (is) and 2 t ( £ ) . We will develop these later in this chapter when we return to consider resonance absorption. First, however, we will generalize these results to the case of A > 1.
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
327
C. Neutron Moderation in Media with A > 1 1. S L O W I N G D O W N W I T H O U T ABSORPTION W e n o w t u r n o u r a t t e n t i o n to the m o r e g e n e r a l case in w h i c h n e u t r o n s slow d o w n via elastic s c a t t e r i n g in a n i n f i n i t e m e d i u m of a r b i t r a r y m a s s n u m b e r A > \ . O n c e a g a i n w e first c o n s i d e r t h e s i t u a t i o n in w h i c h a b s o r p t i o n is i g n o r e d . T h e n the slowing d o w n e q u a t i o n f o r t h e collided f l u x r e s u l t i n g f r o m a m o n o e n e r g e t i c s o u r c e at e n e r g y E0 is j u s t
Zs(E)
=
I f
dE'
2
f f ^ f f (1 -a)E
0
+ t ^ ^ V , (1 -a)E0
E<E0.
(8-44)
If w e o n c e a g a i n write this e q u a t i o n in t e r m s of the collision d e n s i t y FC(E) d i f f e r e n t i a t e w i t h r e s p e c t to E , w e arrive a t
dE
(!-„)£
and
0-«)£
a
T h e a p p e a r a n c e of t h e first t e r m c o r r e s p o n d i n g to t h e u p p e r limit of i n t e g r a t i o n is w h a t f o u l s u s u p , since this t e r m d e p e n d s n o t o n E b u t r a t h e r o n E/a. This 20, 21 e q u a t i o n is r e l a t e d to a differential-difference equation a n d requires an alternative m e t h o d of solution. Its s o l u t i o n is c o m p l i c a t e d b e c a u s e the collision d e n s i t y is in f a c t d i s c o n t i n u o u s (or possesses d i s c o n t i n u o u s derivatives) at energies anE0 ( c o r r e s p o n d i n g to the m i n i m u m e n e r g y a n e u t r o n c a n slow d o w n to in n collisions). T h e s e d i s c o n t i n u o u s " t r a n s i e n t s " in t h e collision d e n s i t y will e v e n t u a l l y s m o o t h o u t to a n a s y m p t o t i c s o l u t i o n f a r b e l o w t h e s o u r c e e n e r g y of t h e f o r m FC(E)~\/E ( i n d e e d , o n e c a n r e a d i l y v e r i f y t h a t s u c h a f u n c t i o n is a s o l u t i o n of E q . (8-44) w h e n t h e source t e r m is n o t p r e s e n t ) . F o r t u n a t e l y this a s y m p t o t i c b e h a v i o r is sufficient f o r m o s t s l o w i n g d o w n studies, since o n e is u s u a l l y c o n c e r n e d w i t h either h y d r o g e n - d o m i n a t e d slowing d o w n ( s u c h as in a L W R ) in w h i c h the t r a n s i e n t s d o n o t arise or w i t h t h e b e h a v i o r of t h e n e u t r o n f l u x f a r b e l o w the fission s o u r c e energies. O n l y f o r i n t e r m e d i a t e m a s s n u m b e r m o d e r a t o r s s u c h a s d e u t e r i u m d o the t r a n s i e n t s c a u s e difficulties. H o w e v e r f o r c o m p l e t e n e s s w e will s k e t c h t h e d e t a i l e d s o l u t i o n 1 of the slowing d o w n e q u a t i o n f o r a r b i t r a r y m a s s n u m b e r A. It is m o s t c o n v e n i e n t to r e c a s t E q . (8-44) in t e r m s of lethargy. H o w e v e r w e m u s t b e c a r e f u l in o u r s t u d y of this e q u a t i o n since a n e u t r o n c a n o n l y g a i n a m a x i m u m lethargy of l n ( l / a ) in a collision. H e n c e t h e s o u r c e t e r m will d i s a p p e a r f r o m the e q u a t i o n f o r lethargies w > l n ( l / a ) . W e first c o n s i d e r o n l y t h e l e t h a r g y r a n g e 0 < w < l n ( l / a ) , in w h i c h case E q . (8-44) b e c o m e s
0 < w
.
(8-46)
W e h a v e d e n o t e d the s o l u t i o n in this r a n g e as 0(«). N o t i c e t h a t to gain a lethargy b e l o w w = l n ( l / a ) , the n e u t r o n w o u l d h a v e to e x p e r i e n c e a t least t w o collisions.
328
/
THE MULTIGROUP DIFFUSION METHOD
Hence we refer to 0 < u < In (1 / a) as the "first collision interval". We will introduce the collision density to write u Cdu'F0(u')eu-"+-^-,
*•(,(«)-"r—
S €—
l-a-'o
"
(1 — a)
0 < w < In — .
(8-47)
a
Now differentiate with respect to u to find dFj
(8-48)
du
The corresponding boundary condition is found, as usual, by examining Eq. (8-47) as u—>0 to find (8-49)
Hence we can readily solve for exp (8-50)
(I-a)
[Note that in terms of energy variables (8-51) To handle higher order collision intervals—say between (n — l ) l n ( l / a ) < u < ( / ? + l ) l n ( l / a ) one must split up the range of integration to write
1
a
J
f
n
i n - l)ln
,du'Fn{u')eu'~u
In — a
n\n
1 a
a
nln — 1
U
(n + l)ln — a
jn _
(8-52) Again we differentiate to find dR
(8-53) We can integrate this to write
F
n («) =
F
n ( "
l n
l
/
«
)
e
*
p
[
(
"
n l n
(8-54) a
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS Since w e k n o w F0(u), For example
F M = S
o
y
l
a x
_laa
/
w e c a n use this e x p r e s s i o n to g e n e r a t e higher o r d e r
j e x p [ ( - ^ )
W
] - 5
0
^ ^ (
W
- l n I ) e x p [ (
T
^ )
329 Fn(u).
W
] .
(8-55) N o t e t h a t t h e r e is a d i s c o n t i n u i t y b e t w e e n F0(u)
a n d Fx(u)
F0(ln 1 / a ) - ^ ( l n 1 / a ) = a / ( l - a).
at w = ln 1 / a : (8-56)
I n a similar m a n n e r , o n e c a n g e n e r a t e the h i g h e r o r d e r collision densities Fn(u). T h e s e f u n c t i o n s (so-called " P l a c z e k f u n c t i o n s " ) a r e illustrated in F i g u r e 8-1. 5 N o t i c e in p a r t i c u l a r h o w these f u n c t i o n s a p p e a r to b e s m o o t h i n g o u t as the n e u t r o n s s u f f e r increasingly m o r e collisions. I n f a c t f o r w » l n l / a , w e c a n use a n a s y m p t o t i c s o l u t i o n to e q u a t i o n (8-46) in w h i c h t h e source t e r m is n e g l e c t e d
f . ( ««)= 0-
Fe(u') du eU U f I x 'jY=a) ~ J tiu —~ In —
, (8"5?)
It is evident t h a t Fc(u) = c o n s t a n t = C is a s o l u t i o n to this e q u a t i o n . H o w e v e r it is
FIGURE 8-1.
The collision density F(u) for A = 2 and 4.
330
/
THE MULTIGROUP DIFFUSION METHOD
n o t e v i d e n t j u s t w h a t r e l a t i o n s h i p this s o l u t i o n h a s t o Fn{u) f o r large n. F u r t h e r m o r e , this h o m o g e n e o u s e q u a t i o n p r o v i d e s n o p r e s c r i p t i o n f o r d e t e r m i n i n g C. P e r h a p s t h e m o s t direct w a y t o g e n e r a t e t h e a s y m p t o t i c s o l u t i o n is t o solve E q . (8-47) u s i n g L a p l a c e - t r a n s f o r m m e t h o d s . 6 W e will a d o p t a s o m e w h a t m o r e p e d e s t r i a n a p p r o a c h b y n o t i n g t h a t t h e slowing d o w n d e n s i t y in t h e l e t h a r g y v a r i a b l e is given b y /•« q(u)=
/*u' + ln—
,_ „
xdu'Fc{u')i
(8-58)
J u — Ina —
^u
If w e s u b s t i t u t e in o u r a s y m p t o t i c f o r m Fc(u) — C, w e f i n d C
ru
I ,du'eu 4-ln. (X
ru' + \n —
I
a
du"e~u
J
u
=C 1 +
a In a I—a
= Ct
(8-59)
H o w e v e r in a n o n a b s o r b i n g m e d i u m , w e k n o w t h a t q{u) m u s t b e c o n s t a n t a n d e q u a l t o t h e s o u r c e r a t e S0 (since t h e r e is n o t h i n g t o p r e v e n t t h e n e u t r o n s f r o m slowing d o w n ) . H e n c e w e m u s t c h o o s e C = or
(8-60) or S0
W e c a n also w r i t e this a s y m p t o t i c s o l u t i o n i n t e r m s of t h e e n e r g y v a r i a b l e as
^ - e ^ f e y
(8 62)
"
H e n c e a s y m p t o t i c a l l y a t least t h e flux c h a r a c t e r i z i n g n e u t r o n s l o w i n g d o w n in a n o n a b s o r b i n g m e d i u m of m a s s n u m b e r A is v e r y similar t o t h a t f o u n d i n h y d r o g e n , w i t h t h e e x c e p t i o n of a n a d d i t i o n a l f a c t o r , t h e m e a n l e t h a r g y loss p e r collision, A s w e h a v e n o t e d , t h e p r i m a r y q u a l i t a t i v e d i f f e r e n c e i n t h e m o r e g e n e r a l s o l u t i o n is t h e o c c u r r e n c e of i r r e g u l a r ( d i s c o n t i n u o u s ) b e h a v i o r n e a r e n e r g i e s a t w h i c h s o u r c e n e u t r o n s a p p e a r or s t r o n g a b s o r p t i o n o c c u r s . A s o n e m o v e s d o w n i n e n e r g y ( u p in l e t h a r g y ) t h e s e t r a n s i e n t s d a m p out, l e a v i n g t h e s m o o t h a s y m p t o t i c b e h a v i o r given b y E q . (8-62). T h e r a p i d i t y w i t h w h i c h this a s y m p t o t i c b e h a v i o r is r e a c h e d d e c r e a s e s w i t h t h e m a s s n u m b e r of t h e m o d e r a t i n g m a t e r i a l ( a l t h o u g h s u c h b e h a v i o r is a l w a y s p r e s e n t in h y d r o g e n ) . M o r e f r e q u e n t l y o n e is c o n c e r n e d w i t h n e u t r o n s l o w i n g d o w n in a m i x t u r e of n u c l i d e s . A l t h o u g h w e will t r e a t this s i t u a t i o n in s o m e d e t a i l in l a t e r sections, w e will n o t e h e r e t h a t o u r earlier analysis c a n b e easily g e n e r a l i z e d t o a m i x t u r e of nuclides by writing
(8-63)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
331
w h e r e 2 s ( i £ ) is t h e t o t a l s c a t t e r i n g cross section a n d \ is the a v e r a g e l e t h a r g y gain p e r collision N
S ^ /= 1
t
/
S
^ /=1
)
.
(8-64)
It s h o u l d b e n o t e d t h a t £(£") is in g e n e r a l e n e r g y (or l e t h a r g y ) - d e p e n d e n t b e c a u s e of t h e e n e r g y d e p e n d e n c e of t h e s c a t t e r i n g cross section. 2. SLOWING D O W N W I T H ABSORPTION W e c o u l d n o w i n t r o d u c e a b s o r p t i o n ( a g a i n a s s u m i n g a n infinitely massive a b s o r b e r ) i n t o E q . (8-44):
:a(£)
+ 2s(is)]
2 (Ef>\( E'} S +S(E), ( l _ a ) E ,
C ^ "dE' • / FT
(8-65)
or in t e r m s of t h e collision d e n s i t y ,
F(E)
=
f JF
a
F{E') dE'
(1
-a)E
7
+S(E).
(8-66)
T h e p r e s e n c e of t h e f a c t o r [ 2 s ( i s ) / 2 t ( . E ) ] m a k e s it very d i f f i c u l t to m a k e m u c h p r o g r e s s t o w a r d a n a n a l y t i c a l s o l u t i o n to this e q u a t i o n , w i t h the e x c e p t i o n of very special cross s e c t i o n b e h a v i o r s a n d e l a b o r a t e a p p r o x i m a t i o n t e c h n i q u e s . 1 ' 2 R a t h e r t h a n discuss s u c h t e c h n i q u e s here, w e will r e c o g n i z e t h a t o n e is usually m o s t i n t e r e s t e d in t h e e f f e c t s of r e s o n a n c e a b s o r p t i o n , a n d so we will p r o c e e d directly to a discussion of t h e v a r i o u s m e t h o d s u s e d to c a l c u l a t e t h e r e s o n a n c e e s c a p e p r o b a b i l i t y . W e will r e t u r n l a t e r to discuss a p p r o x i m a t e t r e a t m e n t s of slowing d o w n w h e n w e c o n s i d e r as well h o w o n e t r e a t s spatial e f f e c t s in n e u t r o n slowing down.
D. Inelastic Scattering T h u s f a r w e h a v e c o n c e r n e d o u r s e l v e s - w i t h n e u t r o n m o d e r a t i o n via elastic collisions in w h i c h a n e u t r o n m e r e l y b o u n c e s off of a n u c l e u s in billiard-ball f a s h i o n , l o s i n g s o m e e n e r g y in t h e p r o c e s s . H o w e v e r for h i g h e r e n e r g y n e u t r o n s ( > 5 0 keV), inelastic s c a t t e r i n g p r o c e s s e s a r e p o s s i b l e in w h i c h a n a p p r e c i a b l e f r a c t i o n of t h e i n c i d e n t n e u t r o n e n e r g y goes i n t o exciting t h e n u c l e u s i n t o a h i g h e r n u c l e a r q u a n t u m state. S u c h s c a t t e r i n g is e x t r e m e l y i m p o r t a n t in h e a v y m a s s nuclei in w h i c h slowing d o w n b y elastic s c a t t e r i n g is negligible. F o r e x a m p l e , if w e c o n s i d e r a 1 M e V n e u t r o n i n c i d e n t o n a 2 3 8 U n u c l e u s , t h e a v e r a g e e n e r g y lost in a n elastic s c a t t e r i n g collision w o u l d b e (1 - a)E0/2 = 0.0085 M e V . By w a y of c o n t r a s t , t h e a v e r a g e e n e r g y lost in a n inelastic s c a t t e r i n g collision is a b o u t 0.6 M e V . Of c o u r s e f o r s u c h inelastic s c a t t e r i n g p r o c e s s e s to o c c u r , the i n c i d e n t n e u t r o n e n e r g y m u s t b e a b o v e t h e t h r e s h o l d c o r r e s p o n d i n g to t h e lowest excited state of the t a r g e t n u c l e u s . F o r light nuclei, this t h r e s h o l d is q u i t e high (4.4 M e V in 1 2 C a n d 6.1 M e V in 1 6 0 , f o r e x a m p l e ) . H o w e v e r in h e a v i e r nuclei t h e inelastic s c a t t e r i n g
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THE MULTIGROUP DIFFUSION METHOD
t h r e s h o l d s a r e m u c h lower (450 k e V in 2 3 N a a n d 45 k e V in 2 3 8 U ) , a n d inelastic processes b e c o m e quite important. Since inelastic s c a t t e r i n g is m u c h m o r e significant f o r f a s t n e u t r o n s in high m a s s - n u m b e r m a t e r i a l s , it m i g h t b e e x p e c t e d to b e of c o n s i d e r a b l e c o n s e q u e n c e in f a s t r e a c t o r s . I n d e e d inelastic s c a t t e r i n g in m a t e r i a l s s u c h as s o d i u m is t h e d o m i n a n t s l o w i n g d o w n m e c h a n i s m in f a s t r e a c t o r s a n d m u s t b e c a r e f u l l y a c c o u n t e d f o r in a n y realistic r e a c t o r analysis. U n f o r t u n a t e l y t h e details of t h e inelastic s c a t t e r i n g cross sections a r e q u i t e c o m p l i c a t e d , a n d a n a l y t i c a l i n v e s t i g a t i o n s such as t h o s e w e h a v e a p p l i e d to elastic s c a t t e r i n g a r e r e s t r i c t e d t o very simple n u c l e a r m o d e l s (e.g., t h e "Weisskopf e v a p o r a tion m o d e l 2 or f e w level models 2 0 ). I n s t e a d , o n e m u s t u s u a l l y p r o c e e d in p r a c t i c e with a b r u t e f o r c e f i n e - s t r u c t u r e m u l t i g r o u p c a l c u l a t i o n . T h a t is, o n e writes the slowing d o w n e q u a t i o n in m u l t i g r o u p f o r m a n d solves it directly f o r m a n y g r o u p s in o r d e r to o b t a i n the i n t r a g r o u p fluxes f o r c a l c u l a t i n g t h e f e w - g r o u p c o n s t a n t s . S u c h f i n e s p e c t r u m c a l c u l a t i o n s m a y r e q u i r e as m a n y as a t h o u s a n d or m o r e m i c r o g r o u p s to a d e q u a t e l y t r e a t the details of inelastic scattering. F u r t h e r m o r e b e c a u s e of t h e r a t h e r large e n e r g y loss e x p e r i e n c e d b y a n e u t r o n in a n inelastic s c a t t e r i n g event, several e n e r g y g r o u p s a r e usually c o u p l e d t o g e t h e r (indirect g r o u p coupling). L a t e r in this c h a p t e r w e will c o n s i d e r in detail h o w inelastic s c a t t e r i n g is t r e a t e d in t h e g e n e r a t i o n of f a s t g r o u p c o n s t a n t s .
II. RESONANCE ABSORPTION (INFINITE MEDIUM) A. Introduction A s n e u t r o n s slow d o w n f r o m fission energies in a n u c l e a r r e a c t o r , they e x p e r i e n c e a n a p p r e c i a b l e p r o b a b i l i t y of b e i n g a b s o r b e d in the n u m e r o u s s h a r p c a p t u r e r e s o n a n c e s w h i c h c h a r a c t e r i z e h e a v y nuclei s u c h as 2 3 8 U or 2 3 2 T h . S u c h r e s o n a n c e a b s o r p t i o n of n e u t r o n s is a n e x t r e m e l y i m p o r t a n t p h e n o m e n o n in n u c l e a r r e a c t o r s . It n o t only a f f e c t s r e a c t o r m u l t i p l i c a t i o n , b u t f u e l b u r n u p a n d b r e e d i n g p e r f o r m a n c e a n d r e a c t o r c o n t r o l c h a r a c t e r i s t i c s as well. O n e c a n d i s t i n g u i s h several d i f f e r e n t types of r e s o n a n c e a b s o r p t i o n in n u c l e a r r e a c t o r s . Of p a r a m o u n t i m p o r t a n c e in t h e r m a l r e a c t o r s is a b s o r p t i o n in t h e wellresolved l o w - e n e r g y r e s o n a n c e s of f u e l m a t e r i a l s such as 2 3 8 U or 2 3 2 T h . T h e analysis of n e u t r o n a b s o r p t i o n in such w e l l - s e p a r a t e d r e s o n a n c e s is s t r a i g h t f o r w a r d , a n d we will c o n s i d e r it in detail in this section. A b o v e several k e V in fertile m a t e r i a l s ( a n d as low a s 50 eV in fissile isotopes), o n e f i n d s t h a t t h e r e s o n a n c e s t r u c t u r e b e c o m e s so finely d e t a i l e d t h a t r e s o n a n c e s c a n n o l o n g e r b e individually resolved. T h e t r e a t m e n t of r e s o n a n c e a b s o r p t i o n in the r e g i o n of s u c h u n r e s o l v e d r e s o n a n c e s is c o n s i d e r a b l y m o r e c o m p l i c a t e d , since it r e q u i r e s the use of n u c l e a r m o d e l s to d e s c r i b e t h e r e s o n a n c e s t r u c t u r e . A l t h o u g h this s u b j e c t is of s o m e i m p o r t a n c e in f a s t r e a c t o r analysis, its c o m p l e x i t y i n d u c e s us to r e f e r the i n t e r e s t e d r e a d e r to m o r e a d v a n c e d studies of r e s o n a n c e a b s o r p t i o n f o r a d e t a i l e d discussion. 7 A n a c c u r a t e t r e a t m e n t of r e s o n a n c e a b s o r p t i o n is essential to r e a c t o r criticality c a l c u l a t i o n s , since this is o n e of t h e p r i m a r y n e u t r o n loss m e c h a n i s m s in b o t h thermal a n d fast reactors. Such a process enters into a multigroup diffusion c a l c u l a t i o n t h r o u g h the f a s t m u l t i g r o u p a b s o r p t i o n cross sections 2 , a n d the
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
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333
a c c u r a t e e s t i m a t e of these g r o u p c o n s t a n t s will b e o n e of o u r p r i n c i p a l c o n c e r n s . I n general, t h e r e will b e a d e p r e s s i o n in the n e u t r o n flux at t h o s e energies in the vicinity of a s t r o n g r e s o n a n c e , a n d the analysis of such flux d e p r e s s i o n s is of c o n s i d e r a b l e i m p o r t a n c e in d e t e r m i n i n g m u l t i g r o u p c o n s t a n t s . Since r e s o n a n c e a b s o r p t i o n in fertile m a t e r i a l ( 2 3 8 U , 2 3 2 T h ) c a n l e a d t o the p r o d u c t i o n of fissile m a t e r i a l ( 2 3 9 Pu, 2 3 3 U ) , a n a c c u r a t e analysis of r e s o n a n c e a b s o r p t i o n is also of i m p o r t a n c e in p r e d i c t i n g c o n v e r s i o n or b r e e d i n g ratios. I n d e e d small i n a c c u r a c i e s in t h e t r e a t m e n t of r e s o n a n c e a b s o r p t i o n c a n p r o p a g a t e sizable e r r o r s in t h e e s t i m a t e of b o t h f u e l d e p l e t i o n a n d fertile-to-fissile c o n v e r s i o n 8 . R e s o n a n c e a b s o r p t i o n is also e x t r e m e l y i m p o r t a n t in d e t e r m i n i n g t h e kinetic b e h a v i o r of the r e a c t o r . T h e a m o u n t of s u c h a b s o r p t i o n d e p e n d s sensitively o n the f u e l t e m p e r a t u r e t h r o u g h the D o p p l e r b r o a d e n i n g m e c h a n i s m . I n f a c t the d o m i n a n t reactivity f e e d b a c k m e c h a n i s m of s i g n i f i c a n c e in very r a p i d p o w e r t r a n s i e n t s is usually t h a t d u e to the D o p p l e r e f f e c t . A r i g o r o u s t r e a m e n t of r e s o n a n c e a b s o r p t i o n w o u l d a t t e m p t to d e t e r m i n e the n e u t r o n f l u x <J>(r,£) in a f u e l lattice cell either b y solving t h e t r a n s p o r t e q u a t i o n directly or u s i n g M o n t e C a r l o t e c h n i q u e s . A l t h o u g h s u c h c a l c u l a t i o n s a r e o c c a s i o n ally p e r f o r m e d , t h e y are f a r t o o expensive f o r use in r o u t i n e design analysis. I n s t e a d o n e u s u a l l y r e m o v e s t h e spatial a n d a n g u l a r d e p e n d e n c e to arrive at a n e q u a t i o n f o r t h e e n e r g y - d e p e n d e n t n e u t r o n f l u x similar t o t h e slowing d o w n e q u a t i o n (8-3). A direct n u m e r i c a l s o l u t i o n of this e q u a t i o n c a n b e , a n d is, c o m m o n l y p e r f o r m e d . H o w e v e r t h e large n u m b e r of c a l c u l a t i o n s typically r e q u i r e d in a r e a c t o r d e s i g n p r e c l u d e t h e extensive use of s u c h direct m e t h o d s in f a v o r of simpler a p p r o x i m a t e s o l u t i o n s of t h e slowing d o w n e q u a t i o n to d e t e r m i n e the n e u t r o n f l u x in t h e vicinity of a r e s o n a n c e . S u c h a p p r o x i m a t e m e t h o d s a r e usually a n a l y t i c a l in n a t u r e , a l t h o u g h v a r i o u s a s p e c t s of the solution m a y r e q u i r e n u m e r i c a l evaluation. W h e n used for appropriate problems, they provide reasonable accuracy w i t h c o n s i d e r a b l y less c a l c u l a t i o n a l e f f o r t t h a n t h e m o r e r i g o r o u s m e t h o d s . I n this section w e will a t t e m p t to i n t r o d u c e s o m e of the m o r e e l e m e n t a r y c o n c e p t s a n d a p p r o x i m a t i o n s u s e f u l in the s t u d y of r e s o n a n c e a b s o r p t i o n . W e will c o n f i n e o u r d i s c u s s i o n to r e s o n a n c e a b s o r p t i o n in a n infinite m e d i u m (consistent w i t h o u r earlier s t u d y of n e u t r o n slowing d o w n ) a n d d e v e l o p the p r i n c i p a l a p p r o x i m a t i o n s u s e f u l in t h e c a l c u l a t i o n of the r e s o n a n c e e s c a p e p r o b a b i l i t y a n d f a s t g r o u p c o n s t a n t s . H o w e v e r t h e results of s u c h i n f i n i t e m e d i u m c a l c u l a t i o n s are of o n l y l i m i t e d utility, since spatial v a r i a t i o n s of the n e u t r o n flux in a f u e l lattice c a n strongly i n f l u e n c e r e s o n a n c e a b s o r p t i o n . W e d e f e r t h e s t u d y of spatially d e p e n d e n t e f f e c t s u n t i l w e c o n s i d e r cell c a l c u l a t i o n t e c h n i q u e s in C h a p t e r 10. O u r g e n e r a l a p p r o a c h is to c o n s i d e r n e u t r o n slowing d o w n in a n infinite, h o m o g e n e o u s m i x t u r e of a h e a v y i s o t o p e c h a r a c t e r i z e d b y a b s o r p t i o n a n d scattering r e s o n a n c e s a n d a m o d e r a t o r m a t e r i a l h a v i n g a c o n s t a n t s c a t t e r i n g cross section (at least over t h e r e s o n a n c e ) a n d a negligible a b s o r p t i o n cross section. I n p a r t i c u l a r w e will s t u d y t h e s o l u t i o n of t h e n e u t r o n slowing d o w n e q u a t i o n in t h e vicinity of a well-isolated r e s o n a n c e . T o illustrate t h e essential i d e a s involved, w e will b e g i n b y c o n s i d e r i n g the s i t u a t i o n in w h i c h t h e m o d e r a t i n g m a t e r i a l is h y d r o g e n a n d s c a t t e r i n g f r o m the a b s o r b e r n u c l e i is n e g l e c t e d . T h i s case is p a r t i c u l a r l y simple, since the n e u t r o n s l o w i n g d o w n e q u a t i o n c a n b e solved exactly w h e n the m a s s n u m b e r of the s c a t t e r e r is u n i t y .
334
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THE MULTIGROUP DIFFUSION METHOD
B. Resonance Absorption in Hydrogen Plus An Infinitely Massive Absorber W e will b e g i n b y s t u d y i n g r e s o n a n c e a b s o r p t i o n d u e t o a n i n f i n i t e l y m a s s i v e a b s o r b e r d i s t r i b u t e d u n i f o r m l y t h r o u g h a n i n f i n i t e m e d i u m of h y d r o g e n . T h e i n f i n i t e a b s o r b e r m a s s implies t h a t all n e u t r o n slowing d o w n d u e t o elastic s c a t t e r i n g will b e d u e t o h y d r o g e n . W e will a l s o i g n o r e inelastic s c a t t e r i n g p r o cesses. W e h a v e a l r e a d y solved t h e n e u t r o n s l o w i n g d o w n e q u a t i o n (8-37) f o r this c a s e in S e c t i o n 8 - I I - A t o f i n d t h e flux
HE)
=
(8-67)
P(E),
EZs(E)
w h e r e t h e r e s o n a n c e e s c a p e p r o b a b i l i t y t o e n e r g y E,p(E),
p(E)
= ex p
-I
dEf
V
is given b y
(8-68)
'
W e will a p p l y this e x a c t e x p r e s s i o n t o c a l c u l a t e t h e r e s o n a n c e e s c a p e p r o b a b i l i t y f o r a single r e s o n a n c e at e n e r g y E0. F o r c o n v e n i e n c e w e will i g n o r e s c a t t e r i n g f r o m t h e infinitely m a s s i v e a b s o r b e r a n d a b s o r p t i o n i n h y d r o g e n , b y a s s u m i n g NnoSH If w e d e n o t e t h e r e s o n a n c e e s c a p e p r o b a b i l i t y f o r this p a r t i c u l a r resonance as p , then we find
p = exp
i
N
dE'
K°y{E')
E'
Nno?
+
(8-69)
NAo*{E')
T h e n o t a t i o n jE i n d i c a t e s t h a t t h e integral is t o b e p e r f o r m e d over energies i n t h e n e i g h b o r h o o d of t h e r e s o n a n c e . T o p r o c e e d f u r t h e r , w e m u s t i n t r o d u c e a n explicit f o r m f o r t h e c a p t u r e cross s e c t i o n o*(E). W e will u s e t h e D o p p l e r - b r o a d e n e d B r e i t - W i g n e r c r o s s section d e v e l o p e d in S e c t i o n 2 - I - D :
(8-70)
W e will first e v a l u a t e p in t h e limit in w h i c h t h e a b s o r b e r c o n c e n t r a t i o n is s u f f i c i e n t l y d i l u t e t h a t t h e s c a t t e r i n g f r o m h y d r o g e n is d o m i n a n t e v e n a t t h e r e s o n a n c e e n e r g y , t h a t is, N A o 0 < g . N H o ^ . T h i s is k n o w n as t h e infinite dilution approximation, a n d it is e q u i v a l e n t to a s s u m i n g t h a t t h e a b s o r b e r c o n c e n t r a t i o n is so dilute tha*t t h e a b s o r p t i o n r e s o n a n c e d o e s n o t p e r t u r b t h e s l o w i n g d o w n f o r m of t h e flux. U s i n g this a p p r o x i m a t i o n in E q . (8-69), w e f i n d t h a t
f dE'
>ex p ^ H ^ s
J
E0
E
A
=P
(8-71)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
335
T o p r o c e e d f u r t h e r , w e n o t e t h a t the m a j o r c o n t r i b u t i o n to the integral c o m e s f r o m t h o s e energies close to r e s o n a n c e , E'~E0. H e n c e we c a n e x t r a c t l / £ ' ~ l / £ 0 to find r
at?'
r
v
^o^v
r
u A
0 0
o
v7'
'0 JEq w h e r e the limits of i n t e g r a t i o n h a v e b e e n e x t e n d e d to infinity since o n l y a m i n o r c o n t r i b u t i o n c o m e s f r o m the w i n g s of the r e s o n a n c e . H e n c e in the i n f i n i t e d i l u t i o n limit irNAa0Ty
p
(8-73)
= exp 2N~^E0
Several f e a t u r e s of this result a r e of interest. F i r s t n o t e t h a t the r e s o n a n c e e s c a p e p r o b a b i l i t y i n c r e a s e s w i t h i n c r e a s i n g m o d e r a t o r d e n s i t y (or cross section), since t h e n the n e u t r o n s slow d o w n t h r o u g h t h e r e s o n a n c e m o r e r a p i d l y . T h i s e f f e c t is p a r t i c u l a r l y i m p o r t a n t in w a t e r m o d e r a t e d r e a c t o r s b e c a u s e it l e a d s to large n e g a t i v e p o w e r c o e f f i c i e n t s of reactivity. T h a t is, i n c r e a s i n g p o w e r a n d h e n c e m o d e r a t o r t e m p e r a t u r e will d e c r e a s e m o d e r a t o r d e n s i t y (usually via s t e a m f o r m a tion). H e n c e t h e r e s o n a n c e e s c a p e p r o b a b i l i t y will d e c r e a s e , t h e r e b y d e c r e a s i n g reactivity. N e x t it s h o u l d b e n o t e d t h a t t h e r e s o n a n c e is m o r e e f f e c t i v e in a b s o r b i n g n e u t r o n s if t h e r e s o n a n c e e n e r g y E0 is lower. T h i s is easily u n d e r s t o o d w h e n it is r e c a l l e d t h a t t h e a s y m p t o t i c collision d e n s i t y b e h a v i o r is as F(E)—1 / E . H e n c e l o w e r e n e r g y n e u t r o n s will e x p e r i e n c e m o r e collisions with a b s o r b e r nuclei a n d t h e r e f o r e a h i g h e r p r o b a b i l i t y of b e i n g a b s o r b e d . F o r this r e a s o n , t h e m o s t s i g n i f i c a n t r e s o n a n c e a b s o r p t i o n in t h e r m a l r e a c t o r s o c c u r s in the l o w e r lying r e s o n a n c e s of fertile m a t e r i a l s s u c h as t h e 6.67 eV r e s o n a n c e in 2 3 8 U . T h e r e s o n a n c e p a r a m e t e r s c h a r a c t e r i z i n g several of t h e lower lying r e s o n a n c e s of 2 3 8 U w h i c h are s i g n i f i c a n t in t h e r m a l r e a c t o r d e s i g n a r e listed in T a b l e 8-2.
TABLE 8-2:
Low-Lying Resonance Data for ^ U
E0
T n (eV)
r y (eV)
a 0 (b)
6.67
.00152
.026
2.16 X 105
1.26
20.90
.0087
.025
3.19 X 10 4
1.95
.174
36.80
.032
.025
3.98 x 10 4
3.65
.306
66.54
.026
.022
2.14 X 10 4
2.26
.554
102.47
.070
.026
1.86 X 10 4
3.98
.850
116.85
.030
.022
1.30 x 10 4
1.32
165.27
.0032
.018
2.41 X 10 3
0.98
1.37
208.46
.053
.022
8.86 X 10 3
2.63
1.73
Tp(eV)
i(l - a^ErfeV) .055
.966
F i n a l l y w e s h o u l d n o t e t h a t a l t h o u g h o*(E) is t e m p e r a t u r e - d e p e n d e n t t h r o u g h the D o p p l e r - b r o a d e n i n g m e c h a n i s m , t h e i n f i n i t e d i l u t i o n l i m i t p ° ° h a s n o t e m p e r a -
336
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THE MULTIGROUP DIFFUSION METHOD
t u r e d e p e n d e n c e . H o w e v e r this m i g h t h a v e b e e n a n t i c i p a t e d since w e h a v e a l r e a d y n o t e d t h a t in this a p p r o x i m a t i o n the n e u t r o n flux is n o t p e r t u r b e d b y the resonance. T h e r e f o r e w e t u r n to the m o r e g e n e r a l case of finite d i l u t i o n . U n f o r t u n a t e l y w e a r e n o w h a m p e r e d in o u r ability to o b t a i n a closed f o r m e v a l u a t i o n of the r e l e v a n t integrals. If w e s u b s t i t u t e t h e f o r m E q . (8-70) f o r a ^ ( E ) i n t o E q . (8-69) a n d e x t e n d the limits of i n t e g r a t i o n , w e f i n d t h a t t h e g e n e r a l result f o r the r e s o n a n c e e s c a p e probability p = exp
(8-74) o
m u s t b e w r i t t e n in t e r m s of a t a b u l a t e d f u n c t i o n 9 / ( f , / ? ) ,
j
a,/?)
I
dx
(8-75)
where (8-76)
102
0 =
»
Ot
I s
5.2
=
10
- s •= O.J
V fov
— — J A i 1 ' i r \\
\
\ \\
ea.
^ s
10
6
8
10
12
14
16
18
; (where (5 = 2 J X 10" 5 ) FIGURE 8-2. The function/(£,/?) [L. Dresner, Resonance Absorption Pergamon, New York (I960)].
in Nuclear
Reactors,
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
337
It is a p p a r e n t f r o m this f o r m t h a t t h e f i n i t e d i l u t i o n r e s o n a n c e integral is i n d e e d t e m p e r a t u r e - d e p e n d e n t . I n F i g u r e 8-2 w e h a v e s h o w n t h e f u n c t i o n • / ( ? , / ? ) p l o t t e d versus f o r v a r i o u s v a l u e s of t h e t e m p e r a t u r e p a r a m e t e r f . F r o m this p l o t it is a p p a r e n t t h a t as t e m p e r a t u r e T increases, f decreases, i m p l y i n g t h a t / ( £ , / $ ) i n c r e a s e s a n d h e n c e t h a t t h e r e s o n a n c e e s c a p e p r o b a b i l i t y p d e c r e a s e s . T h a t is, r e s o n a n c e a b s o r p t i o n i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . T h i s is t h e so-called Doppler effect r e f e r r e d t o earlier in o u r d i s c u s s i o n of t e m p e r a t u r e c o e f f i c i e n t s of reactivity. TABLE 8-3
The Function / ( f , / 3 ) for f = 0.1-1.0 and p = V x 10" 5 +
j
Jtt>fi) £-0.1
£ = 0.2
£ = 0.3
£ = 0.4
£ = 0.5
£ = 0.6
£ = 0.7
£ = 0.8
£ = 0.9
£=1.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13
4.979(2*) 3.532 2.514 1.801 1.307 9.667(1) 7.355 5.773 4.647 3.781 3.045 2.367 1.730 1.164*
4.970(2) 3.517 2.491 1.767 1.257 8.993(1) 6.501 4.777 3.589 2.759 2.153 1.676 1.268 9.081(0)
4.969(2) 3.514 2.487 1.761 1.248 8.872(1) 6.335 4.562 3.328 2.471 1.867 1.423 1.074 7.815(0)
4.968(2) 3.513 2.485 1.759 1.245 8.831(1) 6.278 4.485 3.230 2.354 1.741 1.301 9.718(0) 7.087
4.968(2) 3.513 2.485 1.758 1.244 8.812(1) 6.252 4.450 3.183 2.297 1.675 1.235 9.119(0) 6.629
4.968(2) 3.513 2.484 1.757 1.243 8.802(1) 6.238 4.430 3.158 2.265 1.638 1.194 8.739(0) 6.322
4.967(2) 3.513 2.484 1.757 1.243 8.796(1) 6.230 4.419 3.143 2.245 1.614 1.168 8.484(0) 6.107
4.967(2) 3.513 2.484 1.757 1.243 8.792(1) 6.225 4.412 3.133 2.232 1.598 1.151 8.304(0) 5.950
4.967(2) 3.513 2.484 1.757 1.242 8.790(1) 6.221 4.407 3.126 2.223 1.587 1.138 8.174(0) 5.833
4.967(2) 3.513 2.484 1.757 1.242 8.788(1) 6.218 4.403 3.121 2.217 1.579 1.129 8.077(0) 5.744
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
7.172(0) 4.088 2.204 1.148 5.862(— 1) 2.963 1.490 7.468(~2) 3.739 1.871 9 . 3 5 8 ( - 3) 4.680 2.340 1.170 5.851(-4) 2.925
6.014 3.658 2.067 1.109 5.757( - 1) 2.936 1.483 7.452( - 2) 3.735 1.870 9 . 3 5 6 ( - 3) 4.680 2.340 1.170 5.851 (— 4) 2.926
5.342 3.371 1.966 1.078 5.671 (— 1) 2.913 1.477 7.437( - 2) 3.732 1.869 9.355( —3) 4.679 2.340 1.170 5.851 (— 4) 2.926
4.914 3.169 1.889 1.053 5.599( - 11) 2.894 1.472 7.424( - 2) 3.728 1.868 9.352( - 3) 4.679 2.340 1.170 5.85K-4) 2.926
4.624 3.022 1.829 1.033 5.539( - 1) 2.877 1.468 7 . 4 1 3 < - :I) 3.726 1.868 9.350( - 3) 4.678 2.340 1.170 5.85K-4) 2.926
4.419 2.911 1.781 1.016 5.488( — 1) 2.863 1.464 7.403( - 2) 3.723 1.867 9.349< - 3) 4.678 2.340 1.170 5.851( —4) 2.926
4.268 2.826 1.743 1.002 5.445( - 1) 2.851 1.461 7.395( - 2) 3.721 1.867 9.348( —3) 4.678 2.340 1.170 5.851{ —4) 2.926
4.154 2.759 1.712 9.904( — 1) 5.408 2.840 1.458 7.388( —2) 3.719 1.866 9.346( —3) 4.677 2.340 1.170 5.851 (— 4) 2.926
4.066 2.706 1.687 9 . 8 0 5 ( - 1) 5.376 2.831 1.455 7.381 {— 2) 3.718 1.866 9.345( —3) 4.677 2.340 1.170 5 . 8 5 1 ( - 4) 2.926
3.997 2.663 1.666 9.722( — 1) 5.348 2.823 1.453 7 . 3 7 5 ( - 2) 3.716 1.865 9.344( - 3) 4.677 2.340 1.170 5 . 8 5 1 ( - 4) 2.926
30 31
1.463 1.463 1.463 1.463 1.463 1.463 1.463 1.463 1.463 1.463 7.314( — 5) 7 . 3 1 4 ( - 5) 7.315( - 5) 7 . 3 1 5 ( - :5) 7 . 3 1 5 ( - :5) 7.315( —5) 7 . 3 1 4 ( - 5) 7 . 3 1 4 ( - 5) 7 . 3 1 4 ( - 5) 7 . 3 1 4 ( - 5)
Dresner, Resonance Absorption in Nuclear Reactors, Pergamon, New York (1960). • N u m b e r s in parentheses are powers of 10, which multiply the entry next to which they stand and all unmarked entries below it.
T o u n d e r s t a n d this b e h a v i o r , recall t h a t o u r earlier a n a l y s i s i n d i c a t e d t h a t i n c r e a s e d t e m p e r a t u r e c a u s e s a r e s o n a n c e to b r o a d e n . H o w e v e r since the a r e a u n d e r t h e r e s o n a n c e is essentially ( a l m o s t ) t e m p e r a t u r e - i n d e p e n d e n t , t h e r e s o n a n c e p e a k d r o p s with t e m p e r a t u r e . H o w e v e r t h e b r o a d e n e d r e s o n a n c e i n c r e a s e s the e n e r g y r a n g e over w h i c h a b s o r p t i o n o c c u r s . T h i s e f f e c t o u t w e i g h s t h e slight l o w e r i n g of the r e s o n a n c e p e a k a n d gives rise to a n e n h a n c e d a b s o r p t i o n with increasing temperature.
338
/
THE MULTIGROUP DIFFUSION METHOD
T h i s e f f e c t arises b e c a u s e of t h e p h e n o m e n o n of self-shielding. W e will s h o w later t h a t t h e n e u t r o n flux is d e p r e s s e d (see F i g u r e 8-3) f o r t h o s e e n e r g i e s in t h e n e i g h b o r h o o d of t h e r e s o n a n c e . T h i s e f f e c t is k n o w n as energy self-shielding since t h e s t r o n g a b s o r p t i o n of t h e r e s o n a n c e t e n d s t o shield t h e a b s o r b e r n u c l e i f r o m n e u t r o n s w i t h e n e r g y E — E 0 (the flux d e p r e s s i o n ) . ( W e will later s t u d y a r e l a t e d p h e n o m e n o n k n o w n as spatial self-shielding.) A s t e m p e r a t u r e increases, t h e reso n a n c e p e a k d e c r e a s e s , t h e r e b y d e c r e a s i n g self-shielding a n d t h e f l u x d e p r e s s i o n , a n d i n c r e a s i n g r e s o n a n c e a b s o r p t i o n [i.e., t h e e n e r g y - i n t e g r a t e d r e a c t i o n r a t e
FIGURE 8-3.
Flux depression in the neighborhood of a resonance.
T h i s l a t t e r i n t e r p r e t a t i o n also e x p l a i n s w h y t h e i n f i n i t e d i l u t i o n limit s h o w s n o t e m p e r a t u r e d e p e n d e n c e , since in this c a s e t h e r e is n o f l u x d e p r e s s i o n (the a b s o r b e r c o n c e n t r a t i o n is t o o l o w t o p e r t u r b the flux) a n d h e n c e n o self-shielding.
C. Resonance Integrals W e n o w t u r n o u r a t t e n t i o n t o the m o r e g e n e r a l case of n e u t r o n s l o w i n g d o w n in a n infinite, h o m o g e n e o u s m i x t u r e of a b s o r b e r a n d m o d e r a t o r . It is possible to cast t h e r e s o n a n c e e s c a p e p r o b a b i l i t y c h a r a c t e r i z i n g this p r o b l e m i n t o a f o r m very similar t o t h a t f o r h y d r o g e n e o u s m o d e r a t o r s , E q . (8-68). If w e i m a g i n e t h e n e u t r o n s b e i n g p r o d u c e d b y a s o u r c e of s t r e n g t h S0 at e n e r g y E0, t h e n t h e total a b s o r p t i o n r a t e e x p e r i e n c e d b y n e u t r o n s w h i l e slowing d o w n f r o m e n e r g i e s E0 t o E is j u s t
E
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
339
H e n c e the r e s o n a n c e e s c a p e p r o b a b i l i t y is j u s t S n — neutron absorption rate i r En P ( E ) = = = l - - f J °dE'^(E')
(8-77)
I n a very similar m a n n e r w e c a n d e f i n e the r e s o n a n c e e s c a p e p r o b a b i l i t y pi f o r a n y single, well-isolated r e s o n a n c e l o c a t e d at a n e n e r g y Er T o d o so, we will a s s u m e t h a t all a b s o r p t i o n o c c u r s in a n u m b e r of w e l l - s e p a r a t e d r e s o n a n c e s lying b e l o w the s o u r c e e n e r g y E0. If t h e i n d i v i d u a l r e s o n a n c e s a r e s u f f i c i e n t l y widely s e p a r a t e d , t h e n t h e f l u x o c c u r r i n g j u s t a b o v e a given r e s o n a n c e at e n e r g y Ei will h a v e a s s u m e d the a s y m p t o t i c f o r m , Sir)
-
H-^A^Se
€SsE
zs
+zs
w h e r e S e $ is t h e original s o u r c e s t r e n g t h S0 r e d u c e d b y t h e p r o b a b i l i t i e s f o r all h i g h e r e n e r g y r e s o n a n c e s — t h a t is,
=S
o
n PP
where
Ej>
resonance-escape
(8-79)
j
(See F i g u r e 8-4.) H e n c e w e c a n i d e n t i f y the a b s o r p t i o n p r o b a b i l i t y f o r the reso n a n c e at Et as j u s t Absorption probability = ^eff
f dE2*(E E i
)
(8-80)
340
/
THE MULTIGROUP DIFFUSION METHOD
a n d t h u s t h e r e s o n a n c e e s c a p e p r o b a b i l i t y f o r this p a r t i c u l a r r e s o n a n c e b e c o m e s just
^eff
E
i
T o w r i t e this i n a m o r e u s e f u l f o r m , s u p p o s e w e n o r m a l i z e t h e f l u x <£(£) f a r a b o v e t h e r e s o n a n c e it b e h a v e s as
for
E»E
so
that
(8-82)
r
T h e n t h e r e s o n a n c e e s c a p e p r o b a b i l i t y f o r t h e r e s o n a n c e is j u s t
P i
= l ~ ~ Jf dEoyA(E)
[
(8-83)
T h i s is still n o t s u f f i c i e n t l y c o n v e n i e n t f o r o u r p u r p o s e s , since t o c o m p u t e t h e t o t a l r e s o n a n c e e s c a p e p r o b a b i l i t y f o r a series of r e s o n a n c e s , w e w o u l d h a v e t o m u l t i p l y t h e c o r r e s p o n d i n g pi t o g e t h e r as
/>total= I I / V
(8-84)
i T o facilitate this c a l c u l a t i o n , we first n o t e t h a t in g e n e r a l t h e a b s o r p t i o n p r o b a b i l i t y 1 —p. is q u i t e small. H e n c e w e c a n use t h e e x p a n s i o n
exp[-(l-Jpl.)]~l-(l-/>i)+
p,
(8-85)
to rewrite Pi^ex p
w h e r e w e h a v e d e f i n e d t h e effective
f
X a
resonance
(8-86)
integral
f o r t h e /th r e s o n a n c e a s
dEoyA(E)4>(E),
(8-87)
T h e n t o c o m p u t e t h e t o t a l r e s o n a n c e e s c a p e p r o b a b i l i t y , w e n e e d o n l y a d d the resonance integrals for each resonance
P total = I I / > , = exp
(8-88) -
T h e r e s o n a n c e integral /, is a v e r y u s e f u l q u a n t i t y f o r c h a r a c t e r i z i n g a r e s o n a n c e .
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
341
W e first s h o u l d n o t e t h a t /• h a s t h e d i m e n s i o n s of a m i c r o s c o p i c cross section (barns), if w e r e m e m b e r the n o r m a l i z a t i o n c o n d i t i o n , (£")—\/E f o r E ^ > E i t I n effect, /,. r e p r e s e n t s t h e a v e r a g e a b s o r p t i o n cross section c h a r a c t e r i z i n g the reso n a n c e , a v e r a g e d over t h e f l u x w i t h i n the r e s o n a n c e . W e will f i n d t h a t /(. is largely i n d e p e n d e n t of t h e p r o p e r t i e s of the m o d e r a t o r . T h e r e s o n a n c e integral I t h a s y e t a n o t h e r e x t r e m e l y i m p o r t a n t p r a c t i c a l application since it c a n b e u s e d directly in t h e g e n e r a t i o n of m u l t i g r o u p c o n s t a n t s . T o see this, first recall t h e d e f i n i t i o n of t h e m u l t i g r o u p a b s o r p t i o n cross section f o r a given g r o u p g: fE'~ldE2a(E)(E) =
. f
(8-89)
dE<j>(E)
N o w since t h e r e s o n a n c e s a r e q u i t e n a r r o w , o v e r m o s t of t h e g r o u p energy r a n g e t h e f l u x b e h a v e s a s y m p t o t i c a l l y as
(E'-ldE
= ln(Eg_l/Eg).
(8-90)
F u r t h e r m o r e essentially all n e u t r o n a b s o r p t i o n will o c c u r in t h o s e r e s o n a n c e s c o n t a i n e d in g r o u p g. H e n c e u s i n g o u r earlier d e f i n i t i o n of t h e effective r e s o n a n c e integral, w e f i n d
^A
^
M
2
i^g
M
h
i
NA
-
X
2
i eg
/,
'
<8 9l)
-
w h e r e / E g is w r i t t e n to i n d i c a t e t h a t only those r e s o n a n c e s c o n t a i n e d in t h e g r o u p g c o n t r i b u t e to 2 a g , a n d Au g r e p r e s e n t s t h e l e t h a r g y w i d t h of g r o u p g. H e n c e we c a n g e n e r a t e m u l t i g r o u p c o n s t a n t s c h a r a c t e r i z i n g a b s o r p t i o n directly b y using r e s o n a n c e integrals. ( W e will later p r o v i d e a n a l t e r n a t i v e p r e s c r i p t i o n using pr) T h e r e f o r e " a l l " we n e e d to d o is f i g u r e o u t a w a y to c a l c u l a t e or m e a s u r e the r e s o n a n c e integrals themselves. W e will n o w t u r n o u r a t t e n t i o n t o w a r d several a p p r o x i m a t e s c h e m e s f o r d o i n g this.
D. Approximate Calculations of Resonance Integrals W e will n o w c o n s i d e r in m o r e detail n e u t r o n slowing d o w n t h r o u g h a single resonance characterizing an absorber (denoted by " A " ) distributed uniformly t h r o u g h o u t a m o d e r a t o r ( " M " ) . W e will a s s u m e t h a t t h e n e u t r o n flux has achieved its a s y m p t o t i c b e h a v i o r
342
/
THE MULTIGROUP DIFFUSION METHOD
s i t u a t i o n is
It is possible t o solve this e q u a t i o n n u m e r i c a l l y . 1 0 H o w e v e r w e will d e v e l o p several s i m p l e a p p r o x i m a t e s o l u t i o n s b a s e d o n a s s u m p t i o n s c o n c e r n i n g t h e w i d t h of t h e resonance. L e t us b e g i n b y n o t i n g several f e a t u r e s of t h e cross s e c t i o n s t h a t a p p e a r in this e q u a t i o n . W e c a n effectively i g n o r e t h e a b s o r p t i o n cross s e c t i o n of t h e m o d e r a t o r o v e r t h e r a n g e of t h e r e s o n a n c e t o write 2tM(£ )~2SM(£ ) = 2^,
(8-93)
w h e r e w e h a v e f u r t h e r n o t e d t h a t s c a t t e r i n g f r o m t h e m o d e r a t o r is c h a r a c t e r i z e d b y t h e p o t e n t i a l s c a t t e r i n g cross section 2 ^ w h i c h is essentially c o n s t a n t over t h e n a r r o w r a n g e of t h e r e s o n a n c e . T h e r e s o n a n c e a b s o r b e r is c h a r a c t e r i z e d b y a m o r e c o m p l i c a t e d c r o s s section dependence =
+
+
(8-94)
w h e r e w e h a v e s e p a r a t e d t h e r e s o n a n c e a n d p o t e n t i a l s c a t t e r i n g c o m p o n e n t s of t h e cross section. W e c a n n o w c h a r a c t e r i z e t h e e f f e c t i v e w i d t h of t h e r e s o n a n c e b y d e f i n i n g t h e practical width T p as the e n e r g y r a n g e over w h i c h t h e r e s o n a n c e cross s e c t i o n e x c e e d s t h e n o n r e s o n a n c e c o m p o n e n t ( w h i c h is essentially j u s t 2 p + 2 £ * = 2 p ) . T o e s t i m a t e T p , w e c a n u s e the B r e i t - W i g n e r f o r m u l a t o c o m p a r e - I
+1 ~2
p
(8-95)
T.
(8-96)
or Tp = 2 ( E - E
T y p i c a l l y f o r low-lying r e s o n a n c e s w h i c h is m u c h larger t h a n t h e a c t u a l N e v e r t h e l e s s f o r all r e s o n a n c e s of is m u c h s m a l l e r t h a n t h e a v e r a g e nucleus ^ \
M
0
) ~ } j ^
in 2 3 8 U , 2 0 / 2 p ~ 10 3 . T h u s T p ^ 3 0 r > . 7 eV w i d t h of t h e r e s o n a n c e . interest in r e a c t o r analysis, t h e p r a c t i c a l w i d t h e n e r g y lost in a collision with a m o d e r a t o r
= U l - a
M
) E
0
» T
p t
(8-97)
T h i s m e a n s t h a t m o s t of t h e r a n g e of i n t e g r a t i o n , E<Ef <E/aM, in t h e m o d e r a t o r i n t e g r a l in E q . (8-92) will b e s u f f i c i e n t l y f a r f r o m t h e r e s o n a n c e t h a t t h e flux a s s u m e s its a s y m p t o t i c f o r m , < £ ( £ ) - > ! / E , a n d h e n c e w e c a n r e p l a c e t h e f l u x in t h e
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
343
integral w i t h this a s y m p t o t i c f o r m with negligible e r r o r :
I E
(1~«M)£'
W
(i
/
"
<8-98>
TO-r-
If we s u b s t i t u t e this i n t o E q . (8-92), w e f i n d t h a t o u r slowing d o w n becomes E_ f «a I dE'
[ ^ + ^ ( E ) + ^{E)]
equation
2f(£"W£') +
(8-99)
E
T h e m a j o r a p p r o x i m a t i o n w e will m a k e in o u r s o l u t i o n of this e q u a t i o n will involve o u r t r e a t m e n t of t h e absorber s c a t t e r i n g integral. T h i s a p p r o x i m a t i o n will a g a i n b e b a s e d o n t h e relative w i d t h of t h e r e s o n a n c e , b u t n o w c o m p a r e d to t h e e n e r g y loss s u f f e r e d in a collision w i t h a n a b s o r b e r n u c l e u s . 1. N A R R O W R E S O N A N C E (NR) A P P R O X I M A T I O N W e will first a s s u m e t h a t t h e p r a c t i c a l w i d t h F p of t h e r e s o n a n c e is small c o m p a r e d t o t h e a v e r a g e e n e r g y loss s u f f e r e d in a collision w i t h the absorber nucleus:
A£|A = ( - ^ ) £
0
» r
p
.
(8-100)
T h e n we c a n a p p r o x i m a t e t h e a b s o r b e r s c a t t e r i n g integral as w e d i d t h a t f o r t h e m o d e r a t o r b y r e p l a c i n g
2*(£»(£')
2*
r~dE'(
1 \
2
P
(8 ioi)
I ^'iN^-o^oX f-te)-^- -
I n t h e s e c o n d i n t e g r a l w e h a v e n o t e d t h a t t h e a b s o r b e r s c a t t e r i n g cross section over m o s t of this r a n g e is essentially j u s t t h a t f o r p o t e n t i a l scattering 2 £ . U s i n g t h e s e results in E q . (8-99), w e f i n d + 2?) St(E)4>(£)= H e n c e t h e narrow resonance
approximation
^
^
^
j
•
E
(8-102)
to t h e flux is
i
j
r
-
(8
-103)
W e c a n c a l c u l a t e t h e c o r r e s p o n d i n g e x p r e s s i o n f o r t h e r e s o n a n c e integral in the N R a p p r o x i m a t i o n as ~ 2 ^ + 2 :p (8-104) ™=JEdEo*(E)^K(E)= f
344
/
THE MULTIGROUP DIFFUSION METHOD
This integral c a n now be c o m p u t e d using our D o p p l e r - b r o a d e n e d resonance s h a p e s , j u s t as it w a s f o r h y d r o g e n . Since t h e a v e r a g e e n e r g y loss p e r collision (1 — a A ) £ 0 / 2 i n c r e a s e s w i t h i n c r e a s i n g e n e r g y E0, w e m i g h t e x p e c t t h e N R a p p r o x i m a t i o n t o b e valid f o r higher e n e r g y r e s o n a n c e s . F o r l o w e r e n e r g y r e s o n a n c e s , t h e e n e r g y loss in collisions w i t h a b s o r b e r nuclei will n o l o n g e r b e larger t h a n t h e p r a c t i c a l w i d t h of t h e r e s o n a n c e T p a n d w e m u s t t h e n c o n s i d e r a n alternative approximation. 2. N A R R O W R E S O N A N C E I N F I N I T E MASS ABSORBER ( N R I M ) OR W I D E R E S O N A N C E (WR) A P P R O X I M A T I O N W e n o w c o n s i d e r t h e o t h e r limit in w h i c h t h e p r a c t i c a l w i d t h of t h e r e s o n a n c e is very w i d e c o m p a r e d t o t h e e n e r g y loss s u f f e r e d in a collision w i t h a n a b s o r b e r nucleus = (
J
^ ) ^ o « r
,
P
(8-105)
(although we continue to treat the resonance width as n a r r o w c o m p a r e d to m o d e r a t o r e n e r g y loss A E | M ) . T h i s is o c c a s i o n a l l y k n o w n as t h e wide resonance case, a l t h o u g h m o r e f r e q u e n t l y o n e o b t a i n s t h e c o r r e s p o n d i n g a p p r o x i m a t i o n t o t h e a b s o r b e r s c a t t e r i n g i n t e g r a l b y a s s u m i n g t h e a b s o r b e r t o b e infinitely m a s s i v e s u c h t h a t n e u t r o n s s u f f e r n o e n e r g y loss i n a collision w i t h t h e a b s o r b e r . T h a t is, w e t a k e t h e limit a s A->oo or aA = (A — \ / A + 1) 2 ->1 t o f i n d E
dE'HrAvA/ (E')<j>(E')
,
A
• a j
r ^
dF'
i f .
(8-106) T h e n o u r s l o w i n g d o w n e q u a t i o n [Eq. (8-99)] b e c o m e s 2m
W e c a n t h e n solve f o r t h e f l u x in the N R I M a p p r o x i m a t i o n as 2M
a n d t h e c o r r e s p o n d i n g r e s o n a n c e integral a s 'M rNRIM
I
dE
A/j7\
(8-109)
N o t i c e t h a t t h e N R a n d N R I M a p p r o x i m a t i o n s d i f f e r o n l y in t h e w a y in w h i c h s c a t t e r i n g f r o m t h e a b s o r b e r nuclei is t r e a t e d . If w e n o t e t h a t u s u a l l y t h e a b s o r b e r d e n s i t y is s u f f i c i e n t l y l o w t h a t t h e n in f a c t t h e o n l y d i f f e r e n c e b e t w e e n t h e t w o f o r m s f o r I lies in t h e s u b t r a c t i o n of t h e a b s o r b e r s c a t t e r i n g cross section f r o m t h e t o t a l cross s e c t i o n in t h e d e n o m i n a t o r of t h e N R I M e x p r e s s i o n f o r
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
345
If the a b s o r b e r c o n c e n t r a t i o n is v e r y small, 2 t ( £ ) — 2 ^ * , a n d w e p a s s t o the i n f i n i t e d i l u t i o n f o r m of t h e r e s o n a n c e integral
/» = f
(8-110)
By c o m p a r i n g E q s . (8-104) a n d (8-109) w i t h (8-110), o n e f i n d s t h a t I00 is larger t h a n either / N R o r / N R I M . T h i s is d u e to the self-shielding e f f e c t . If w e e x a m i n e e i t h e r of the a p p r o x i m a t e f o r m s , E q . (8-103) or (8-108), for t h e f l u x §(E) w i t h i n the r e s o n a n c e , w e n o t e t h a t t h e f l u x d e c r e a s e s as 2 A ( £ " ) i n c r e a s e s — t h a t is, flux depression o c c u r s . T h i s f l u x d e p r e s s i o n l o w e r s the effective a b s o r p t i o n in the r e s o n a n c e a n d h e n c e I is similarly d e c r e a s e d . W e will f i n d later in C h a p t e r 10 t h a t spatial self-shielding d u e to f u e l l u m p i n g h a s t h e s a m e e f f e c t o n r e s o n a n c e a b s o r p tion since it also r e d u c e s I . O n e c a n g o f u r t h e r a n d i n t r o d u c e t h e D o p p l e r - b r o a d e n e d B r e i t - W i g n e r line s h a p e s to c a l c u l a t e these r e s o n a n c e integrals in t e r m s of r e s o n a n c e p a r a m e t e r s . It is c u s t o m a r y ( a l t h o u g h n o t a l w a y s c o r r e c t 1 1 ) to neglect i n t e r f e r e n c e s c a t t e r i n g s u c h t h a t o u r cross sections b e c o m e
(8"
r
m
)
T h e n inserting these i n t o o u r expressions f o r t h e r e s o n a n c e integrals yields
(2M + 2A)r / N R
=
\
AE O
nrim=
2sMr .
2m + 2
a (8 112)
'
P = - j rArO- '
and 7
J { S
^
=
2sMr —
(
8
-
1
1
3
)
w h e r e t h e d e f i n i t i o n of t h e is given in E q . (8-75). H e r e (3 is t h e p a r a m e t e r essentially c h a r a c t e r i z i n g the d i l u t i o n of the a b s o r b e r , while f c h a r a c t e r i z e s t h e t e m p e r a t u r e . A s fi or b e c o m e s large, b o t h the N R a n d N R I M f o r m s a p p r o a c h the i n f i n i t e d i l u t i o n limit /NR(
—>oo) = / N R I M (
—>co) = / 0 0 =
2
.
(8-114)
EQ
A s in t h e case of h y d r o g e n e o u s m o d e r a t o r s , w e f i n d t h a t t h e r e is n o t e m p e r a t u r e d e p e n d e n c e in this limit. If w e r e c o g n i z e t h a t the t e m p e r a t u r e r a n g e of m o s t interest in r e a c t o r a p p l i c a t i o n s c o r r e s p o n d s to 0.1 < f < 1.0, t h e n w e f i n d t h a t unless 1.6 X 1 0 " 4 < / ? < 2 . 6 , t h e r e will b e n o t e m p e r a t u r e d e p e n d e n c e of / ( a n d h e n c e of p). F o r the r a n g e 2 ^ / N A ~ 10-10 3 b p e r a b s o r b e r a t o m e n c o u n t e r e d in t h e r m a l r e a c t o r design, o n e c a n e s t i m a t e the d e r i v a t i v e of / with r e s p e c t to t e m p e r a t u r e as | ^ ~ 1 0 -
4
/ ° C ^ ^ ~ - 1 0 -
5
/ ° C .
(8-115)
346 /
THE MULTIGROUP DIFFUSION METHOD
H e n c e in this p a r a m e t e r r a n g e , D o p p l e r b r o a d e n i n g will c a u s e a s i g n i f i c a n t dec r e a s e in p w i t h i n c r e a s i n g t e m p e r a t u r e . T h e N R a p p r o x i m a t i o n is c o m m o n l y u s e d f o r all b u t t h e lowest lying r e s o n a n c e s f o r w h i c h t h e N R I M a p p r o x i m a t i o n is m o r e s a t i s f a c t o r y . I n T a b l e 8-4 we h a v e given e x a m p l e s 1 2 of s u c h c a l c u l a t i o n s f o r several r e s o n a n c e s i n 2 3 8 U . T h e s e a r e c o m p a r e d w i t h d i r e c t n u m e r i c a l c a l c u l a t i o n s of t h e r e s o n a n c e integral. A l t h o u g h n e i t h e r a p p r o x i m a t i o n s e e m s t o yield s a t i s f a c t o r y results f o r t h e r e s o n a n c e s at i n t e r m e d i a t e e n e r g y , 1 3 w h e n l a r g e n u m b e r s of r e s o n a n c e s a r e a c c o u n t e d for, t h e e r r o r s t e n d t o a v e r a g e o u t . F o r c o n v e n i e n c e , w e h a v e also s u m m a r i z e d t h e results of t h e s e v a r i o u s s c h e m e s f o r h a n d l i n g r e s o n a n c e a b s o r p t i o n i n T a b l e 8-5.
TABLE 8-4:
Resonance Absorption Calculations12
Resonance Energy £ 0 ( e V )
0.2376 (21%) 0.07455 (+10.4%) 0.04739 ( - 1 8 . 6 % ) 0.010163 ( + 5.9%) 0.001114 ( + 0.9%) 0.009035 ( - 1.5%) 0.005086 ( - 2 8 . 6 % ) 0.004440 ( - 1 1 . 6 % )
6.67 21.0 36.9 81.3 90. 117.5 192. 212.
TABLE 8-5:
(1 ~Pi) NRIA
(1 -Pt) NR
(1-A) Exact
0.1998 (1.8%) 0.07059 ( + 4.5%) 0.06110 ( + 5.0%) 0.008109 ( - 1 5 . 5 % ) 0.000998 ( - 9 . 6 % ) 0.009500 ( + 3.6%) 0.01228 (73.0%) 0.007689 ( + 53.1%)
0.1963 0.06755 0.05820 0.009596 0.001104 0.009170 0.007119 0.005021
Resonance Integrals for Single Resonances Hydrogen + Infinite Mass Absorption
NR Approximation
NRIM Approximation
Homogeneous Infinite dilution
o0Tyvr Zero temperature
7
"
2E0
CT0ry7T
Homogeneous
a0TyTr 7
"
2 E0
2E0
(y0ry7r
j _ a°ry
2 E0a
2E
2 E0a
\/ac-b2/4
°
Finite dilution
w
i.
Zero temperature
N
c = a-\
^Mp
-°O
J a /
a
=1
2m
(2^ + 2^) Ty 2m + 2A
Finite temperature
B'~ B~
NHO?T
rr^
r.
Homogeneous Finite dilution
+
r
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
347
W e will r e t u r n later in C h a p t e r 10 to discuss t h e m o d i f i c a t i o n s r e q u i r e d in these results in o r d e r t o a c c o u n t f o r h e t e r o g e n e i t i e s (i.e., f u e l l u m p i n g ) . N o w , however, w e will c o n t i n u e with o u r s t u d y of n e u t r o n slowing d o w n in a n e f f o r t to t a k e a c c o u n t of spatial e f f e c t s (i.e., finite m e d i a ) .
III. NEUTRON SLOWING DOWN IN FINITE MEDIA A. The Lethargy-Dependent Px Equations W e n o w t u r n o u r a t t e n t i o n to t h e d e t e r m i n a t i o n of f a s t n e u t r o n s p e c t r a w h e n t h e spatial d e p e n d e n c e of the n e u t r o n f l u x m u s t b e c o n s i d e r e d . Since the effects of a n i s o t r o p i c s c a t t e r i n g a r e usually significant, w e will b e g i n o u r s t u d y with the Px e q u a t i o n s (4-153) a n d (4-154) r a t h e r t h a n t h e e n e r g y - d e p e n d e n t d i f f u s i o n e q u a t i o n . T h i s will allow a f a r m o r e c o n s i s t e n t t r e a t m e n t of t h e spatial a n d e n e r g y variables t h a n t h a t p r o v i d e d b y e n e r g y - d e p e n d e n t d i f f u s i o n t h e o r y . It will b e s u f f i c i e n t to c o n s i d e r t h e s t e a d y - s t a t e f o r m of these e q u a t i o n s in only a single spatial v a r i a b l e x since o u r i n t e n t is to d e v e l o p a r a t h e r s i m p l e t r e a t m e n t of t h e spatial d e p e n d e n c e t h a t w e c a n use in t h e g e n e r a t i o n of f a s t g r o u p c o n s t a n t s . It will p r o v e c o n v e n i e n t to r e w r i t e t h e Px e q u a t i o n s in t h e l e t h a r g y v a r i a b l e u as N
|£+2t(M)$(*,ii)= 2 7=1 ~
j
fU^4o(u^u)
N
+2t(M)/(jC,u)= 2
7
^ X ^ " ) *
(*>"')>
(8"116)
w h e r e 2g0(w'—»w) a n d 2* 1 («'—»w) a r e t h e i s o t r o p i c a n d linearly a n i s o t r o p i c c o m p o n e n t s of t h e d i f f e r e n t i a l cross section c h a r a c t e r i z i n g elastic s c a t t e r i n g f r o m i s o t o p i c species " i . " ( U n l e s s o t h e r w i s e i n d i c a t e d , w e will a l w a y s use the cross s e c t i o n n o t a t i o n 2 S to r e f e r o n l y to elastic s c a t t e r i n g in this section.) It s h o u l d b e n o t i c e d t h a t w e h a v e restricted these e q u a t i o n s to the d e s c r i p t i o n of n e u t r o n slowing d o w n b y t r u n c a t i n g t h e u p p e r limit of i n t e g r a t i o n at u' = u ( t h a t is, i g n o r i n g upscattering). F o r t h e m o m e n t , w e will i n c l u d e the e f f e c t s of inelastic s c a t t e r i n g a n d fission in the s o u r c e t e r m N
JTi J °
N
J
°
(8-117) a n d f o c u s o u r a t t e n t i o n o n a p p r o x i m a t e t r e a t m e n t s of slowing d o w n via elastic scattering. A l t h o u g h inelastic s c a t t e r i n g c a n p l a y a very i m p o r t a n t role in n e u t r o n slowing d o w n ( p a r t i c u l a r l y in f a s t r e a c t o r s ) , it is n o t n e a r l y as susceptible to a p p r o x i m a t e t r e a t m e n t s as elastic s c a t t e r i n g a n d usually r e q u i r e s a direct m u l t i g r o u p t r e a t m e n t , discussion of w h i c h will b e d e f e r r e d until later in this section. It s h o u l d also b e n o t e d t h a t w e h a v e explicitly i g n o r e d a n y a n i s o t r o p i c c o m p o n e n t s of inelastic s c a t t e r i n g or fission b y setting t h e s o u r c e t e r m 5 , ( x , w ) = 0 in the s e c o n d P ,
348
/
THE MULTIGROUP DIFFUSION METHOD
e q u a t i o n (8-116). O u r neglect of this a n i s o t r o p i c s o u r c e c o m p o n e n t is m o t i v a t e d b y recalling t h a t b o t h of t h e s e r e a c t i o n s involve c o m p o u n d n u c l e u s f o r m a t i o n in h e a v y nuclei a n d h e n c e s h o u l d b e a d e q u a t e l y d e s c r i b e d as p r o c e s s e s t h a t a r e i s o t r o p i c in the L A B system. T o p r o c e e d f u r t h e r , w e will n o w i n t r o d u c e t h e s l o w i n g d o w n d e n s i t i e s c h a r a c t e r i z i n g elastic s c a t t e r i n g f r o m e a c h i s o t o p i c species b y r e w r i t i n g E q . (8-18) in t e r m s of t h e l e t h a r g y v a r i a b l e as ql0(x9u)
ru r oo = / du' I du"2\ ( m ' - > m " ) $ ( j c , k ' ) . •'O Ju °
W e h a v e i n s e r t e d t h e s u b s c r i p t " 0 " t o d i s t i n g u i s h ql0(x,u) q u a n t i t y d e f i n e d in t e r m s of t h e c u r r e n t d e n s i t y J(x,u):
q[(x,u)=
("du'
J
0
(8-118)
f r o m a closely r e l a t e d
(0°du'%l(u'^u")J(x9u').
(8-119)
W e c a n m a k e t h e s e d e f i n i t i o n s a bit m o r e explicit b y r e c a l l i n g t h a t t h e elastic s c a t t e r i n g p r o c e s s e s of g r e a t e s t c o n c e r n in n u c l e a r r e a c t o r a n a l y s i s i n v o l v e s - w a v e scattering, for which we c a n write
—— r—, (l-a,-)
u — I n —
0,
(8-120)
otherwise.
O n e c a n similarly s h o w t h a t f o r s - w a v e scattering,
m-^-m
-
2
,
u — I n —
otherwise.
0,
(8-121) T h u s w e c a n u s e t h e s e cross s e c t i o n s t o w r i t e t h e s l o w i n g d o w n densities in m o r e explicit f o r m . F o r e x a m p l e ,
ql0(xiu)=
I
du'
•A/— In—
• f Ju-in!
I
du"
Ju eu~u-a: du't\(u')${xyu')
(8-122)
OLi
T o i m p l e m e n t t h e s e d e f i n i t i o n s , w e will d i f f e r e n t i a t e t h e s l o w i n g d o w n densities w i t h r e s p e c t t o u: H
du
= fXdu"^So(u^u")4>(x,u)-
f d u ' ^ u ' ^ u ^ u ' ) ,
(8-123)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
349
or noting that
J
r oo du"^{u-*u") = Vlu),
(8-124)
= S ^ M X j c , U) - f j d u ' ^ ( u ' ^ u ^ x , « ' ) .
(8-125)
U
we find
I n a v e r y similar m a n n e r w e c a n f i n d
-r— = ju 01 .2s(w)/ ( x , u) — / duf^ls{u'->u)J{x,u'), 1 ou Jq
(8-126)
where we have noted that /* 00
/
du"VSi(u^u")
= ji0^(u).
(8-127)
W e c a n n o w use E q s . (8-125) a n d (8-126) to r e p l a c e t h e elastic s c a t t e r i n g t e r m s in t h e Px e q u a t i o n s (8-116) w i t h t h e derivatives of t h e elastic slowing d o w n densities: N
+ ^neM^*'
+ S0(X9 U),
") = - 2 i=l
x
H e r e w e h a v e d e f i n e d t h e nonelastic
S ^ m ) = 2t(«) -
a n d the transport
v
(8-128) ^
cross section
(«) - 2t(tt) -
2 2j(«) 1=1
(8-129)
cross s e c t i o n
2tr(«) = 2 t ( « ) - M 0 2 f ( M ) = 2 t ( w ) - 2 1=1
(8-130)
T h e Px e q u a t i o n s in t h e f o r m (8-128) will serve a s o u r p o i n t of d e p a r t u r e f o r the s t u d y of n e u t r o n slowing d o w n in finite m e d i a . W e w o u l d stress t h a t t h e only essential a p p r o x i m a t i o n u s e d in d e r i v i n g these e q u a t i o n s involves t h e a s s u m p t i o n t h a t t h e a n g u l a r n e u t r o n f l u x
350
/
THE MULTIGROUP DIFFUSION METHOD
v a r i a b l e s are f o r m a l l y d e f i n e d in t e r m s of t h e f l u x a n d c u r r e n t b y E q s . (8-118) a n d (8-119), it will p r o v e c o n v e n i e n t to seek a l t e r n a t i v e a p p r o x i m a t e e q u a t i o n s f o r the slowing d o w n densities t h a t c a n b e u s e d to a u g m e n t t h e Px e q u a t i o n s (8-128).
B. Approximate Treatments of Neutron Slowing Down—Continuous Slowing Down Theory I n m o s t s c h e m e s u s e d to g e n e r a t e f a s t n e u t r o n s p e c t r a , elastic s c a t t e r i n g is t r e a t e d u s i n g a p p r o x i m a t e m e t h o d s t h a t a r e usually r e f e r r e d to collectively as continuous slowing down theory. I n s u c h m e t h o d s , a n a p p r o x i m a t e d i f f e r e n t i a l e q u a t i o n is d e v e l o p e d f o r e a c h slowing d o w n d e n s i t y b y e x p a n d i n g t h e collision densities 2*(w')
b y e x a m i n i n g t h e p a r t i c u l a r l y simple case of s l o w i n g d o w n w h i c h A = 1 a n d h e n c e a = 0. T h i s case is a p a r t i c u l a r l y p o i n t since it c a n b e t r e a t e d exactly. If w e set a = 0 in E q s . we find
(8-131)
H o w e v e r of m o r e r e l e v a n c e is t h e f a c t t h a t b y d i f f e r e n t i a t i n g these expressions, w e c a n i d e n t i f y t w o simple d i f f e r e n t i a l e q u a t i o n s ,
(8-132)
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351
w h i c h c a n b e u s e d to s u p p l e m e n t t h e Px slowing d o w n e q u a t i o n s (8-128). It s h o u l d b e stressed t h a t while t h e s e e q u a t i o n s a r e exact, t h e y only a p p l y to t h e case of hydrogen. 2. T H E A G E A P P R O X I M A T I O N F o r t h e m o r e g e n e r a l c a s e of n o n h y d r o g e n e o u s m o d e r a t o r s , o n e c a n n o t o b t a i n s u c h d i f f e r e n t i a l e q u a t i o n s exactly c h a r a c t e r i z i n g t h e slowing d o w n densities. H o w e v e r essentially all t r e a t m e n t s of elastic s c a t t e r i n g a t t e m p t t o o b t a i n a p p r o x i m a t e d i f f e r e n t i a l e q u a t i o n s f o r qx0(x,u) a n d q\(x,u) in the f o r m H
• (8-133)
3*1
.
T o a c c o m p l i s h this o n e first n o t e s t h a t in m a n y cases the elastic s c a t t e r i n g collision density «)<£(«) is a slowly v a r y i n g f u n c t i o n of lethargy. F o r e x a m p l e , f o r z e r o a b s o r p t i o n in a n i n f i n i t e m e d i u m f a r b e l o w s o u r c e energies, w e k n o w t h a t the a s y m p t o t i c f o r m of the collision d e n s i t y is in f a c t c o n s t a n t , 2).(w)<|>(«) = S 0 / ! ( . T h u s if the s y s t e m of interest is n o t t o o strongly a b s o r b i n g , a n d if t h e collision interval is n o t too l a r g e — t h a t is, if A is l a r g e — w e a r e t e m p t e d to e x p a n d the collision density a p p e a r i n g in t h e integrals of E q s . (8-118) a n d (8-119) as T a y l o r e x p a n s i o n s a b o u t u'= u a n d r e t a i n o n l y l o w - o r d e r t e r m s : Z i ( « > ( * , « ' ) » u ) + ( « ' - u ) f - [ 2 j ( « M x , u ) J, (8-134) 2 j ( « ' ) / (x,«')
= 2i(«)/
(*, u) + (u'-u)-^[
2 \ { u ) J (x, u) ].
If we s u b s t i t u t e t h e first of these e x p a n s i o n s i n t o E q . (8-118), w e f i n d +
(8-135)
where we have identified a. In 4
f
du'
a.
= 1
a.-
1 — a, '
(8-136)
Ju-In—
at=
I Ju-inl
du'
1 — a.
2(1-a,)
(8-137)
It is c o m m o n to t r u n c a t e E q . (8-135) a f t e r o n l y o n e t e r m t o write (8-138) T h i s result is k n o w n a s t h e age approximation
o r s o m e t i m e s as continuous
slowing
352 /
THE MULTIGROUP DIFFUSION METHOD
down theory. T h e f o r m e r n a m e will o n l y b e c o m e a p p a r e n t i n S e c t i o n 8-1V-C. T h e l a t t e r n a m e arises b e c a u s e t h e neglect of h i g h e r o r d e r t e r m s i n t h e e x p a n s i o n of t h e collision d e n s i t y e f f e c t i v e l y i m p l i e s t h a t n e u t r o n s lose o n l y a n i n f i n i t e s i m a l a m o u n t of e n e r g y in e a c h collision. T o p r o c e e d f u r t h e r , w e will a s s u m e t h a t w e c a n r a t h e r a r b i t r a r i l y set q\(x9u)
= 09
(8-139)
t h a t is, w e will n e g l e c t a n i s o t r o p i c e n e r g y e x c h a n g e . T h e n t h e s e c o n d of e q u a t i o n s [Eq. (8-128)] c a n b e solved f o r
/ ( x , u) = - [ 3 2 » r ' ^
= -.D
(u) ^
,
(8-140)
w h i c h yields j u s t F i c k ' s law. T h i s c a n t h e n b e s u b s t i t u t e d i n t o t h e first e q u a t i o n (8-128) t o f i n d
(8-H1)
w h e r e w e h a v e u s e d t h e a g e a p p r o x i m a t i o n f o r all i s o t o p e s in t h e s y s t e m . T h i s is k n o w n as t h e age-diffusion equation. It is s o m e t i m e s a l s o r e f e r r e d t o a s inconsistent Pj theory s i n c e w e h a v e n o t t r e a t e d t h e slowing d e n s i t y q\{x9 u) i n t h e s a m e f a s h i o n as w e t r e a t e d ql${x9 u). T h e a g e - d i f f u s i o n e q u a t i o n is p r i m a r i l y of historical i m p o r t a n c e in r e a c t o r a n a l y s i s since it is r e s t r i c t e d t o slowing d o w n in h e a v y m a s s m o d e r a t o r s s u c h as g r a p h i t e a n d w o u l d c e r t a i n l y n o t a p p l y t o h y d r o g e n e o u s m o d e r a t o r s . It is of i n t e r e s t s i m p l y b e c a u s e it c a n b e solved exactly i n c e r t a i n special cases (as w e will see in S e c t i o n 8-IV-C). W e c a n p a r t i a l l y alleviate t h e restriction t o h e a v y m a s s m o d e r a t o r s b y using t h e e x a c t e q u a t i o n s (8-132) t o d e s c r i b e h y d r o g e n m o d e r a t i o n a n d t h e n u s e t h e a g e a p p r o x i m a t i o n ( i n c o n s i s t e n t />,) o n l y to d e s c r i b e n o n h y d r o g e n e o u s species (A > 1). T h i s s c h e m e , w h i c h is o n e of t h e c o m m o n m e t h o d s u s e d i n c o m p u t i n g f a s t s p e c t r a in L W R s , is k n o w n as t h e Selengut-Goertzel method. It is a relatively s i m p l e m a t t e r t o r e m o v e t h e i n c o n s i s t e n c y p r e s e n t in t h e a g e a p p r o x i m a t i o n [i.e., E q . (8-139)] b y s u b s t i t u t i n g E q . (8-134) i n t o t h e d e f i n i t i o n of q\{x9u) given b y E q . (8-119) t o f i n d q\(x9u)^^(u)J(x9u)
(8-142)
where
{ ^^^ C1 —
In
-K l ) ) _ (1 _
y^ArK
_
ln
l)) (8-143)
If E q s . (8-138) a n d E q . (8-142) a r e b o t h u s e d in t h e Px e q u a t i o n s , o n e a r r i v e s a t t h e consistent P1 approximation.
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
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353
O n e o t h e r m o d i f i c a t i o n o c c a s i o n a l l y e n c o u n t e r e d in the t r e a t m e n t of n e u t r o n slowing d o w n in w e a k l y a b s o r b i n g s y s t e m s involves r e t a i n i n g b o t h t e r m s in the e x p a n s i o n (8-135) t o w r i t e q0(x9 u) = ££s(w)<J>(x, u) + a—
[2s(w)(*,«)]
a €2g(«)*(*, « ) + f - ! ;
«)]
(8-144)
w h e r e w e h a v e r e i n t r o d u c e d E q . (8-138) i n t o t h e s e c o n d t e r m t o a c h i e v e a c o n s i s t e n t s e c o n d - o r d e r e x p a n s i o n . N o w in a n infinite, sourceless m e d i u m , E q . (8-128) c a n b e w r i t t e n as -^=2ne(M)
(8-145)
If w e s u b s t i t u t e E q . (8-144) i n t o E q . (8-145), w e c a n solve f o r *„(*,«) However we can approximate
|2ne(W))
(8-146)
— I (see P r o b l e m 8-29) to write q0(x9u)a&t(u)
(8-147)
T h i s f o r m of the a g e a p p r o x i m a t i o n is f r e q u e n t l y u s e d i n s t e a d of o u r earlier f o r m given b y E q . (8-138). F o r w e a k a b s o r p t i o n , b o t h expressions yield very similar results. 3. T H E G R U E L I N G - G O E R T Z E L A P P R O X I M A T I O N T h u s f a r w e h a v e p r o v i d e d p r e s c r i p t i o n s f o r h a n d l i n g n e u t r o n slowing d o w n in either v e r y h e a v y (age t h e o r y ) or h y d r o g e n e o u s (exact) m o d e r a t o r s . H o w e v e r n e i t h e r of t h e s e s c h e m e s is a d e q u a t e f o r light, n o n h y d r o g e n e o u s m a t e r i a l s — w i t h d e u t e r i u m 2 D b e i n g a p r i m e villian. H e n c e w e a r e m o t i v a t e d to c o n s t r u c t a n a l t e r n a t i v e to t h e a g e a p p r o x i m a t i o n f o r n o n h y d r o g e n e o u s m e d i a . T o a c c o m p l i s h this, w e will use the T a y l o r e x p a n s i o n of t h e collision densities to write a p p r o x i m a t e e q u a t i o n s f o r qlQ(x,u) a n d q\(x,u) in the f o r m s E q . (8-133). T o illustrate h o w this m a y b e a c c o m p l i s h e d , s u p p o s e w e use t h e e x p a n s i o n E q . (8-135) to c a l c u l a t e H
i
+
(8-148)
3w T o cast this i n t o t h e f o r m of E q . (8-133), w e c a n i g n o r e t e r m s in 3 2<j>/du2 as b e i n g of h i g h e r o r d e r a n d c h o o s e A 0/ so t h a t t e r m s in d<j>/du cancel. T h i s yields 1
a , ( l + l n — + ^-ln 2 — ) a \ <*/ 2 iJ 1 — a. I 1 + In W
(8-149)
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THE MULTIGROUP DIFFUSION METHOD
A very similar c a l c u l a t i o n will yield the s e c o n d of e q u a t i o n s (8-133) w i t h t h e c h o i c e (1 + Y,)a f (1 + Y,-) 4
a f ^ l / a - ^ l n l / a ^ l ) 7i
1
(1 -
3
Yi)[8-av"/
2
(ln21 / o i - 4 l n 1 / a , + 8)]jft - f i u =
(8-150)
A u g m e n t i n g t h e set of Px e q u a t i o n s (8-128) w i t h E q . (8-133) yields w h a t is k n o w n as t h e Greuling-Goertzel approximation. Since h i g h e r o r d e r t e r m s in t h e collision d e n s i t y e x p a n s i o n h a v e b e e n r e t a i n e d in this s c h e m e , t h e G r e u l i n g - G o e r t z e l a p p r o x i m a t i o n will p r o v i d e a f a r m o r e a c c u r a t e t r e a t m e n t of i n t e r m e d i a t e m a s s m o d e r a t o r s s u c h as 2 D o r 4 Be t h a n t h a t p r o v i d e d b y a g e t h e o r y . T h i s a p p r o x i m a t i o n a l s o h a s t h e p r o p e r t y t h a t it r e d u c e s t o t h e e x a c t e q u a t i o n s d e s c r i b i n g h y d r o g e n in t h e limit A—>1. 4. S U M M A R Y O F T H E Px S L O W I N G D O W N E Q U A T I O N S It is c o n v e n i e n t t o s u m m a r i z e t h e s e v a r i o u s a p p r o x i m a t e n e u t r o n s l o w i n g d o w n b y w r i t i n g t h e Px e q u a t i o n s as
treatments
of
NHj du
dx
du
+
S0(x,u), (8-151)
1 3<J> 3 dx
/ x /
x du
du
w h e r e t h e s l o w i n g d o w n densities a r e given b y t h e e q u a t i o n s 3?OH du dqH 3 -g
—
( e x a c t t r e a t m e n t of H )
U
Ki
Xlt—^—I-
(8-152) + 4oH'ix9
u) =
u
\
= fixiJ(x>u)> ( a p p r o x i m a t e t r e a t m e n t of N H ) .
N o t e w e h a v e s e p a r a t e d t h e t r e a t m e n t of h y d r o g e n o u s ( H ) f r o m n o n h y d r o g e n e o u s ( N H ) slowing d o w n . T h e various approximations we have discussed c a n then be c h a r a c t e r i z e d b y a p a r t i c u l a r c h o i c e of t h e c o e f f i c i e n t s A 0/ , X w fi 0 i , a n d f} u . W e h a v e s u m m a r i z e d t h e s e c h o i c e s in T a b l e 8-6 b e l o w :
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS TABLE 8-6:
/
355
Choice of Parameters for Continuous Slowing Down Models
Poi
^0/
Pit
Age approximation Selengut-Goertzel
0
0
0
Consistent age approximation
0
0
fi^S 0
Eq. (8-149)
Eq. (8-150)
Greuling-Goertzel (s-wave scattering)
T h e s e e q u a t i o n s c a n b e r a t h e r easily solved o n a digital c o m p u t e r p r o v i d e d the spatial d e p e n d e n c e c a n b e simplified.
5. T R E A T M E N T O F T H E SPATIAL D E P E N D E N C E IN T H E />, S L O W I N G DOWN EQUATIONS R e c a l l t h a t w e a r e g o i n g to use t h e s o l u t i o n of t h e Px slowing d o w n e q u a t i o n s t o g e n e r a t e f a s t g r o u p c o n s t a n t s . H e n c e w e only n e e d a c r u d e m e t h o d to a c c o u n t f o r gross l e a k a g e effects. I n p a r t i c u l a r w e will a s s u m e t h a t t h e l e t h a r g y a n d spatial d e p e n d e n c e of t h e f l u x a r e s e p a r a b l e a n d f u r t h e r m o r e c h a r a c t e r i z e t h e spatial d e p e n d e n c e of e a c h of t h e v a r i a b l e s b y a simple b u c k l i n g m o d e . F o r e x a m p l e , the f l u x w o u l d b e w r i t t e n as
(8-153)
w i t h similar expressions f o r c u r r e n t , source, a n d slowing d o w n densities. T h a t is, w e a p p r o x i m a t e t h e spatial d e p e n d e n c e b y a single F o u r i e r m o d e , exp (iBx). (This t u r n s o u t to b e s o m e w h a t m o r e c o n v e n i e n t t h a n c h o o s i n g a slab g e o m e t r y m o d e s u c h as c o s ^ x since it results in d i f f e r e n t spatial v a r i a t i o n f o r t h e f l u x a n d t h e c u r r e n t w h e n t h e real p a r t of t h e s o l u t i o n s is t a k e n . ) H e r e t h e p a r a m e t e r B will c h a r a c t e r i z e t h e l e a k a g e in e a c h r e g i o n of t h e c o r e in w h i c h t h e n e u t r o n s p e c t r u m 4>(u) is t o b e c a l c u l a t e d , a n d m u s t b e c h o s e n f r o m o t h e r c o n s i d e r a t i o n s . F o r e x a m p l e , if w e w e r e c o n s i d e r i n g a b a r e , h o m o g e n e o u s r e a c t o r w e k n o w the flux would indeed be separable, a n d B would b e determined by
V ^ ( r ) + 5 V ( r ) = 0.
(8-154)
M o r e generally, t h e flux will n o t b e s e p a r a b l e in r a n d w, b u t B 2 c a n b e r e g a r d e d as c h a r a c t e r i z i n g the a v e r a g e c o r e leakage. I n t h e e v e n t t h a t a n a c c u r a t e e s t i m a t e is r e q u i r e d , o n e c o u l d first e s t i m a t e B2, c a l c u l a t e g r o u p c o n s t a n t s , use these in a m u l t i g r o u p d i f f u s i o n c a l c u l a t i o n of t h e flux, a n d t h e n use this flux to c a l c u l a t e a n e w B2~(V2
356
/
THE MULTIGROUP DIFFUSION METHOD
s e p a r a b l e f o r m s E q . (8-153) i n t o t h e Pl e q u a t i o n s t o arrive a t dq" iBJ («) + Sne(u)
dq™> - ^
+
S0(u),
1
+ 2tr(W)/ ( „ ) = - _ -
(8-155)
^
_ i
a u g m e n t e d b y t h e set dq?
and
H e n c e w e n o w h a v e a r r i v e d a t a set of simple, f i r s t - o r d e r d i f f e r e n t i a l e q u a t i o n s t h a t w e c a n i n t e g r a t e n u m e r i c a l l y . W e s h o u l d recall h e r e t h a t inelastic s c a t t e r i n g a n d fission h a v e b e e n i n c l u d e d in t h e s o u r c e t e r m as N
SQ(u)=
2 7-i
N
f du'2\a(u'-*u)
J
2 * ( « ) 72i
r d u ' v ^ u ' M u ^ + S ^ u ) . (8-157)
6. A N A L T E R N A T I V E T R E A T M E N T O F SPATIAL D E P E N D E N C E — T H E METHOD15 T h u s f a r w e h a v e c o n s i d e r e d a p p r o x i m a t i n g t h e s p a t i a l d e p e n d e n c e of t h e s o l u t i o n s t o t h e P , s l o w i n g d o w n e q u a t i o n s b y a single m o d e of t h e f o r m e x p T h e r e is a n a l t e r n a t i v e s c h e m e t h a t b y p a s s e s t h e Px e q u a t i o n s entirely, a n d f o r c o m p l e t e n e s s w e will s k e t c h its d e r i v a t i o n h e r e . T o d o so, w e m u s t r e t u r n t o t h e s t e a d y - s t a t e t r a n s p o r t e q u a t i o n , w r i t t e n in its o n e - d i m e n s i o n a l f o r m , E q . (4-48), as 8
/ A — +2 T (W)
r + 1
ru
d f i ' j du/1s(u/^ui/x,-^iJ,)(p(x,ii\u/)
+
S(x,ii9u). (8-158)
N o w a s s u m e a n a n g u l a r f l u x a n d s o u r c e spatial d e p e n d e n c e of t h e f o r m similar to t h a t in E q . (8-153):
S(u)
p
(8-159)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
357
( N o t e t h a t w e h a v e a s s u m e d a n i s o t r o p i c s o u r c e c o n s i s t e n t with o u r a s s u m p t i o n s c o n c e r n i n g fission a n d inelastic scattering.) T h e n w e f i n d
earlier
+1
[ 2 t ( w ) - iB{x]
dy! j " du'2s(u'^>u9ii'-->n)
S(u) /x', u') + ^ y ^ .
(8-160)
A t this p o i n t , if w e m u l t i p l i e d b y 1 or /x a n d i n t e g r a t e d over /x a n d t h e n e x p a n d e d cp— l/2<j>(u) + 3/2fiJ(u), w e w o u l d r e d e r i v e the P , e q u a t i o n s . I n s t e a d s u p p o s e we first divide b y ( 2 t — iBfi) a n d f u r t h e r m o r e a s s u m e o n l y linearly a n i s o t r o p i c scattering to write:
-1
\
f"du'2s0(u'^u)
+ ^ L J0 N o w multiply by equations:
fdu'i:ll(u'->u)j(u')+
(8-161)
1 a n d /x a n d i n t e g r a t e over /x to f i n d the c o u p l e d p a i r of
a(u) r
r
+ ^ [ 1 -A(u)]
J
(") = i
f"du'2sl(u'^u)J(u'),
[1 - •A (") ] f0"du' 32,(w)
2
s0(«'->«)*(«')
+
S0(«)
ru
B t a n "- 1 [ 5 / 2 , ( m ) ] A(u)
£/2t(«)
(8-162)
*
A f t e r s o m e r e a r r a n g i n g , o n e c a n rewrite these e q u a t i o n s as iBJ (u) + 2 t ( u)
2 s 0 ( w ' - > u)$(u')
+ S 0 (M). (8-163)
iB 4>(t#) + y ( w ) 2 t ( w ) / (w) = f"du' 3 •'o
2 s 1 (m'->m)./ («'),
where
? ( « ) - —
— - 1 + 1 5 ( 2 ; tan
'(t)
•
(8-164)
358
/
THE MULTIGROUP DIFFUSION METHOD
U p o n r e w r i t i n g t h e s c a t t e r i n g integrals i n t e r m s of slowing d o w n densities, o n e c a n r e c o g n i z e t h a t t h e Bx equations a b o v e a r e very similar t o t h e P , e q u a t i o n s , e x c e p t f o r t h e a p p e a r a n c e of t h e f a c t o r y. F o r b a r e s l a b geometries, t h e Bx e q u a t i o n s a r e f o u n d t o yield slightly m o r e a c c u r a t e s p e c t r a . I n f a c t , if o n l y linearly a n i s o t r o p i c s c a t t e r i n g is p r e s e n t , t h e Bx e q u a t i o n s p r o v i d e a n e x a c t t r e a t m e n t of a n g u l a r effects. H o w e v e r f o r m o r e c o m p l i c a t e d geometries, as well as f o r large t h e r m a l r e a c t o r c o r e s in w h i c h l e a k a g e is relatively small, b o t h m e t h o d s yield v e r y similar results. T h e m o r e r e c e n t t e n d e n c y h a s b e e n t o g e n e r a t e g r o u p c o n s t a n t s u s i n g t h e Bx e q u a t i o n s . H o w e v e r b e c a u s e of their simplicity ( a n d m a t h e m a t i c a l similarity), w e will c o n t i n u e o u r d i s c u s s i o n u s i n g t h e Px e q u a t i o n s f o r t h e c a l c u l a t i o n of f a s t n e u t r o n spectra.
IV. FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS W e n o w t u r n o u r a t t e n t i o n t o w a r d t h e a p p l i c a t i o n of t h e Px a n d Bx e q u a t i o n s t o t h e c a l c u l a t i o n of f a s t n e u t r o n s p e c t r a s u i t a b l e f o r t h e g e n e r a t i o n of f a s t g r o u p c o n s t a n t s t o b e u s e d in f e w - g r o u p d i f f u s i o n c o d e s . W e will s t u d y t h r e e d i f f e r e n t s c h e m e s t h a t c a n b e u s e d t o g e n e r a t e f a s t g r o u p c o n s t a n t s . T h e first s u c h s c h e m e will b e t h e n u m e r i c a l s o l u t i o n of t h e Px (or B x ) e q u a t i o n s b y discretizing t h e s e e q u a t i o n s o n a f i n e g r o u p m e s h (typically 5 0 - 1 0 0 g r o u p s ) . T h i s s c h e m e w a s originally d e v e l o p e d f o r t h e a n a l y s i s of L W R s , a n d w h e n a p p l i e d t o t h e g e n e r a t i o n of g r o u p c o n s t a n t s f o r h y d r o g e n e o u s systems, it is k n o w n as t h e M U F T techn i q u e . 1 6 F o r m a n y y e a r s it a n d its v a r i o u s m o d i f i c a t i o n s s u c h as t h e G A M codes 1 7 h a v e b e e n t h e m a i n s t a y of t h e r m a l r e a c t o r design. F o r this r e a s o n , a n d also b e c a u s e t h e t e c h n i q u e s u s e d in t h e M U F T - G A M c o d e s t o solve t h e Px (or Bx) e q u a t i o n s a r e c o m m o n t o o t h e r f a s t s p e c t r u m codes, w e will c o n s i d e r this s c h e m e in s o m e detail. W e will a l s o c o n s i d e r t h e c a l c u l a t i o n of f a s t r e a c t o r s p e c t r a in w h i c h inelastic s c a t t e r i n g is of p r i m a r y i m p o r t a n c e a n d f o r w h i c h a d i r e c t s o l u t i o n of t h e Px e q u a t i o n s over a n u l t r a f i n e g r o u p s t r u c t u r e is r e q u i r e d . T h e n f i n a l l y f o r c o m p l e t e n e s s ( a n d h i s t o r i c a l s e n t i m e n t ) , w e will b r i e f l y d e v e l o p t h e a n a l y t i c a l aged i f f u s i o n m o d e l ( a l t h o u g h s u c h a m o d e l h a s v e r y little r e l e v a n c e t o m o d e r n d a y reactor calculations).
A. MUFT-GAM TYPE FAST SPECTRUM CALCULATIONS L e t us b e g i n b y c o n s i d e r i n g t h e Px e q u a t i o n s a p p l i e d t o d e s c r i b e n e u t r o n slowing d o w n i n a m i x t u r e of h y d r o g e n p l u s a single h e a v i e r i s o t o p e of m a s s n u m b e r A. ( T h e m o d i f i c a t i o n s of t h e m e t h o d w e will d e s c r i b e t o h a n d l e m o r e g e n e r a l m i x t u r e s of m o d e r a t o r s is s t r a i g h t f o r w a r d . ) T o b e specific, w e will utilize t h e a g e a p p r o x i m a t i o n t o d e s c r i b e slowing d o w n f r o m this h e a v y i s o t o p e c o u p l e d w i t h a n e x a c t t r e a t m e n t of h y d r o g e n — t h a t is, w e will use t h e S e l e n g u t - G o e r t z e l d e s c r i p t i o n . It s h o u l d b e stressed t h a t t h e p r o c e d u r e w e will o u t l i n e w o u l d a p p l y e q u a l l y well to t h e o t h e r a p p r o x i m a t e t r e a t m e n t s of s l o w i n g d o w n t h a t w e h a v e d i s c u s s e d (e.g., t h e c o n s i s t e n t Px a p p r o x i m a t i o n or t h e G o e r t z e l - G r e u l i n g m e t h o d ) .
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
359
/
T h e Px e q u a t i o n s c h a r a c t e r i z i n g a single spatial m o d e c a n t h e n b e w r i t t e n as
iB . , v +
j - - ^ . L du
du
3
du
H e r e w e h a v e f o u n d it c o n v e n i e n t to s e p a r a t e t h e r e m o v a l cross s e c t i o n i n t o t h r e e parts: = " s m o o t h " a b s o r p t i o n cross section, = " r e s o n a n c e " a b s o r p t i o n cross section, r oo
J
du'Z^u—.>w') = inelastic scattering cross section. o R e m e m b e r t h a t r e m o v a l d u e to elastic s c a t t e r i n g h a s a l r e a d y b e e n a c c o u n t e d f o r in t h e t e r m s i n v o l v i n g t h e s l o w i n g d o w n density. T o solve t h e s e e q u a t i o n s , w e will n o w discretize t h e m into m u l t i g r o u p f o r m . W e will c h o o s e a f i n e g r o u p s t r u c t u r e in the l e t h a r g y variables as s h o w n in F i g u r e 8-5. W e will l a b e l these f i n e g r o u p s w i t h a s u b s c r i p t n to a v o i d c o n f u s i o n with the c o a r s e g r o u p s t r u c t u r e d e n o t e d earlier b y g. T h e set of e q u a t i o n s (8-165) will n o w b e a v e r a g e d over t h e f i n e g r o u p s t r u c t u r e in a m a n n e r very similar to t h e m u l t i g r o u p e q u a t i o n s d e v e l o p e d in C h a p t e r 7. F o r e x a m p l e , t h e g r o u p a v e r a g e s of d e p e n d e n t v a r i a b l e s s u c h as
n
un—\
Simple d i f f e r e n c e expressions c a n b e u s e d f o r derivatives: du
f
dq fa=(?n-
T h e fine group constants characterizing smoothly varying (nonresonant) sections a r e d e t e r m i n e d b y simple a v e r a g i n g : 2
-. = daV P u n Ju
n
un-1
cross
(8 168) n
"
_x
or n
(8-167)
un> _ j
360
/
THE MULTIGROUP DIFFUSION METHOD
= 0
-i-u
10 MeV w o
1 MeV
0.1 MeV
--5
ln W
10 keV
I or
--
1 keV
- - 1 0
100 eV
10 eV
--15
- I 0) o
--
1 eV
--
0.1 eV
f— 0.02 eV
—20
FIGURE 8-5.
Typical few-group structures.
T h e g r o u p c o n s t a n t s r e p r e s e n t i n g r e s o n a n c e a b s o r p t i o n in t h e g r o u p c a n either b e c a l c u l a t e d in t e r m s of t h e r e s o n a n c e integrals c h a r a c t e r i z i n g r e s o n a n c e s in t h e group: 2
r a
"
= ~ A A un
(8-170) Un
or in t e r m s of t h e r e s o n a n c e e s c a p e p r o b a b i l i t y pn f o r t h e g r o u p b y n o t i n g : P" "n-1
•'I/
=
+
(8-171)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
/
361
H e n c e the m u l t i g r o u p f o r m of the slowing d o w n e q u a t i o n s (8-165) c a n b e w r i t t e n as: iBJnAun
+
un + 2in^„A
un + {\~Pn)[
+
] ~ K
= X „Au„ + Au n
«
+ <_,)AM„/2+ § (,« -
J
_ 4
2
~ Ion -.) - «
~ <
,)-
h
( q A + < j A w „ / 2 = ( | A 2 f )^„AW„,
„ = i,...,
N
W e h a v e n o w a r r i v e d at a set of 5N a l g e b r a i c e q u a t i o n s f o r the u n k n o w n s Jn, (/>„, = q ^ , q " , a n d qo n . T o solve these e q u a t i o n s , w e b e g i n by n o t i n g #(£> = 0 (since n o n e u t r o n s c a n slow d o w n i n t o t h e u p p e r m o s t g r o u p ) . W e c a n t h e n solve f o r Jx a n d
_
( *-%{u)$(u)du
2 n= 1
nnAun
7
'
j
«u)du
2
(8"173)
^
n=1
or 54 1 ^
Dx =
.
(8-174)
2 n—
S o m e t i m e s a n a l t e r n a t i v e s c h e m e is u s e d t o c a l c u l a t e t h e g r o u p - d i f f u s i o n coefficients. T h i s p r o c e d u r e c o m e s f r o m r e q u i r i n g t h a t Dg b e d e f i n e d s u c h t h a t for a g r o u p of l e t h a r g y w i d t h A u f duJ(x,w) •J A u
= -Dg-j-
f ClX J
du<j>(x,u).
(8-175)
If w e u s e o u r single Lgle bb\u c k l i n g m o d e f o r m s f o r J ( x , u) a n d <j>(x, w), w e f i n d t h a t Dg is then defined by 54
2
JMn
2 <#>„aw„
n=1
362 /
THE MULTIGROUP DIFFUSION METHOD
I n s u m m a r y t h e n , M U F T - G A M - t y p e Schemes g e n e r a t e a f a s t n e u t r o n s p e c t r u m a n d c a l c u l a t e f e w - g r o u p c o n s t a n t s b y u s i n g t h e Px (or Bx) slowing d o w n e q u a t i o n s , a p p r o x i m a t i n g the spatial d e p e n d e n c e b y a single spatial m o d e c h a r a c t e r i z e d b y a n " e q u i v a l e n t b a r e c o r e b u c k l i n g " B 2 , t r e a t i n g elastic s c a t t e r i n g b y a c o n t i n u o u s slowing d o w n m o d e l (an exact t r e a t m e n t for h y d r o g e n a n d either the age approximation or the Goertzel-Greuling model for materials with mass n u m b e r A > 1), a n d h a n d l i n g inelastic s c a t t e r i n g u s i n g a m u l t i g r o u p t r a n s f e r m a t r i x . W e h a v e n o t d w e l t in d e t a i l o n t h e t r e a t m e n t of r e s o n a n c e a b s o r p t i o n , b e c a u s e such c a l c u l a t i o n s will usually c o r r e c t t h e i n f i n i t e m e d i u m r e s o n a n c e e s c a p e p r o b a b i l i t i e s w e c o n s i d e r e d earlier f o r f u e l l u m p i n g ( h e t e r o g e n e o u s ) e f f e c t s . W e will r e t u r n to discuss this in s o m e d e t a i l in C h a p t e r 10. M o s t typically t h e M U F T s c h e m e is u s e d to g e n e r a t e either o n e - f a s t - g r o u p or t h r e e - f a s t - g r o u p c o n s t a n t s f o r L W R c a l c u l a t i o n s w i t h t h e g r o u p s t r u c t u r e as s h o w n in F i g u r e 8-5 ( w e h a v e also i n d i c a t e d t h e t h e r m a l g r o u p , w h i c h m u s t b e h a n d l e d b y m e t h o d s to b e d i s c u s s e d in t h e n e x t c h a p t e r ) . A n a l t e r n a t i v e t o t h e M U F T f a s t s p e c t r u m p r o c e d u r e is t h a t utilized in t h e G A M c o d e a n d its o f f s p r i n g , in w h i c h a d i r e c t f i n e - g r o u p s o l u t i o n of the slowing d o w n e q u a t i o n (in t h e Bx or Px a p p r o x i m a t i o n is solved). T y p i c a l l y of t h e o r d e r of 100 l e t h a r g y g r o u p s a r e used, v a r y i n g in w i d t h Aw f r o m 0.1 to 0.25. I n m o s t respects (aside f r o m its direct m u l t i g r o u p t r e a t m e n t of elastic scattering), G A M s p e c t r u m c o d e s a r e v e r y similar t o M U F T c o d e s a n d a r e c o m m o n l y u s e d to g e n e r a t e f a s t group constants for H T G R s .
B. Fast Group Constant Generation for Fast Reactor Calculations T h e g e n e r a t i o n of f e w - g r o u p c o n s t a n t s f o r f a s t r e a c t o r a p p l i c a t i o n s involves m u c h m o r e in t h e w a y of direct or b r u t e f o r c e t e c h n i q u e s t h a n c o m p a r a b l e t h e r m a l r e a c t o r c a l c u l a t i o n s . T o a large d e g r e e this is b e c a u s e of t h e relatively limited e x p e r i e n c e in f a s t r e a c t o r design a n d h e n c e t h e desire f o r as a c c u r a t e a c a l c u l a t i o n a s possible. It also arises, h o w e v e r , b e c a u s e of t h e i n c r e a s e d c o m p l e x i t y of the variety of n u c l e a r p r o c e s s e s t h a t i n f l u e n c e a f a s t r e a c t o r s p e c t r u m — p a r t i c u l a r l y t h e e f f e c t s of r e s o n a n c e a b s o r p t i o n a n d inelastic scattering. I n a t h e r m a l r e a c t o r , the s t r o n g elastic s c a t t e r i n g p r e s e n t in t h e s y s t e m a n d t h e f a c t t h a t m o s t fissions are i n d u c e d b y t h e r m a l n e u t r o n s will t e n d to w a s h o u t m a n y of these details. H o w e v e r in a f a s t r e a c t o r essentially all fissions a r e i n d u c e d b y n e u t r o n s w i t h energies in j u s t t h o s e r a n g e s in w h i c h inelastic s c a t t e r i n g a n d r e s o n a n c e a b s o r p t i o n are m o s t important. A t t h e p r e s e n t time, t h e m o s t g e n e r a l class of c o d e s f o r g e n e r a t i n g f e w - g r o u p c o n s t a n t s f o r f a s t r e a c t o r c a l c u l a t i o n s i n c l u d e s the M C 2 c o d e a n d its v a r i a n t s . 1 8 T h i s c o d e b e g i n s w i t h e v a l u a t e d cross s e c t i o n d a t a ( E N D F / B ) a n d p r o c e e d s to p e r f o r m a s e q u e n c e of g r o u p a v e r a g e s f r o m u l t r a f i n e g r o u p s with A w ~ 1 / 1 2 0 , to f i n e g r o u p s w i t h A w — 1 / 4 , a n d finally to b r o a d g r o u p s w i t h A w — 1 / 2 to 1. I n t h e m o r e a d v a n c e d versions of this c o d e , a d i r e c t m u l t i g r o u p s p e c t r u m c a l c u l a t i o n is p e r f o r m e d at h i g h e r energies, w h e r e a s a c o n t i n u o u s slowing d o w n c a l c u l a t i o n is u s e d to h a n d l e elastic s c a t t e r i n g at lower energies. T h e c o n t i n u o u s slowing d o w n c a l c u l a t i o n u s e s a n i m p r o v e d G o e r t z e l - G r u e l i n g a p p r o x i m a t i o n similar to t h a t w e d i s c u s s e d in t h e p r e v i o u s section. Spatial d e p e n d e n c e is h a n d l e d in either the Px or Bx a p p r o x i m a t i o n w i t h a g a i n a single spatial m o d e c h a r a c t e r i z i n g l e a k a g e w i t h a n effective b u c k l i n g B2. A n u m b e r of r a t h e r s o p h i s t i c a t e d t e c h n i q u e s a r e u s e d to h a n d l e r e s o n a n c e a b s o r p t i o n — p a r t i c u l a r l y f o r t h e o v e r l a p p i n g or u n r e s o l v e d res-
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
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363
o n a n c e r e g i o n s [the isolated r e s o n a n c e s a r e m o s t typically h a n d l e d b y t h e n a r r o w resonance ( N R ) approximation]. A s w e h a v e m e n t i o n e d , c o d e s such as M C 2 a r e m u c h t o o e l a b o r a t e f o r daily use in r e a c t o r design a n d a r e i n t e n d e d m o r e f o r the e v a l u a t i o n of c a l c u l a t i o n a l t e c h n i q u e s a n d c o m p a r i s o n w i t h critical e x p e r i m e n t s . A s m o r e e x p e r i e n c e is a c q u i r e d in t h e design of cores f o r c o m m e r c i a l f a s t r e a c t o r s , less e l a b o r a t e ( a l t h o u g h sufficiently a c c u r a t e ) t e c h n i q u e s will evolve in a m a n n e r similar to the M U F T - G A M s c h e m e s u s e d in t h e r m a l r e a c t o r design. W e h a v e n o t e d t h a t in t h e r m a l r e a c t o r s t h e m u l t i g r o u p c o n s t a n t s are sensitive f u n c t i o n s of c o r e c o m p o s i t i o n a n d c o r e o p e r a t i n g c o n d i t i o n s a n d h e n c e m u s t b e r e c a l c u l a t e d m a n y times in a c o r e design. I n f a s t r e a c t o r s t h e e x p e n s e of a direct c a l c u l a t i o n of t h e m u l t i g r o u p c o n s t a n t s c o u p l e d with a s o m e w h a t lower sensitivity to core e n v i r o n m e n t h a s m o t i v a t e d the d e v e l o p m e n t of sets of u n i v e r s a l microscopic m u l t i g r o u p c o n s t a n t s w h i c h c a n b e u s e d f o r a n y f a s t r e a c t o r c o r e design. T h e m o s t w e l l - k n o w n of such sets of fast g r o u p c o n s t a n t s i n c l u d e the Y O M set, 2 2 t h e H a n s e n - R o a c h or L A S L 16-group set, 2 3 a n d the B o n d a r e n k o or A B N set. 2 4 T h e e f f e c t s of c o r e e n v i r o n m e n t o n t h e m u l t i g r o u p c o n s t a n t s a r e t h e n t a k e n i n t o a c c o u n t b y u s i n g so-called " / - f a c t o r s " 2 5 ( s o m e t i m e s also r e f e r r e d to as B o n d a r e n k o self-shielding f a c t o r s ) w h i c h a r e d e f i n e d in such a w a y t h a t if a e f f is t h e g r o u p c o n s t a n t c h a r a c t e r i z i n g t h e p a r t i c u l a r c o r e of interest, while a a v is t h e universal group constant, then a
eff=/aav
•
H e r e , a a v w o u l d essentially c o r r e s p o n d to t h e i n f i n i t e d i l u t i o n cross section (similar t o the i n f i n i t e d i l u t i o n r e s o n a n c e integral). T h e set of u n i v e r s a l m i c r o s c o p i c g r o u p c o n s t a n t s is typically c a l c u l a t e d u s i n g a v e r y simple i n t r a g r o u p f l u x — f o r e x a m p l e , a 1 / E collision d e n s i t y . T h e / - f a c t o r t h e n t a k e s i n t o a c c o u n t the d e t a i l e d int r a g r o u p f l u x b e h a v i o r d u e to r e s o n a n c e self-shielding, f o r e x a m p l e . T h e / - f a c t o r s a r e u s u a l l y t a b u l a t e d as f u n c t i o n s of c o n c e n t r a t i o n a n d t e m p e r a t u r e a n d i n c l u d e d with t h e cross section sets.
C. Age-Diffusion Theory F o r historical c o m p l e t e n e s s , let u s r e t u r n to c o n s i d e r t h e a g e - d i f f u s i o n e q u a t i o n in a bit m o r e detail. R e c a l l t h a t this e q u a t i o n t a k e s t h e f o r m - D (w)V2(r, u) + 2 a ( w » ( r , u) = - ~
+ S (r, u)
(8-177)
where (8-178) N o w w e k n o w t h a t this e q u a t i o n is o n l y valid f o r n e u t r o n slowing d o w n in m o d e r a t o r s with r e a s o n a b l y large m a s s n u m b e r s (such as graphite). N e v e r t h e l e s s , a g e - d i f f u s i o n t h e o r y h a s b e e n t h o r o u g h l y s t u d i e d in t h e p a s t b e c a u s e o n e c a n o b t a i n a n explicit solution of this e q u a t i o n in c e r t a i n simplified cases. F o r c o n v e n i e n c e , w e will first c o n s i d e r t h e solution of t h e a g e - d i f f u s i o n e q u a t i o n in the a b s e n c e of a source. If w e t h e n s u b s t i t u t e E q . (8-178) i n t o (8-177), we f i n d
D(u)
S.(u)
3q
(8-179)
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THE MULTIGROUP DIFFUSION METHOD
Let us i n t r o d u c e a n integrating f a c t o r t o eliminate the a b s o r p t i o n t e r m b y defining a new d e p e n d e n t variable,
, 2.(«0
(8-180)
T h e n if we substitute this f o r m i n t o Eq. (8-179), we f i n d
$2,(11)
y Q = 3q . h au
(8-181)
T o solve this e q u a t i o n , w e next m a k e a c h a n g e of i n d e p e n d e n t variable b y defining
) i
f
(8-182)
and note 3 3w
3t 3 3w 3T
D
(u) 3 £2 S (W) 3T *
(8-183)
If we n o w use this n e w i n d e p e n d e n t variable in Eq. (8-181), we f i n d t h a t satisfies V 2
M
q(r,r)
(8-184)
3r *
N o t i c e that the f o r m of this e q u a t i o n is very similar to t h a t of the t i m e - d e p e n d e n t d i f f u s i o n e q u a t i o n . F o r this reason, r is k n o w n as the Fermi age in a n a l o g y to the time variable t even t h o u g h its units are c m 2 r a t h e r t h a n sec. By n o t i n g the similarity b e t w e e n the age e q u a t i o n (8-184) a n d t h e timed e p e n d e n t d i f f u s i o n equation, w e c a n a d a p t o u r entire earlier analysis to solve this e q u a t i o n . W e w o n ' t d o this here, however, b e c a u s e age theory is primarily of historical interest a n d is only of limited utility in p r e s e n t - d a y reactor design. W e will, however, generalize o u r results to include a b s o r p t i o n b y n o t i n g
#(r,T) =
/
Sa(«0 du'
= t)/?(W),
(8-185)
w h e r e we h a v e identified the exponential f a c t o r as j u s t the r e s o n a n c e escape probability p(u) to lethargy u. T o illustrate these ideas m o r e clearly, let us consider the e x a m p l e of a n infinite p l a n a r source emitting n e u t r o n s of energy E0 at the center of a n o n a b s o r b i n g slab of width a (as we h a v e seen, a b s o r p t i o n c a n be easily included) [see Figure 8-6]. W e will d e f i n e the lethargy variable with respect to the source energy E0 so that u = 0 c o r r e s p o n d s to the source energy. If we also n o t e
T(
J
E CE0 dE D ( ) E $2,(2?) ' F
(8-186)
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS/388 S08(X)8(T)
t h e n w e c a n w r i t e t h e a p p r o p r i a t e f o r m of t h e a g e e q u a t i o n a s
a
2
q
ax subject t o :
aq
2
v
( a ) initial c o n d i t i o n :
7
= S8 ( x )
(b) boundary conditions:
q( - a / 2 , r ) = 0 =
q(a/2,t)
T o solve this, w e will try a n e i g e n f u n c t i o n e x p a n s i o n ( n o t i n g b y s y m m e t r y t h a t o n l y even eigenfunctions need be used) q(x,T)=-2fn(T)cosBnx,
(8-188) a
n
If w e s u b s t i t u t e this i n t o t h e a g e e q u a t i o n ~ 2 B%n(r)cmBnx= n
^ 1£ cosBnx>
2
(8"189)
n
a n d use o r t h o g o n a l i t y w e f i n d
§
=
(8-190)
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THE MULTIGROUP DIFFUSION METHOD
T h e s o l u t i o n t o this e q u a t i o n is j u s t /«(T)=
e x p — B„t.
(8-191)
H e n c e the g e n e r a l s o l u t i o n to E q . 8-187 is q(x,r)=
*2Anexp(~B*T)cosBnx.
(8-192)
n
N o w u s i n g t h e initial c o n d i t i o n a t r = 0, w e r e q u i r e q(x, 0 ) = 2 A n c o s B n x = S 8 ( x ) .
(8-193)
n
I n t h e u s u a l m a n n e r w e c a n n o w m u l t i p l y b y c o s f i m x , i n t e g r a t e over x, a n d use o r t h o g o n a l i t y t o e v a l u a t e all An t o f i n d t h e s o l u t i o n exp(-5„2T)cos^x.
ci
(8-194)
n
J u s t as t h e f u n d a m e n t a l m o d e (« = 1) will d o m i n a t e t h e s o l u t i o n of t h e timed e p e n d e n t d i f f u s i o n e q u a t i o n , w e f i n d t h a t f o r large v a l u e s of r , t h a t is, f o r n e u t r o n e n e r g i e s f a r b e l o w t h e s o u r c e e n e r g y E0, t h e s o l u t i o n of t h e a g e - d i f f u s i o n e q u a t i o n approaches 9 c
q{x,T)~^txp(-Bg\)cosBgx,
Bg = Bv
(8-195)
T h i s is a very u s e f u l expression, since it essentially d e s c r i b e s t h e s p a t i a l d i s t r i b u t i o n of n e u t r o n s s l o w i n g d o w n t o t h e r m a l energies in a b a r e , h o m o g e n e o u s a s s e m b l y f r o m fission s o u r c e s . Since t h e r e is n o a b s o r p t i o n i n o u r m o d e l e d p r o b l e m , w e c a n i n t e r p r e t t h e d e c r e a s e in q(x,r) w i t h i n c r e a s i n g a g e as b e i n g d u e t o n e u t r o n l e a k a g e w h i l e slowing d o w n . N o t e in p a r t i c u l a r t h a t if all of t h e s o u r c e n e u t r o n s h a d initially b e e n d i s t r i b u t e d in t h e f u n d a m e n t a l m o d e cos Bgx s h a p e ( w h i c h w o u l d c o r r e s p o n d , f o r e x a m p l e , t o t h e a p p e a r a n c e of fission n e u t r o n s in a b a r e , critical s l a b r e a c t o r ) , t h e n t h e loss of n e u t r o n s d u r i n g slowing d o w n w o u l d b e d e s c r i b e d b y t h e e x p ( — B g r ) f a c t o r . T h a t is, w e c a n i d e n t i f y t h e f a s t n o n l e a k a g e p r o b a b i l i t y -P F N L i n t r o d u c e d earlier ( C h a p t e r 3) in t h e six-factor f o r m u l a f o r k a s j u s t P F N L = exp(-JBg2Tth),
(8-196)
w h e r e r t h is t h e a g e c o r r e s p o n d i n g to t h e r m a l energies (usually t a k e n as —1 eV): CEo T
*-
v J? .
D(E) eJE-
(8
"197)
Of c o u r s e this e x p r e s s i o n f o r P F N L is valid o n l y in t h e a g e a p p r o x i m a t i o n a s a p p l i e d to a bare, u n i f o r m reactor. T h e a g e - d i f f u s i o n e q u a t i o n c a n a l s o b e s o l v e d in several o t h e r s t a n d a r d
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
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367
g e o m e t r i e s . F o r e x a m p l e , the slowing d o w n d e n s i t y resulting f r o m a p l a n e source in a n infinite, n o n a b s o r b i n g m e d i u m is
—
=
—
'
(
8
-
1
9
8
)
V47TT
while t h a t r e s u l t i n g f r o m a p o i n t s o u r c e in s u c h a m e d i u m is
Sex
p(^f)
(4
TTT)
T h i s last result c a n b e u s e d to d e m o n s t r a t e t h a t r = i
1 / 6 the a v e r a g e ( c r o w - f l i g h t d i s t a n c e ) 2 f r o m the point where a neutron enters a s y s t e m w i t h E0 to the p o i n t at w h i c h it slows d o w n t o a n age T.
(8-200)
O n e usually c a l c u l a t e s o r m e a s u r e s t h e a g e - t o - t h e r m a l energies r t h or m o r e c o m m o n l y , the a g e - t o - i n d i u m r e s o n a n c e at 1.45 eV, r I N (since this is the m o s t c o m m o n e x p e r i m e n t a l m e a s u r e m e n t ) . I n T a b l e 8-7, w e h a v e listed the a g e T^ f o r several c o m m o n moderators, along with other parameters characterizing thermal neutron b e h a v i o r in these m o d e r a t o r s . TABLE 8-7:
Moderator
Diffusion Parameters for Some Common Moderators
Density (g/cm 3 )
D(cm)
H2O D2O Be Graphite
1.00 1.10 1.85 1.60
0.16 0.87 0.50 0.84
Reactor Type
r(cm)
PWR BWR HTGR LMFBR GCFR
1.8 2.2 12.0 5.0 6.6
2a(cnrl) 0.0197 2.9x10" 5 l.OxlO" 3 2.4X10" 4
40 50 300
L(cm)
r(cm 2 )
2.85 170 21 59
26 131 102 368
M(cm) 5.84 170 23 62
Diameter <M>
Diameter
( cm) 6.6 7.3 21
56 50 40
190 180 63 35 35
-
-
-
-
-
-
If we recall t h a t t h e t h e r m a l n o n l e a k a g e p r o b a b i l i t y is ^ T N L = (1 + L 2 B 2 ) ~ \ a n d n o t e f u r t h e r t h a t f o r m o s t larger p o w e r r e a c t o r s , B 2 is sufficiently small t h a t b o t h L 2 B 2 a n d t B 2 a r e m u c h less t h a n o n e , t h e n w e c a n write the t o t a l n o n l e a k a g e
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THE MULTIGROUP DIFFUSION METHOD
p r o b a b i l i t y as P
NL = ^ T N l / W =
eX +
P( "
r
B
1
(1 + L ^
2
) (l + O
w h e r e w e h a v e d e f i n e d t h e neutron
l )
1 2
{1 + L%
migration M2=L2
1
+ TB?)
area M2
(8-201)
(1+
as
+ t.
(8-202)
T h e migration length M c a n b e i n t e r p r e t e d as 1 / V 6 of t h e r m s d i s t a n c e a n e u t r o n travels f r o m its a p p e a r a n c e as a f a s t fission n e u t r o n to its c a p t u r e as a t h e r m a l n e u t r o n (see P r o b l e m 8-42). W e h a v e listed t h e m i g r a t i o n lengths f o r typical m o d e r a t o r s in T a b l e 8-7. T h e m i g r a t i o n l e n g t h p r o v e s u s e f u l as a scale w i t h w h i c h to c h a r a c t e r i z e t h e size of a r e a c t o r core. F o r e x a m p l e , a m o d e r n L W R is t y p i c a l l y ~ 5 0 M in d i a m e t e r , w h e r e a s a c o m p a r a b l e H T G R c o r e is a b o u t 4 0 M. I n this sense, e v e n t h o u g h t h e p h y s i c a l size of t h e H T G R is t w o to t h r e e t i m e s t h a t of a L W R , its " n e u t r o n i c " size is a c t u a l l y s o m e w h a t smaller. B o t h r e a c t o r t y p e s a r e c e r t a i n l y f a r m o r e loosely c o u p l e d in a n e u t r o n i c sense t h a n f a s t r e a c t o r c o r e s ( — 3 0 L) a n d h e n c e a r e m o r e s u s c e p t i b l e to spatial p o w e r t r a n s i e n t s .
D. Some Additional Comments on Fast Spectrum Calculations A s w e h a v e seen, m o s t s c h e m e s f o r g e n e r a t i n g f a s t g r o u p c o n s t a n t s first p e r f o r m a f i n e m u l t i g r o u p s o l u t i o n of the n e u t r o n slowing d o w n e q u a t i o n s (in the Px or Bx a p p r o x i m a t i o n ) to o b t a i n t h e f a s t n e u t r o n s p e c t r u m , a n d t h e n a v e r a g e m i c r o s c o p i c cross section d a t a o v e r this s p e c t r u m to o b t a i n t h e d e s i r e d f e w - g r o u p c o n s t a n t s . M i c r o s c o p i c cross s e c t i o n d a t a e n t e r this c a l c u l a t i o n in t w o d i f f e r e n t w a y s . First, the m i c r o s c o p i c cross sections a v a i l a b l e f r o m n u c l e a r d a t a files s u c h as E N D F / B m u s t b e c o n v e r t e d i n t o f i n e m u l t i g r o u p c o n s t a n t s in o r d e r to allow t h e c a l c u l a t i o n of t h e fast n e u t r o n s p e c t r u m . T h e n these d a t a m u s t b e a v e r a g e d over this s p e c t r u m to g e n e r a t e the f e w g r o u p c o n s t a n t s themselves. S u c h b a s i c n u c l e a r d a t a are usually given in the f o r m of p o i n t w i s e d a t a a l o n g w i t h v a r i o u s i n t e r p o l a t i o n s p e c i f i c a t i o n s in a d a t a set s u c h as E N D F / B . H e n c e the first step in a f a s t s p e c t r u m c a l c u l a t i o n is to p r e p a r e a l i b r a r y of f a s t f i n e - g r o u p c o n s t a n t s b y a v e r a g i n g these b a s i c cross section d a t a over t h e g r o u p s to b e u s e d in t h e f a s t s p e c t r u m c a l c u l a t i o n . U s u a l l y either a 1/ E or a fission s p e c t r u m is u s e d as a w e i g h t i n g f u n c t i o n . T h e s e f a s t libraries a r e typically s u p p l i e d as p a r t of a given f a s t s p e c t r u m c o d e , since their g e n e r a t i o n f r o m a b a s i c n u c l e a r d a t a set c a n b e r a t h e r expensive. A s c h e m a t i c of this p r o c e d u r e is s h o w n in F i g u r e 8-7. A r a t h e r i m p o r t a n t a s p e c t of s u c h cross section libraries involves t h e p r e p a r a t i o n of r e s o n a n c e cross section d a t a . I n the r e s o l v e d r e s o n a n c e region, the t e c h n i q u e s w e h a v e d i s c u s s e d (e.g., t h e N R or N R I M m e t h o d s ) c a n b e u s e d to c a l c u l a t e the r e s o n a n c e i n t e g r a l s c h a r a c t e r i z i n g the r e s o n a n c e s in e a c h of the f i n e g r o u p s . H o w e v e r in the r e g i o n of u n r e s o l v e d r e s o n a n c e s m o r e e l a b o r a t e s c h e m e s m u s t b e u s e d . A l t h o u g h a r a t h e r c r u d e t r e a t m e n t of t h e u n r e s o l v e d r e s o n a n c e r e g i o n is usually s u f f i c i e n t in t h e r m a l r e a c t o r c a l c u l a t i o n s , in a f a s t r e a c t o r as m u c h as 40% of the r e s o n a n c e a b s o r p t i o n c a n o c c u r in u n r e s o l v e d r e s o n a n c e s , a n d h e n c e a d e t a i l e d c a l c u l a t i o n is r e q u i r e d .
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FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
FIGURE 8-7.
369
Generation of cross section libraries from E N D F / B .
I n m o s t f a s t s p e c t r u m codes, o n e h a s the o p t i o n of s p e c i f y i n g the g r o u p s t r u c t u r e c h a r a c t e r i z i n g the f e w - g r o u p c o n s t a n t s t h a t h e wishes to g e n e r a t e . In general o n e wishes to use t h e smallest n u m b e r of c o a r s e g r o u p s c o n s i s t e n t with t h e desired a c c u r a c y in t h e c o r r e s p o n d i n g m u l t i g r o u p d i f f u s i o n c a l c u l a t i o n . In fact, the n e c e s s a r y g r o u p s t r u c t u r e will usually d e p e n d o n the type of c a l c u l a t i o n of interest. F o r e x a m p l e , t h e c a l c u l a t i o n of integral q u a n t i t i e s such as the c o r e m u l t i p l i c a t i o n f a c t o r c a n u s u a l l y b e a c c o m p l i s h e d with f e w e r g r o u p s t h a n a n e s t i m a t e of the p o w e r - p e a k i n g in a f u e l a s s e m b l y . T h e lower cutoff of t h e f a s t s p e c t r u m is u s u a l l y c h o s e n high e n o u g h so t h a t u p s c a t t e r i n g o u t of the t h e r m a l g r o u p s c a n b e neglected. I n a L W R , this cutoff is typically c h o s e n at a r o u n d 0.6 eV. H o w e v e r in a n H T G R , it m u s t b e c h o s e n s o m e w h a t h i g h e r (typically 2.38 eV) b e c a u s e t h e higher m o d e r a t o r t e m p e r a t u r e in s u c h r e a c t o r s c a u s e s a p p r e c i a b l y larger u p s c a t t e r i n g . T h e r e m a i n i n g g r o u p s t r u c t u r e is usually c h o s e n such t h a t d i f f e r e n t p h y s i c a l p h e n o m e n a t e n d to be isolated within a given g r o u p . F o r e x a m p l e , in a t h r e e - f a s t - g r o u p calculation, the highest energy g r o u p is u s u a l l y c h o s e n to c o n t a i n all of t h e fission s p e c t r u m , the m i d d l e g r o u p is c h a r a c t e r i z e d b y elastic slowing d o w n , a n d m o s t of the r e s o n a n c e a b s o r p t i o n is c o n f i n e d to t h e lowest e n e r g y g r o u p . (See F i g u r e 8-5). A f a s t s p e c t r u m c a l c u l a t i o n n o t o n l y supplies f a s t f e w - g r o u p c o n s t a n t s a n d the f a s t n e u t r o n s p e c t r u m , b u t also will p r o v i d e t h e r a t e at w h i c h n e u t r o n s slow d o w n i n t o t h e t h e r m a l e n e r g y r a n g e . T h i s latter i n f o r m a t i o n is n e e d e d to c o m p l e t e the d e t e r m i n a t i o n of f e w - g r o u p c o n s t a n t s b y g e n e r a t i n g a t h e r m a l n e u t r o n s p e c t r u m a n d t h e n a v e r a g i n g cross section d a t a over this s p e c t r u m to o b t a i n t h e r m a l g r o u p constants.
REFERENCES 1. J . H . F e r z i g e r a n d P. F . Z w e i f e l , The
Reactors, M.I.T. Press, Cambridge (1966).
Theory
of Neutron
Slowing
Down
in
Nuclear
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THE MULTIGROUP DIFFUSION METHOD
2. M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam (1966). 3. Ibid., Chapter VIII. 4. A. Radowsky, (Ed.), Naval Reactors Physics Handbook, Vol. I, Selected Techniques, USAEC Report (1964), Chapter 2. 5. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, University of Chicago Press (1958), p. 290. 6. J. H. Ferziger and P. F. Zweifel, The Theory of Neutron Slowing Down in Nuclear Reactors, M.I.T. Press, Cambridge (1966) pp. 73-78; R. E. Marshak, Rev. Mod. Phys. 19, 185 (1947). 7. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, Princeton, N.J., (1970), pp. 439-443. 8. P. E. Meyer and P. F. Zweifel, Trans. Am. Nucl Soc. 13, 305 (1970). 9. M. E. Rose, W. Miranker, P. Leak, and G. Rabinowitz, A Table of the Integral \j/(x,t), BNL-257 (T-40), (1953); WAPD-SR-506 (1954); K. K. Seth and R. H. Tabony, USAEC Document TID-21304 (1964). 10. L. W. Nordheim, Nucl. Sci. Eng. 12, 457 (1962). 11. Y. Ishiguro, Nucl. Sci. Eng. 49, 526 (1972); R. Goldstein, Nucl. Sci. Eng. 49, 526 (1972). 12. K. T. Spinney, BNL-433 (1960). 13. P. E. Meyer, Approximate Methods for the Calculation of Resonance Absorption, University of Michigan Ph.D. Dissertation, 1974. 14. G. Goertzel and E. Greuling, Nucl. Sci. Eng. 7, 69 (1960). 15. H. Hurwitz and P. F. Zweifel, J. Appl. Phys. 26, 923 (1956). 16. H. Bohl, Jr., E. M. Gelbard, and G. H. Ryan, MUFT-4, A Fast Neutron Spectrum Code, Report WAPD-TM-22 (1957). 17. G. D. Joanou, E. J. Leshan, and J. S. Dudek, GAM-1 a Consistent Px Multigroup Code for the Calculation of Fast Neutron Spectra and Multigroup Constants, General Atomic Report GA-1850 (1961). 18. B. J. Toppel, A. L. Rago, and D. M. O'Shea, MC 2 A Code to Calculate Multigroup Cross Sections, ANL-7318 (1967). 19. J. R. Lamarsh, An Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, Mass. (1966). 20. M. Lineberry and N. Corngold, Nucl. Sci. Eng. 53, 153 (1974). 21. R. Bellman and K. Cooke, Differential-Difference Equations, Academic, New York (1962). 22. S. Yiftah, D. Okrent, and P. A. Moldauer, Fast Reactor Cross Sections, Pergamon, New York (1960). 23. I. I. Bondarenko, et. al., Group Constants for Nuclear Reactor Calculations, Consultants Bureau, New York (1964). 24. Reactor Physics Constants, 2nd Ed., USAEC Document ANL-5800 (1963), pp. 568-579. 25. M. Segev, Nucl. Sci. Eng. 56, 72 (1975).
PROBLEMS 8-1
8-2
8-3
The lower cutoff energy Ec for the fast region is usually chosen such that there is negligible upscattering above this energy. If the moderator is modeled as a free proton gas [cf. Eq. (2-107)], compute the cutoff energy Ec such that less than 0.1% of the thermal region neutrons will be upscattered above Ec. Plot this cutoff energy versus moderator temperature T. Determine the neutron flux <j>(E) resulting from an arbitrary source in an infinite hydrogeneous medium by: (a) solving the infinite medium slowing down equation with this general source term, and then (b) using the solution obtained for a monoenergetic source as a Green's function for the more general problem. (a) Derive an expression for the neutron balance in a nuclear reactor in terms of the
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS scalar flux
/
371
in terms of
(c) Prove that in a purely hydrogeneous medium and in the steady state one may write
8-4 8-5
where S represents the sources. You may use the lethargy variable u rather than the energy E if you prefer. Throughout, assume E^>kT. Transform the infinite medium slowing down equation (assuming only elastic scattering) from the energy variable to the lethargy variable. Repeat the derivation of an expression for the neutron slowing down density in an infinite moderating medium by first calculating P(E'>E), the probability that a neutron with energy E' scatters below an energy E in a collision. Then integrate the product of this quantity with the collision density 2 S ( £")<£(£') over all E > E to find q(E).
8-6
Show that the neutron continuity equation can be written quite generally in terms of the slowing down density q(r,Ej) as ^
8-7
Solve the coupled slowing down equations for an infinite hydrogeneous medium 2a(w)(w)= —
8-8
8-9
8-10 8-11
+ V- J + 2 a ( £ )
H-S(w);
+q(w) = 2s(w)(w) for a monoenergetic source at leth-
argy u — 0. Calculate the mean lethargy loss per collision f, the moderating power, and the moderating ratio for H 2 0 and D 2 0 at energies of 1 eV and 100 eV. Estimate the necessary cross section data from BNL-325. Consider neutrons slowing down in an infinite medium with no absorption. (a) By equating the number of collisions in a lethargy interval du with the number of collisions in a time dt, obtain an expression for the time required for neutrons to slow down to thermal energies. (b) Calculate the time required for fission neutrons to slow down to thermal in graphite and water. Assume 2 f = 0.385 c m - 1 while 2 " 2 ° = 0.528 c m - 1 (constant in energy). Demonstrate that for large mass numbers, ^ +22 / 3 '
Show that the average increase in lethargy f in an elastic (s-wave) collision is one-half the maximum lethargy increase. 8-12 Calculate the multigroup transfer scattering elements for hydrogen under the assumption that 2 s (m) and the intragroup flux <j>(u) can be taken as constant. Assume the groups are equally spaced with a lethargy width Au. 8-13 Demonstrate that the multigroup transfer scattering elements characterizing elastic scattering for directly coupled groups of lethargy width Aug are given by = 2 s g £ g / A u g (Hint: Balance the scattering rates into and out of each group.) 8-14 As an example of a nondirectly coupled multigroup calculation, consider a scheme in which all groups are of the same lethargy width Aw and in which neutrons are able to skip at most one group. Assume that only s-wave elastic scattering need be considered and that the scattering cross section 2 s (w) can be treated as lethargy-independent. Calculate the elements of the scattering transfer cross section for all g' and g, assuming a constant intragroup flux <£(w) = <£>. Also verify that the sum of these transfer cross sections is indeed equal to the group-averaged scattering cross section 2sg.
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THE MULTIGROUP DIFFUSION METHOD
8-15
Determine the neutron flux
8-22
Consider a medium in which the average lethargy gain £ and the scattering cross section 2 S are independent of energy. A common expression for p{E) is
When, if ever, is this expression (a) exact, (b) a good approximation, and (c) a poor approximation? Assume that the resonance region consists of two widely spaced resonances at EX and E 2 , such that EX - E2*>olEX. Within and A E 2 , 2 a / 2 s = o o , and everywhere else 2 a / 2 s = 0. What is the overall resonance escape probability if: (a)
8-23
8-24
E
X
- A E
X
^ > A E
X
,
and
E
2
-
AE2^>AE2;
(b) EX — AEX — ^EX, and E 2 — AE2 = AE2. Compute the resonance escape probabilities for a N n / N v = 1.0 mixture of 2 3 8 U and hydrogen at temperatures 0°K and 500°C for the 6.67 eV, 21 eV, and 208 eV resonances of 2 3 8 U as given in Table 8-2. (a) Develop an accurate equation for the resonance escape probability for an infinite homogeneous mixture of nonabsorbing moderator of mass A and nonscattering
FAST SPECTRUM CALCULATIONS AND FAST GROUP CONSTANTS
8-25
8-26 8-27 8-28 8-29 8-30 8-31
8-32 8-33
8-34
8-35
8-36 8-37 8-38
8-39 8-40
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373
absorber whose absorption cross section is infinite over the range Ex to E2 and zero at all other energies. The resonance region is several orders of magnitude below the source energy, (b) What is the resonance escape probability when E2=otExl Calculate the escape probabilities for r = 3 0 0 ° K , 600°K, and 1200°K for both the 6.67 eV and 208 eV resonances in 2 3 8 U, assuming that 2 ™ / N A = 10 and 1000 b per 238 U atom. Use the N R I M approximation for the lower resonance and the N R approximation for the higher resonance. (Ignore interference scattering terms.) Estimate the probability that a 2 MeV neutron emitted in an infinite medium of water will not be absorbed while slowing down to 1 eV. Demonstrate explicitly that for any capture resonance, dI/dT>0 and hence dp/dT < 0. (Use either the N R or N R I M expression for the effective resonance integral.) Derive the equations (8-132) for the slowing down densities qo(x,u) and in hydrogen. Show that a / — £ by plotting both parameters as functions of a for 0 < a < 1. Explicitly derive the expressions for A0, Xx, and fix in the Goertzel-Greuling approximation. Derive the Bx equations by explicitly calculating the first two angular moments of the transport equation as described in Section 8-III-B and then rearranging these results into the form of Eq. (8-163). Using the group-averaging schemes discussed in Section 8-IV, develop in detail the fine multigroup slowing down equations (8-172) from the set (8-165). Determine the following quantities in terms of the fine-group M U F T representation: (a) source rate, (b) slowing down rate to thermal, (c) leakage rate, and (d) absorption rate. Then calculate the slowing down probability, leakage probability, and absorption probability. Verify that these latter probabilities all add to unity. Consider an infinite slab of thickness L composed of pure atomic hydrogen with a number density N H . The slab contains an external source of the form S(x,u) = x(w)exp(/5x) where x( M ) is the 2 3 5 U fission spectrum and x = 0 denotes the midplane of the slab. Write a computer program that will solve the P x equations for this problem. Your program should tabulate <#>(«), J ( u ) , #o(M)> a n < l ?i( M )i it should also calculate the fast leakage and absorption probabilities and the age to thermal. Run your program for several slab thicknesses and compare the results. Use the M U F T 54 group scheme and a pointwise representation of the fission spectrum as given by Eq. (2-112), and use BNL-325 to obtain a pointwise representation of Determine the slowing down spectrum of fission neutrons slowing down in an infinite medium of water using a MUFT-type fast spectrum code. Repeat this calculation for an infinite medium of D 2 0 . Determine the slowing down density established by a monoenergetic plane source at the origin of an infinite moderating medium as given by age-diffusion theory. Determine the slowing down density resulting from a point source in an infinite moderating medium (using age-diffusion theory). Sources of monoenergetic fast neutrons are distributed in a moderating slab as S(x)= S0cos(7rx/a), where a is the width of the slab. Using age-diffusion theory, determine the slowing density in the slab. Then calculate the average probability that a source neutron leaks out of the slab while slowing down. Age theory for hydrogen fails principally because the collision density may not be slowly varying over the limits of the scattering integral. Why not? Consider a "zero-temperature" reactor, namely, a reactor in which all nuclei are at rest and thus slowing down theory is valid for all lethargies. Assume that the reactor is bare with a geometric buckling and that all fission neutrons are born with zero lethargy. Then using age-diffusion theory, derive the criticality relation for this reactor
374
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THE MULTIGROUP DIFFUSION METHOD in the form: 1=
8-41 8-42 8-43
f00 *(i#)2 f («) / du
p(u)exp[-BgMu)].
Estimate the fast and thermal nonleakage probabilities for each of the reactor types listed in Table 8-7. Use core sizes from Appendix H. Demonstrate that A / 2 = L 2 + T is indeed 1 / 6 the mean-square distance a neutron will travel from its origin in fission to its eventual absorption as a thermal neutron. Consider an effectively infinite medium in which a monoenergetic pulse of fast neutrons is instantaneously injected at time / = 0 uniformly throughout the medium. Using age theory, determine the time-dependent slowing down density q(u,t).
9 Thermal Spectrum Calculations and Thermal Group Constants
W e n o w t u r n o u r a t t e n t i o n to t h e g e n e r a t i o n of m u l t i g r o u p c o n s t a n t s characterizing l o w - e n e r g y n e u t r o n s . O n c e again t h e general a p p r o a c h will b e to d e v e l o p m e t h o d s f o r d e t e r m i n i n g the detailed energy d e p e n d e n c e or s p e c t r u m of such n e u t r o n s in t h o s e situations in which the spatial d e p e n d e n c e c a n b e i g n o r e d (or t r e a t e d in a c r u d e m a n n e r ) a n d t h e n to a v e r a g e m i c r o s c o p i c cross section d a t a over this energy s p e c t r u m . A d e t a i l e d investigation of t h e n e u t r o n energy s p e c t r u m below several eV b e c o m e s q u i t e involved d u e to t h e c o m p l i c a t e d n a t u r e of t h e n e u t r o n scattering process. F o r a t s u c h low energies we c a n n o longer ignore t h e t h e r m a l m o t i o n of t h e nuclei, as we did in the s t u d y of n e u t r o n slowing d o w n . T h e energy of such slow or t h e r m a l n e u t r o n s is c o m p a r a b l e as well to the b i n d i n g energy of the a t o m s in m o l e c u l a r or crystalline materials, a n d h e n c e the n e u t r o n will t e n d to interact with a n a g g r e g a t e of a t o m s r a t h e r t h a n with a single nucleus. T h e s e f e a t u r e s greatly c o m p l i c a t e t h e d e t e r m i n a t i o n of cross sections c h a r a c t e r i z i n g the scattering of t h e r m a l n e u t r o n s , a n d such cross sections will exhibit a r a t h e r involved d e p e n d e n c e o n b o t h n e u t r o n energy a n d scattering angle. M o r e o v e r t h e r m a l n e u t r o n cross sections d e p e n d , in a detailed m a n n e r , o n t h e t e m p e r a t u r e a n d physical state (i.e., solid, liquid, o r gas) of t h e scattering m e d i u m , unlike fast n e u t r o n cross sections, w h i c h d e p e n d p r i m a r i l y o n t h e n u c l e a r species. T h e c o m p l i c a t e d n a t u r e of such cross sections r e n d e r s t h e d e t e r m i n a t i o n of t h e r m a l n e u t r o n spectra r a t h e r involved, a n d o n e usually m u s t resort to direct n u m e r i c a l m e t h o d s . T h e s t u d y of t h e r m a l n e u t r o n b e h a v i o r is c u s t o m a r i l y r e f e r r e d to as neutron thermalization,1-4 Actually, however, the s u b j e c t of n e u t r o n t h e r m a l i z a t i o n c a n b e classified i n t o t w o s e p a r a t e p r o b l e m s : (a) t h e c a l c u l a t i o n of cross sections c h a r a c t e r i z i n g t h e r m a l n e u t r o n scattering in v a r i o u s m a t e r i a l s a n d (b) t h e use of these cross sections in t h e d e t e r m i n a t i o n of t h e energy s p e c t r u m c h a r a c t e r i z i n g 375
376
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THE MULTIGROUP DIFFUSION METHOD
l o w - e n e r g y n e u t r o n s (e.g., f o r u s e in d e t e r m i n i n g t h e r m a l g r o u p c o n s t a n t s f o r f e w - g r o u p d i f f u s i o n c a l c u l a t i o n s ) . T h e s e s u b j e c t s a r e quite i n t e r r e l a t e d since t h o s e f e a t u r e s c h a r a c t e r i s t i c of t h e r m a l n e u t r o n cross sections m a y h a v e a r a t h e r signific a n t i n f l u e n c e o n t h e d e t e r m i n a t i o n of t h e r m a l s p e c t r a . F o r e x a m p l e , t h e f a c t t h a t t h e kinetic e n e r g y of a l o w - e n e r g y n e u t r o n is c o m p a r a b l e t o t h e t h e r m a l e n e r g y of a t o m i c m o t i o n m e a n s t h a t m i c r o s c o p i c t h e r m a l n e u t r o n cross sections a c t u a l l y m u s t b e r e g a r d e d as a v e r a g e s over t h e t h e r m a l d i s t r i b u t i o n of n u c l e a r speeds a n d h e n c e a r e t e m p e r a t u r e - d e p e n d e n t , as w e s a w in C h a p t e r 2. F u r t h e r m o r e , it will b e possible f o r t h e n e u t r o n t o gain e n e r g y — t h a t is, t o u p s c a t t e r — i n a s c a t t e r i n g collision w i t h a m o v i n g n u c l e u s . T h i s will c o m p l i c a t e t h e n u m e r i c a l s o l u t i o n of t h e f i n e - s t r u c t u r e m u l t i g r o u p e q u a t i o n s used to determine thermal spectra. Of similar i m p o r t a n c e is t h e f a c t t h a t t h e e n e r g y of t h e r m a l n e u t r o n s is c o m p a r a b l e to t h e c h e m i c a l b i n d i n g e n e r g y of t h e s c a t t e r i n g nuclei (e.g., in a m o l e c u l e or a crystal lattice). H e n c e t h e n u c l e u s will n o l o n g e r recoil freely, a n d t h u s b i n d i n g will b e c o m e s i g n i f i c a n t in d e t e r m i n i n g t h e e n e r g y a n d a n g l e c h a n g e of a n e u t r o n in a collision. I n f a c t o n e c a n s h o w t h a t t h e s c a t t e r i n g cross s e c t i o n of a b o u n d n u c l e u s is s o m e w h a t larger t h a n t h a t of a f r e e n u c l e u s b y a n a m o u n t t h a t is r o u g h l y a b o u n d = ( l + l / ^ 4 ) 2 a f r e e , w h e r e A is t h e m a s s n u m b e r of t h e s c a t t e r i n g n u c l e u s . 2 F o r e x a m p l e , in h y d r o g e n e o u s m a t e r i a l s f o r w h i c h A — 1, w e w o u l d f i n d t h a t t h e a c t u a l l o w - e n e r g y s c a t t e r i n g cross s e c t i o n is s o m e f o u r t i m e s larger t h a n t h e f r e e - a t o m s c a t t e r i n g cross s e c t i o n a f r e e . Of r e l a t e d s i g n i f i c a n c e a r e inelastic s c a t t e r i n g p r o c e s s e s in w h i c h t h e i n t e r n a l states of t h e s c a t t e r i n g s y s t e m (e.g., m o l e c u l a r v i b r a t i o n a n d r o t a t i o n o r crystal lattice v i b r a t i o n ) a r e excited b y n e u t r o n s c a t t e r i n g collisions. ( S u c h t h e r m a l inelastic s c a t t e r i n g p r o c e s s e s s h o u l d n o t b e c o n f u s e d w i t h n u c l e a r inelastic s c a t t e r i n g in w h i c h t h e n u c l e u s itself is excited i n t o a h i g h e r q u a n t u m state. T h e latter p r o c e s s is of little c o n c e r n f o r t h e low energies c h a r a c t e r i z i n g t h e r m a l n e u t r o n s . ) Inelastic s c a t t e r i n g gives rise t o a c o m p l i c a t e d cross s e c t i o n d e p e n d e n c e o n e n e r g y a n d angle. F o r v e r y l o w energies, t h e n e u t r o n w a v e l e n g t h is c o m p a r a b l e t o t h e i n t e r a t o m i c s p a c i n g of t h e s c a t t e r i n g m a t e r i a l . H e n c e t h e n e u t r o n w a v e f u n c t i o n e x p e r i e n c e s d i f f r a c t i o n e f f e c t s (just a s light is d i f f r a c t e d ) . H o w e v e r , w e s h o u l d a d d t h a t while s u c h c o h e r e n t i n t e r f e r e n c e e f f e c t s c a n b e q u i t e i m p o r t a n t in d e t e r m i n i n g t h e very low e n e r g y b e h a v i o r of n e u t r o n s c a t t e r i n g cross sections, t h e y a r e r a r e l y of i m p o r t a n c e in n u c l e a r r e a c t o r b e h a v i o r . N e e d l e s s t o say, s u c h c o n s i d e r a t i o n s greatly c o m p l i c a t e t h e s u b j e c t of n e u t r o n t h e r m a l i z a t i o n . T h e d e t e r m i n a t i o n of t h e r m a l n e u t r o n cross sections is e x t r e m e l y c o m p l i c a t e d a n d c a n b e r e g a r d e d as essentially a s u b f i e l d of statistical m e c h a n i c s a n d solid a n d l i q u i d - s t a t e physics. Since t h e m e a s u r e m e n t a n d c a l c u l a t i o n of s u c h cross sections c a n infer information a b o u t the microscopic structure a n d dynamics of m a t e r i a l s , t h e r m a l n e u t r o n s c a t t e r i n g (or thermal neutron spectroscopy) has r e c e i v e d a n e x h a u s t i v e t r e a t m e n t i n t h e scientific literature, a n d w e will n o t a t t e m p t t o d u p l i c a t e t h a t t r e a t m e n t in this text. 4 - 6 F o r t u n a t e l y most of t h e c o m p l i c a t e d details of t h e r m a l n e u t r o n c r o s s section b e h a v i o r a r e of s e c o n d a r y c o n c e r n in n u c l e a r r e a c t o r analysis. I n d e e d in m o s t large thermal power reactors the neutron energy s p e c t r u m is s u f f i c i e n t l y well " t h e r m a l i z e d " t h a t r a t h e r c r u d e m o d e l s of t h e n e u t r o n s c a t t e r i n g p r o c e s s a r e s u f f i c i e n t f o r t h e g e n e r a t i o n of t h e r m a l g r o u p c o n s t a n t s . F o r e x a m p l e , i n L W R
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377
c a l c u l a t i o n s , t h e core c a n f r e q u e n t l y b e m o d e l e d as a n ideal gas of p r o t o n s (totally i g n o r i n g b i n d i n g a n d d i f f r a c t i o n effects) s e e d e d with a u n i f o r m l y d i s t r i b u t e d a b s o r b e r c o r r e s p o n d i n g to t h e f u e l . S u c h a m o d e l is p a r t i c u l a r l y u s e f u l f o r g e n e r a t ing t h e t h e r m a l g r o u p c o n s t a n t s to b e u s e d in survey or p a r a m e t r i c studies (e.g., c o r e lifetime studies). Of course, m o r e d e t a i l e d studies of such large, h e t e r o g e n e o u s r e a c t o r cores c h a r a c t e r i z e d b y s t r o n g t e m p e r a t u r e g r a d i e n t s r e q u i r e m o r e e l a b o r a t e m o d e l s of t h e s c a t t e r i n g m a t e r i a l . T h i s is p a r t i c u l a r l y t r u e f o r t h e analysis of r e a c t o r s utilizing solid m o d e r a t o r s , such as the H T G R , w h i c h uses g r a p h i t e as a m o d e r a t o r . H o w e v e r , in all cases t h e m o d e l s of t h e p h y s i c a l a n d c h e m i c a l s t r u c t u r e of the s c a t t e r i n g m a t e r i a l u s e d to c a l c u l a t e t h e r m a l n e u t r o n cross sections f o r n u c l e a r r e a c t o r analysis a r e very c r u d e i n d e e d w h e n c o m p a r e d to t h e sophisticated t h e o r i e s a n d m e a s u r e m e n t s t h a t exist f o r the i n t e r p r e t a t i o n of t h e r m a l n e u t r o n scattering data. H e n c e o u r c o n c e r n in this c h a p t e r will b e to first illustrate s o m e of the simple i d e a s i n v o l v e d in n e u t r o n t h e r m a l i z a t i o n a n d t h e n to d e v e l o p t h o s e tools f o r m i n g t h e basis f o r t h e r m a l s p e c t r a c a l c u l a t i o n s a n d u s e d to g e n e r a t e t h e r m a l g r o u p constants.
I. GENERAL FEATURES OF THERMAL NEUTRON SPECTRA A. Thermal Equilibrium A l t h o u g h the d e t a i l e d f o r m of t h e cross section c h a r a c t e r i z i n g t h e r m a l n e u t r o n s c a t t e r i n g is e x t r e m e l y c o m p l i c a t e d , d e p e n d i n g as it d o e s o n the t e m p e r a t u r e a n d p h y s i c a l s t r u c t u r e of t h e s c a t t e r i n g m a t e r i a l , t h e r e a r e several simple g e n e r a l f e a t u r e s of such cross section b e h a v i o r with i m p o r t a n t i m p l i c a t i o n s f o r t h e r m a l n e u t r o n spectra. S u p p o s e w e o n c e a g a i n c o n s i d e r a s i t u a t i o n in w h i c h w e i m a g i n e s o u r c e s of n e u t r o n s d i s t r i b u t e d u n i f o r m l y t h r o u g h o u t a n infinite m e d i u m s u c h t h a t spatial a n d t i m e d e p e n d e n c e c a n b e i g n o r e d . T h e n w e recall t h a t the n e u t r o n c o n t i n u i t y e q u a t i o n (4-79) simplies to C oo [ S a ( £ ) + 2 J ( £ ) ] < # > ( £ ) = f dE'Zs(E'^E)
+ S(E).
(9-1)
N o t i c e t h a t w e h a v e a l l o w e d the r a n g e of i n t e g r a t i o n to e x t e n d to E = 0, since we n o w wish to a n a l y z e the low e n e r g y b e h a v i o r of the n e u t r o n flux. T h e first u s e f u l p r o p e r t y of t h e r m a l n e u t r o n cross s e c t i o n s — a n d i n d e e d , all d i f f e r e n t i a l s c a t t e r i n g cross sections—is essentially j u s t a d e f i n i t i o n : r oo
(9-2)
J(\
H o w e v e r e v e n this f a m i l i a r p r o p e r t y tells us s o m e t h i n g interesting. F o r if we i n t e g r a t e E q . (9-1) over all e n e r g y a n d use t h e relation E q . (9-2), w e f i n d the balance relation /•CO
I
f 00
dE 2 a ( £ )
-A)
dES(E).
(9-3)
T h a t is, f o r a s t e a d y - s t a t e f l u x (£) to b e possible in a n i n f i n i t e m e d i u m , w e
378
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THE MULTIGROUP DIFFUSION METHOD
r e q u i r e t h a t t h e r a t e a t w h i c h s o u r c e n e u t r o n s a p p e a r b e j u s t e q u a l to t h e r a t e a t w h i c h n e u t r o n s a r e a b s o r b e d (since t h e r e is n o leakage). If w e r e c o g n i z e t h a t in t h e r m a l i z a t i o n p r o b l e m s all s o u r c e n e u t r o n s will a p p e a r as fission n e u t r o n s slowing d o w n i n t o t h e t h e r m a l e n e r g y r a n g e , a n d t h e a b s e n c e of u p s c a t t e r i n g a b o v e the t h e r m a l c u t o f f e n e r g y Ec implies t h a t E q . (9-1) h o l d s w i t h t h e u p p e r limit t r u n c a t e d at Ec, t h e n o n e c a n rewrite E q . (9-3) as (see P r o b l e m s 9-2 a n d 9-3)
T o t a l r a t e of a b s o r p t i o n _ f ^ ^ ' S (E}
(E
r a t e at w h i c h s ow * c ) ~ d o w n into thermal , range/cm3. neutrons
(g 4\ ^ " '
T h e d i f f e r e n t i a l s c a t t e r i n g cross section o r s c a t t e r i n g k e r n e l 2 S ( £ ' — > £ ) c a n b e s h o w n to possess a n o t h e r v e r y i m p o r t a n t p r o p e r t y : i
)
M
(
E
'
)
= vZs(E-^E')M
(E),
(9-5)
w h e r e M(E) is t h e M a x w e l l - B o l t z m a n n d i s t r i b u t i o n f u n c t i o n c h a r a c t e r i z i n g the energies of t h e p a r t i c l e s of a n i d e a l gas at t e m p e r a t u r e T:
E q . (9-5) is k n o w n as t h e principle of detailed balance, a n d it m u s t b e satisfied b y a n y n e u t r o n cross s e c t i o n c h a r a c t e r i z i n g n e u t r o n s c a t t e r i n g f r o m a s y s t e m of n u c l e i in t h e r m a l e q u i l i b r i u m at a t e m p e r a t u r e T ( r e g a r d l e s s of their s t r u c t u r e or d e t a i l e d d y n a m i c s ) . T h i s p r o p e r t y is essentially a c o n s e q u e n c e of t h e laws of statistical m e c h a n i c s c h a r a c t e r i z i n g t h e s c a t t e r i n g m a t e r i a l . 7 It implies a n e x t r e m e l y i m p o r t a n t c o n s e q u e n c e f o r t h e n e u t r o n e n e r g y s p e c t r u m in a r e a c t o r , as w e will n o w demonstrate. S u p p o s e w e c o n s i d e r t h e special case in w h i c h w e set t h e a b s o r p t i o n a n d s o u r c e t e r m s in o u r i n f i n i t e m e d i u m e q u a t i o n , E q . (9-1), to z e r o : c 00
Z s ( E ) < t > ( E ) = f dE'Zs(E'^E)
(9-7)
T h e n w e c l a i m t h a t the s o l u t i o n to this e q u a t i o n , r e g a r d l e s s of the d e t a i l e d f o r m of t h e s c a t t e r i n g cross section, m u s t b e j u s t t h e n e u t r o n f l u x c h a r a c t e r i z i n g n e u t r o n s in t h e r m a l e q u i l i b r i u m a t t h e s a m e t e m p e r a t u r e T as the s c a t t e r i n g m e d i u m : p ( - - | ) ,
(9-8)
w h e r e n0 is t h e n e u t r o n n u m b e r d e n s i t y in t h e m e d i u m . T h i s result is a c o n s e q u e n c e of the p r i n c i p a l of d e t a i l e d b a l a n c e , f o r w e c a n use E q . (9-5) to d e m o n s t r a t e t h a t M is i n d e e d a s o l u t i o n of E q . (9-7): /• 00 2S(£)«?>M(£) = / J o
dE'^{E'^E)v'n0M{E')
= ( X d E ' ^ { E - ^ E ' ) v n 0 M { E ) = ^s(E)<$>u{E). Jr\
(9-9)
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
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379
H e n c e t h e p r i n c i p l e of d e t a i l e d b a l a n c e e n s u r e s t h a t t h e e q u i l i b r i u m s p e c t r u m of t h e n e u t r o n s (in t h e a b s e n c e of a b s o r p t i o n , sources, leakage, etc.) will b e a " M a x w e l l i a n " c h a r a c t e r i z e d b y t h e m o d e r a t o r t e m p e r a t u r e T — t h a t is, t h a t the n e u t r o n s will e v e n t u a l l y c o m e i n t o t h e r m a l e q u i l i b r i u m w i t h t h e m o d e r a t o r nuclei. I n this sense, t h e n e u t r o n s b e h a v e as a very d i l u t e gas t h a t will g r a d u a l l y c o m e i n t o t h e r m a l e q u i l i b r i u m — n a m e l y , " t h e r m a l i z e " — w i t h the s y s t e m t h r o u g h w h i c h it is d i f f u s i n g . F r o m t h e p r o p e r t i e s of the M a x w e l l - B o l t z m a n n d i s t r i b u t i o n f u n c t i o n w e k n o w t h a t t h e m o s t p r o b a b l e n e u t r o n e n e r g y a n d s p e e d in s u c h a s i t u a t i o n are t h e n given in t e r m s of t h e s y s t e m t e m p e r a t u r e T a s : Most probable energy = £
T
M o s t p r o b a b l e s p e e d = vT=
= k T = 8.62X 1 0 " 5 r ( e V )
y
= 1.28X 1 0 4 V r ( c m / s e c ) .
(9-10)
F o r e x a m p l e , at T = 2 9 3 ° K , £ T = .025 eV while vr = 22x 10 5 c m / s e c . ( A l t h o u g h at typical r e a c t o r c o r e o p e r a t i n g t e m p e r a t u r e s , T= 5 9 0 ° K c o r r e s p o n d s to ET= .051 eV a n d t>T = 3.1 x 10 5 c m / s e c . ) T h e d e t a i l e d b a l a n c e p r o p e r t y o n l y g u a r a n t e e s t h a t t h e n e u t r o n s will b e in t h e r m a l e q u i l i b r i u m w i t h t h e m e d i u m if t h e r e a r e n o m e c h a n i s m s p r e s e n t t h a t t e n d to i n t r o d u c e n o n e q u i l i b r i u m b e h a v i o r . Of course, in a n u c l e a r r e a c t o r c o r e even f o r a very t h e r m a l r e a c t o r t h e n e u t r o n d i s t r i b u t i o n will n e v e r b e precisely in t h e r m a l e q u i l i b r i u m b e c a u s e of o n e of t h e f o l l o w i n g e f f e c t s : (a) p r e s e n c e of a b s o r p t i o n , (b) p r e s e n c e of sources, (c) l e a k a g e of n e u t r o n s , or (d) time d e p e n d e n c e . T h e s e e f f e c t s all act t o p e r t u r b the n e u t r o n d i s t r i b u t i o n a w a y f r o m t h e r m a l e q u i l i b r i u m . I n f a c t , n e u t r o n t h e r m a l i z a t i o n c a n physically b e r e g a r d e d as a c o m p e t i t i o n p r o c e s s b e t w e e n t h e m o d e r a t o r a t o m s a t t e m p t i n g to " t h e r m a l i z e " the neutron distribution into thermal equilibrium with them, a n d those effects tending t o p e r t u r b or distort the t h e r m a l n e u t r o n s p e c t r u m f r o m this e q u i l i b r i u m distribution.
B. Nonequilibrium Thermal Spectra T h e p r e s e n c e of a b s o r p t i o n or l e a k a g e or a slowing d o w n s o u r c e c a n a c t to d i s t o r t the t h e r m a l s p e c t r u m , t h a t is, t h e s o l u t i o n <£(.£) to E q . (9-1), f r o m a M a x w e l l i a n f l u x <£ M (£). W e will c o n s i d e r later p r o c e d u r e s f o r c a l c u l a t i n g t h e r m a l s p e c t r a to a n y desired a c c u r a c y b y solving E q . (9-1) directly in Section 9-III. H o w e v e r it is u s e f u l to give a s o m e w h a t m o r e qualitative discussion of n o n e q u i l i b r i u m n e u t r o n s p e c t r a at this p o i n t to lay the g r o u n d w o r k f o r this s u b s e q u e n t discussion. C o n s i d e r first t h e a d d i t i o n of a n a b s o r p t i o n t e r m t o E q . (9-7). O n e t h e n f i n d s t h a t t h e t h e r m a l n e u t r o n s p e c t r u m will b e slightly s h i f t e d to h i g h e r energies, a l m o s t as if its t e m p e r a t u r e w e r e effectively i n c r e a s e d b y t h e a d d i t i o n of a b s o r p t i o n (see F i g u r e 9-1). T h i s " a b s o r p t i o n h e a t i n g " of t h e s p e c t r u m c a n b e easily u n d e r s t o o d , h o w e v e r , w h e n it is recalled t h a t m o s t a b s o r p t i o n cross sections b e h a v e essentially as 1 / E x / 1 f o r l o w energies. H e n c e t h e l o w e r e n e r g y n e u t r o n s in t h e s p e c t r u m will t e n d to b e p r e f e r e n t i a l l y d e p l e t e d b y a b s o r p t i o n , a n d this l e a d s to a n e f f e c t i v e shift (at least in t h e n o r m a l i z e d s p e c t r u m ) to h i g h e r energies. E x a c t l y t h e o p p o s i t e e f f e c t o c c u r s w h e n o n e a c c o u n t s f o r n e u t r o n leakage. T h i s c a n b e m o d e l e d b y a d d i n g a t e r m of t h e f o r m D ( E ) B } to t h e total cross section in
Absorption
Maxwellian flux
(b)
Leakage
A vD(E)
(c) Slowing d o w n source
(d) T h e r m a l
resonance
A
E
0.001
0.01
0.1
1
£(eV)
FIGURE 9-1.
380
Effects of nonequilibrium perturbations on thermal neutron flux spectra.
10
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
/
381
E q . (9-1). N o t e t h a t this l e a k a g e t e r m a p p e a r s as a n effective a b s o r p t i o n . N o w since t?Z)(£') = t ; [ 3 2 t r ( £ , ) ] ~ 1 t e n d s to i n c r e a s e w i t h i n c r e a s i n g energy, o n e f i n d s t h a t h i g h e r e n e r g y n e u t r o n s will t e n d t o leak m o r e r a p i d l y f r o m t h e system. T h i s c a u s e s t h e e q u i l i b r i u m s p e c t r u m to s h i f t to lower e n e r g i e s — t h a t is, to a " d i f f u s i o n cooling" effect. O b v i o u s l y t h e p r e s e n c e of a s o u r c e t e r m in E q . (9-1) will also p e r t u r b t h e s p e c t r u m . F o r e x a m p l e , o n e is m o s t c o m m o n l y c o n c e r n e d w i t h a s o u r c e c o r r e s p o n d i n g t o n e u t r o n slowing d o w n f r o m h i g h e r energies. I n this case o n e e x p e c t s t h e n e u t r o n s p e c t r u m to b e h a v e a s 1 / E f o r energies a b o v e several eV. Several o t h e r e f f e c t s a c t to p e r t u r b the t h e r m a l s p e c t r u m a w a y f r o m e q u i l i b r i u m . F o r e x a m p l e , c e r t a i n n u c l i d e s s u c h as 1 3 5 Xe, 2 3 5 U , or 2 3 9 P u , h a v e low-lying reso n a n c e s in t h e t h e r m a l r a n g e . T h e n e u t r o n f l u x will b e d e p r e s s e d in the vicinity of t h e s e r e s o n a n c e s , as s h o w n in F i g u r e 9-1. S u c h low-lying r e s o n a n c e e f f e c t s are p a r t i c u l a r l y i m p o r t a n t t o w a r d t h e e n d of c o r e life in r e a c t o r s f u e l e d w i t h lowe n r i c h m e n t u r a n i u m , since t h e a p p r e c i a b l e i n v e n t o r y of t h e p l u t o n i u m p r o d u c e d c a n c a u s e a s t r o n g d i s t o r t i o n of t h e t h e r m a l s p e c t r u m .
C. Effective Neutron Temperature Models W e h a v e seen t h a t t h e a d d i t i o n of a b s o r p t i o n (or l e a k a g e ) to a s y s t e m t e n d s to s h i f t t h e e q u i l i b r i u m n e u t r o n s p e c t r u m to h i g h e r (or lower) energies, m u c h as w o u l d o c c u r if o n e w e r e to c h a n g e t h e t e m p e r a t u r e c h a r a c t e r i z i n g the M a x w e l l B o l t z m a n n d i s t r i b u t i o n . F o r t h a t r e a s o n , early studies of n e u t r o n t h e r m a l i z a t i o n r e p r e s e n t e d t h e n e u t r o n s p e c t r u m b y a M a x w e l l i a n w i t h a n effective t e m p e r a t u r e Tn=£T:*>9
I exp
E
\
/8£rn\1/2
Since a b s o r p t i o n w a s t h e d o m i n a n t loss m e c h a n i s m in large t h e r m a l r e a c t o r cores, t h e effective neutron temperature Tn w a s m o d e l e d as d e p e n d i n g o n l y o n the relative a b s o r p t i o n in the c o r e
(9-12)
w h e r e T w a s the m o d e r a t o r t e m p e r a t u r e a n d A w a s a d i m e n s i o n l e s s c o e f f i c i e n t t h a t h a d to b e d e t e r m i n e d e m p i r i c a l l y b y p e r f o r m i n g e x p e r i m e n t a l s p e c t r u m m e a s u r e m e n t s o n t h e a s s e m b l y of interest (typically, ^ 4 ~ 1 . 2 - 1 . 8 ) . It s h o u l d b e n o t e d t h a t the p a r a m e t e r V t h a t a p p e a r s in this expression is j u s t t h e inverse of the m o d e r a t i n g r a t i o t h a t m e a s u r e s the e f f e c t i v e c o m p e t i t i o n b e t w e e n a b s o r p t i o n a n d s c a t t e r i n g in determining the thermal spectrum. T h e simple e f f e c t i v e t e m p e r a t u r e m o d e l given b y E q . (9-11) is i n c o r r e c t at high energies since it fails to yield t h e " 1 / £ " b e h a v i o r c a u s e d b y a slowing d o w n source. T h i s c a n b e easily c o r r e c t e d b y a d d i n g to the m o d e l a slowing d o w n t e r m A
(E/kTn) E
(9-13)
w h e r e A ( E / k T ) is a " j o i n i n g f u n c t i o n " c h a r a c t e r i z i n g the t r a n s i t i o n b e t w e e n the
382
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THE MULTIGROUP DIFFUSION METHOD
M a x w e l l i a n a n d t h e s l o w i n g d o w n s p e c t r u m (see F i g u r e 9-2) while A is a n o r m a l i z a t i o n f a c t o r given b y
T h e e f f e c t i v e n e u t r o n t e m p e r a t u r e m o d e l p l a y e d a n i m p o r t a n t role i n early r e a c t o r a n a l y s i s a n d still is u s e f u l in o b t a i n i n g a q u a l i t a t i v e u n d e r s t a n d i n g of t h e r m a l r e a c t o r s p e c t r a . It is i n t e r e s t i n g t o n o t e t h a t if i n d e e d t h e n e u t r o n f l u x c o u l d b e c h a r a c t e r i z e d b y a M a x w e l l i a n <j>M(E, Tn) a t a t e m p e r a t u r e Tni t h e n t h e total flux characterizing the thermal group would be just J r £
r oo O / dE
c
dE
o
(9-15)
w h e r e w e h a v e i d e n t i f i e d t h e m o s t p r o b a b l e n e u t r o n s p e e d vT~ \ f l k T J m . H e n c e w e f i n d t h a t <J>T d e p e n d s o n t h e e f f e c t i v e n e u t r o n t e m p e r a t u r e Tn. A n i n t e r e s t i n g a p p l i c a t i o n of this result is t o t h e c a l c u l a t i o n of t h e r e a c t i o n r a t e s c h a r a c t e r i z i n g t h e r m a l n e u t r o n s . F i r s t recall t h a t t h e e f f e c t i v e cross s e c t i o n s w h i c h m u s t b e u s e d w o u l d b e a v e r a g e d over a d i s t r i b u t i o n of n u c l e a r s p e e d s a t s o m e t e m p e r a t u r e T. T h a t is, t h e e f f e c t i v e r e a c t i o n r a t e d e p e n d s b o t h o n t h e c o r e t e m p e r a t u r e T a n d t h e n e u t r o n t e m p e r a t u r e Tn:
(EcdE2(E,T)
F=
(9-16)
N o w m o s t t h e r m a l a b s o r p t i o n cross sections b e h a v e as 1 / v in t h e t h e r m a l n e u t r o n r a n g e . I n C h a p t e r 2, w e f o u n d t h a t w h e n a v e r a g e d over t h e d i s t r i b u t i o n of n u c l e a r s p e e d s , t h e e f f e c t i v e a b s o r p t i o n cross s e c t i o n c h a r a c t e r i z i n g a 1 / v a b s o r b e r w a s i n d e p e n d e n t of t e m p e r a t u r e a n d d e p e n d e d o n n e u t r o n s p e e d v a s
v
FIGURE 9-2.
'
=
2a(£0H v
The joining function, A ( £ / k T ) .
v
(9-17)
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
/
383
w h e r e 2 a = UqE^^q) is j u s t t h e a b s o r p t i o n f r e q u e n c y e v a l u a t e d a t a n y r e f e r e n c e s p e e d v0. H e n c e the a b s o r p t i o n r a t e c h a r a c t e r i z i n g t h e r m a l n e u t r o n s is j u s t
= f ^ d E v ^ E , T)n0M J o
(E, Tn) =
Za(E0)v0n0
= Wo)*o>
(9-18)
w h e r e
T
(9-19)
W e c a n also express the t h e r m a l g r o u p c o n s t a n t c h a r a c t e r i z i n g a \ / v a b s o r b e r in a M a x w e l l i a n f l u x in t e r m s of S a ( £ ' 0 ) as
2
a t h
=^(^)
1 / 2
S
a
(£
0
).
(9-20)
N o t e t h a t o n e m u s t t a k e c a r e to distinguish b e t w e e n the 2200 m / s e c cross section S a ( £ ' 0 ) a n d t h e t h e r m a l l y - a v e r a g e d cross section at a m b i e n t t e m p e r a t u r e , 2 a h = ( 7 7 / 4 ) 1 / 2 E a ( £ 0 ) , w h i c h d i f f e r b y s o m e 12%. O n e c a n m o d i f y these results to c a l c u l a t e t h e M a x w e l l i a n - a v e r a g e d t h e r m a l g r o u p c o n s t a n t s of n o n - l / u a b s o r b e r s in this f o r m b y i n s e r t i n g a n o n - 1 / v f a c t o r ga(rn)
to
write
/— ( T S a ^ ^ ^ r j ^ j
2a(£0).
(9-21)
S u c h n o n - 1 / v f a c t o r s h a v e b e e n t a b u l a t e d f o r a n u m b e r of isotopes of interest. 1 0 H o w e v e r , since we will m a k e little use of the effective t e m p e r a t u r e m o d e l in our s t u d y of t h e r m a l n e u t r o n spectra, w e will a v o i d a f u r t h e r discussion of such n o n - 1 / v corrections. T h e r e a r e several i m p o r t a n t r e a s o n s f o r a b a n d o n i n g the effective n e u t r o n t e m p e r a t u r e m o d e l in f a v o r of a m o r e s o p h i s t i c a t e d t r e a t m e n t of t h e r m a l n e u t r o n s p e c t r a . First, this r a t h e r simple m o d e l r e q u i r e s a g o o d d e a l of empirical guess w o r k in d e t e r m i n i n g t h e a b s o r p t i o n h e a t i n g c o e f f i c i e n t A a n d t h e transition f u n c t i o n &(E/kT). F u r t h e r m o r e o n e f i n d s in p r a c t i c e t h a t this m o d e l is i n a d e q u a t e w h e n t h e r e is either s t r o n g a b s o r p t i o n (i.e., T > 0 . 1 ) or w h e n the s p e c t r u m is a p p r e c i a b l y i n f l u e n c e d b y r e s o n a n c e a b s o r p t i o n in t h e t h e r m a l r a n g e . Since T for a typical L W R r a n g e s f r o m a v a l u e of 0.2 f o r b e g i n n i n g of c o r e life (with zero void f r a c t i o n c o o l a n t ) to as high as 0.6 at t h e e n d of c o r e life (with high-quality s t e a m
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THE MULTIGROUP DIFFUSION METHOD
c o n d i t i o n s ) , it is a p p a r e n t t h a t t h e e f f e c t i v e t e m p e r a t u r e m o d e l h a s little r e l e v a n c e t o this class of r e a c t o r s . I n a d d i t i o n , t h e p r e s e n c e of r e s o n a n c e a b s o r b e r s in t h e t h e r m a l r a n g e (e.g., 2 3 9 P u ) t h a t c a n b u i l d u p t o a p p r e c i a b l e levels n e a r t h e e n d of c o r e life also a c t t o i n v a l i d a t e this m o d e l .
D. An Overview of Techniques for Calculating Thermal Spectra M o s t s c h e m e s f o r c a l c u l a t i n g t h e r m a l n e u t r o n s p e c t r a directly solve infinite m e d i u m spectrum equation for the thermal energy range 2t(£W£)=
J
(EcdE'29(E'^>E)
+ S(E),
0 <E<EC.
the
(9-22)
H e r e S(E) is u s u a l l y t a k e n as a slowing d o w n source, a n d t h e cutoff e n e r g y Ec is t a k e n s u f f i c i e n t l y h i g h t h a t n o a p p r e c i a b l e u p s c a t t e r i n g o u t of t h e t h e r m a l r a n g e occurs. If t h e cross s e c t i o n s a r e k n o w n , t h e n E q . (9-22) c a n b e w r i t t e n in a f i n e - s t r u c t u r e m u l t i g r o u p f o r m a n d solved directly. T h e r e a r e several difficulties w i t h s u c h a b r u t e f o r c e s o l u t i o n , h o w e v e r . First, t h e p o i n t w i s e r e p r e s e n t a t i o n of t h e d e t a i l e d cross section d a t a i n a n u c l e a r d a t a file s u c h a s E N D F / B is q u i t e e x t e n s i v e a n d c a n r e q u i r e large a m o u n t s of d a t a h a n d l i n g ( p a r t i c u l a r l y f o r t h e s c a t t e r i n g kernels). I n f a c t , it is s o m e t i m e s p r e f e r a b l e t o directly g e n e r a t e t h e t h e r m a l c r o s s s e c t i o n d a t a f r o m s u i t a b l e t h e o r e t i c a l m o d e l s of t h e s c a t t e r i n g m a t e r i a l f o r t h e p a r t i c u l a r p r o b l e m of interest. F u r t h e r m o r e t h e f a c t t h a t a p p r e c i a b l e u p s c a t t e r i n g o c c u r s in t h e t h e r m a l r a n g e m e a n s t h a t t h e m u l t i g r o u p r e p r e s e n t a t i o n of E q . (9-22) involves full s c a t t e r i n g m a t r i c e s , r a t h e r t h a n t h e l o w e r t r i a n g u l a r s t r u c t u r e s t h a t a p p e a r w h e n t r e a t i n g n e u t r o n s l o w i n g d o w n . T h i s implies t h a t t h e m u l t i g r o u p t h e r m a l i z a t i o n e q u a t i o n s m u s t b e solved s i m u l t a n e o u s l y r a t h e r t h a n successively f r o m u p p e r to lower energy groups, as were the multigroup slowing d o w n equations. Such a s o l u t i o n u s u a l l y r e q u i r e s iterative t e c h n i q u e s (similar t o t h o s e u s e d in m u l t i d i m e n sional diffusion calculations). H o w e v e r in c e r t a i n i n s t a n c e s o n e c a n m o d e l t h e s c a t t e r i n g m a t e r i a l i n s u c h a w a y so as t o r e d u c e t h e i n t e g r a l e q u a t i o n (9-22) t o a s i m p l e s e c o n d - o r d e r d i f f e r e n tial e q u a t i o n , w h i c h is c o n s i d e r a b l y easier t o solve. O n e very p o p u l a r s u c h m o d e l r e p r e s e n t s t h e m o d e r a t o r a s a n i d e a l gas of u n b o u n d h y d r o g e n a t o m s (or p r o t o n s ) w h i c h is in t h e r m a l e q u i l i b r i u m at a given t e m p e r a t u r e T. B e c a u s e t h e p r o t o n gas m o d e l is e x t r e m e l y u s e f u l f o r m o d e l i n g t h e t h e r m a l n e u t r o n b e h a v i o r in L W R s , w e will c o n s i d e r its d e v e l o p m e n t i n s o m e detail in this c h a p t e r .
II. APPROXIMATE MODELS OF NEUTRON THERMALIZATION A. The Proton Gas (Wigner-Wilkins) Model11
12
P e r h a p s t h e simplest d e s c r i p t i o n of n e u t r o n t h e r m a l i z a t i o n y i e l d i n g results of s u f f i c i e n t a c c u r a c y f o r u s e in r e a c t o r d e s i g n is t h a t w h i c h m o d e l s t h e r e a c t o r c o r e a s a p r o t o n gas ( m a s s n u m b e r A = 1) i n t h e r m a l e q u i l i b r i u m at a t e m p e r a t u r e T. S u c h a m o d e l o b v i o u s l y i g n o r e s b o t h c h e m i c a l b i n d i n g a n d d i f f r a c t i o n . H o w e v e r it does describe upscattering a n d has been proven remarkably successful for generating t h e r m a l s p e c t r a u s e f u l in L W R design. T o b e m o r e specific, w e will c o n s i d e r t h e s o l u t i o n of t h e i n f i n i t e m e d i u m s p e c t r u m e q u a t i o n (9-22) w i t h t h e s o u r c e t e r m o m i t t e d . W e will i n c l u d e t h e
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
/
385
p r e s e n c e of a s o u r c e b y d e m a n d i n g t h a t
erf
V^ '
SH E'
E'>
A
E (9-23)
{E'-E) exp
E'<E
kT
I n E q . (9-23) w e h a v e d e f i n e d t h e free atom scattering cross section of h y d r o g e n , = W e c a n c a l c u l a t e t h e m a c r o s c o p i c s c a t t e r i n g cross s e c t i o n b y i n t e g r a t i n g 1L (E'—>E) o v e r all f i n a l n e u t r o n energies t o f i n d 2.(£)= f
C C
d E ' 2
s
( E - ^ E ' ) =
(9-24)
(ZZ/v)V(E),
where we have defined
=
^
+
{
^
f
•
(9-25)
W e n o t e d t h e f o r m of t h e d i f f e r e n t i a l s c a t t e r i n g cross section f o r a p r o t o n g a s in F i g u r e 2-15. I n p a r t i c u l a r , w e f o u n d t h a t it a p p r o a c h e d t h e u s u a l slowing d o w n form E'>E 0,
E'<E
(9-26)
f o r large E'^>kT. It s h o u l d also b e n o t e d f r o m E q . (9-23) t h a t 2 s ( £ ' - > £ ) h a s a d i s c o n t i n u o u s first derivative a t E=E\ I n f a c t , t h e s t r u c t u r e of t h e scattering k e r n e l H ^ E ' ^ E ) is v e r y similar t o t h a t w e e n c o u n t e r e d f o r the G r e e n ' s f u n c t i o n s of s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n s (cf. Section 5-II). T h a t is, E q . (9-22) is of the form a (x)4>(x) = fbdx' Ja w h e r e G(x,x')
(9'27)
G (x, *>(*')
is t h e G r e e n ' s f u n c t i o n of a d i f f e r e n t i a l o p e r a t o r
LG (x9x')
= a{x) ^ 2 dx
+ b(x)^~
+ c{x)G dx
(*,*')
- 8 (x - x').
(9-28)
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THE MULTIGROUP DIFFUSION METHOD
H e n c e if o n e c a n d e t e r m i n e L , t h e n b y o p e r a t i n g w i t h L o n E q . (9-27) as cb LA(X)(X) = | d:C/LG(X,X')^>(A:') Jn
o n e c a n c o n v e r t t h e o r i g i n a l integral e q u a t i o n differential equation:
= ^>(X),
(9-29)
into a second-order
ordinary
(9-30) T h i s p r o c e d u r e w a s utilized b y W i g n e r a n d W i l k i n s 1 1 t o c o n v e r t E q . (9-22) i n t o a differential equation for $ = $(E)/[EM{E)]>: 1 dx [ P(x)
d
dx
[V(x)
+ T ] t ( x ) \ + lw(x)[V(x)
4
+T]-
$(x)
=0
(9-31)
where W(x)
=
X'
X
P(x)
P\x)
P (x) = E-*2 +
r=
VTT
K T )
=
(— V kT
I (9-32)
x erf (A:), Na
y
VTNTJOI1
25
E q . (9-31) is k n o w n n a t u r a l l y e n o u g h as t h e Wigner-Wilkins equation. T o it o n e a p p e n d s t h e b o u n d a r y c o n d i t i o n s t h a t t h e f l u x v a r i a b l e v a n i s h as x—>0 [ i s - » 0 ] a n d b e n o r m a l i z e d t o t h e slowing d o w n s o u r c e f o r x > l [E^>kT]. T h e W i g n e r - W i l k i n s e q u a t i o n h a s b e e n s t u d i e d a n a l y t i c a l l y in e x h a u s t i v e detail u s i n g a w i d e v a r i e t y of a p p r o x i m a t i o n t e c h n i q u e s — n o n e of w h i c h really c o n c e r n us h e r e . 1 Of m u c h m o r e i n t e r e s t is t h e f a c t t h a t s u c h s e c o n d - o r d e r O D E s c a n b e very easily solved o n a digital c o m p u t e r . T o f a c i l i t a t e this solution, o n e first i n t r o d u c e s a c h a n g e of d e p e n d e n t v a r i a b l e t o recast E q . (9-31) i n t o a n o n l i n e a r first-order d i f f e r e n t i a l e q u a t i o n (a Ricatti e q u a t i o n ) m o r e s u i t a b l e f o r n u m e r i c a l i n t e g r a t i o n : dJ dE
S(E) + 2 kT
__ IE
J(E)
—-J'itf),
(9-33)
where
*
{E)=
exp q i e )
] [ ' ]] 1' -
p{E>)+1 _ 1+E /kT
f*iTlF[•
(9 34)
while
(9-35) S a ( E ) + DB £ ? ( £ ); = P + - ^ e r f x + lf x 2 \ VkT ki
^fr
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS H e r e w e h a v e m o d i f i e d this d e r i v a t i o n t o i n c l u d e a n o n - l / u m o d e l e d l e a k a g e w i t h a D(E)B2 term where Z)(^) = (32tr)-1=[3(Ea + Es(l-/I0))]-1.
/
387
absorption
and
(9-36)
A l t h o u g h t h e e q u a t i o n f o r J(E) looks r a t h e r c o m p l i c a t e d it is in f a c t q u i t e simple t o solve n u m e r i c a l l y . T h e r m a l s p e c t r u m c o d e s b a s e d o n s o l u t i o n s of the W i g n e r - W i l k i n s e q u a t i o n a r e o c c a s i o n a l l y r e f e r r e d to as S O F O C A T E - t y p e m e t h o d s 1 2 ( a f t e r t h e early t h e r m a l s p e c t r u m c o d e s u s e d f o r L W R calculations). I n m o s t of t h e s e c o d e s , E q . (9-33) is first solved f o r J(E) n u m e r i c a l l y using M i l n e ' s p r e d i c t o r - c o r r e c t o r m e t h o d . 1 3 T o get this m e t h o d s t a r t e d , a n a s y m p t o t i c s o l u t i o n f o r E<.kT is u s e d . O n c e J(E) is k n o w n , (E) is t h e n d e t e r m i n e d f r o m E q . (9-34) u s i n g a n u m e r i c a l i n t e g r a t i o n (e.g., t r a p e z o i d a l rule). T y p i c a l l y o n t h e o r d e r of 5 0 - 6 0 m e s h p o i n t s a r e f o u n d to yield s u f f i c i e n t a c c u r a c y o v e r the interval 0 < £ < 1 eV. (It is a m u s i n g to n o t e t h a t E q . (9-33) is i n t e g r a t e d f r o m l o w e r to h i g h e r energies, in c o n t r a s t t o f a s t s p e c t r u m c a l c u l a t i o n s w h i c h a l w a y s p r o c e e d d o w n w a r d in e n e r g y — o r u p w a r d s in lethargy, t h e m o r e c o n v e n i e n t v a r i a b l e f o r slowing d o w n c a l c u l a t i o n s . ) T h e W i g n e r - W i l k i n s e q u a t i o n c a n b e a p p l i e d to m i x t u r e s of s c a t t e r i n g isotopes b y simply r e p l a c i n g t h e f r e e a t o m cross s e c t i o n b y a n e f f e c t i v e s c a t t e r i n g cross section c h a r a c t e r i z i n g t h e m i x t u r e : (9-37) j I n f a c t s o m e t i m e s ( a l t h o u g h very i n f r e q u e n t l y ) t h e W i g n e r - W i l k i n s e q u a t i o n is a p p l i e d b y this m o d i f i c a t i o n to t h e a n a l y s i s of n o n h y d r o g e n e o u s m e d i a in t h e h o p e t h a t t h e a c t u a l s p e c t r u m will n o t b e t o o sensitive to t h e details of the s c a t t e r i n g process. I n F i g u r e (9-3) w e h a v e illustrated t h e s h a p e of t h e W i g n e r - W i l k i n s s p e c t r u m f o r a l / i ; a b s o r b e r , a slowing d o w n source, a n d a t h e r m a l r e s o n a n c e (recall F i g u r e 9-1). It s h o u l d b e n o t e d t h a t this m o d e l yields n o t only t h e c o r r e c t s o l u t i o n s f o r 2a—»0[<|>—>M] a n d AT[—»l/is], b u t also a c c o u n t s f o r b o t h a b s o r p t i o n " h e a t i n g " e f f e c t s as well as f l u x d e p r e s s i o n in t h e vicinity of a t h e r m a l r e s o n a n c e .
10
X
0) > <2 O) cc
0.001
0.01
0.1 N e u t r o n energy e V
FIGURE 9-3.
Calculated thermal flux shapes.
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THE MULTIGROUP DIFFUSION METHOD
O n e can n o w generate thermal group constants by simply performing n u m e r i c a l l y t h e a v e r a g e s of t h e m i c r o s c o p i c cross section d a t a over these c o m p u t e d thermal spectra. It is very i m p o r t a n t t o t a k e i n t o a c c o u n t t h e spatial h e t e r o g e n e i t i e s p r e s e n t in a reactor core w h e n generating thermal group constants because the thermal n e u t r o n m f p is f r e q u e n t l y s m a l l c o m p a r e d to t h e d i s t a n c e s c h a r a c t e r i z i n g t h e c o r e lattice s t r u c t u r e . F o r t u n a t e l y t h e r e a r e p r o c e d u r e s a v a i l a b l e b y w h i c h o n e c a n m a k e simple m o d i f i c a t i o n s to i n f i n i t e m e d i u m s p e c t r u m c a l c u l a t i o n s t o a c c o u n t f o r these spatial effects, t h e r e b y a v o i d i n g a d e t a i l e d m u l t i g r o u p c a l c u l a t i o n of n e u t r o n d i f f u s i o n . W e will r e t u r n to discuss t h e s e m e t h o d s in m o r e d e t a i l in C h a p t e r 10. It s h o u l d b e n o t e d t h a t t h e p r o t o n g a s m o d e l effectively c o n v e r t s a n integral e q u a t i o n (9-22) i n t o a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n . S u c h a result is of s o m e significance w h e n considering n u m e r i c a l solutions, since these differential e q u a t i o n s a r e c o n s i d e r a b l y easier t o solve t h a n t h e full m a t r i x e q u a t i o n s t h a t arise f r o m discretizing t h e i n t e g r a l e q u a t i o n directly. I n p a r t i c u l a r it allows u s to c a l c u l a t e t h e n u m e r i c a l s o l u t i o n at o n e e n e r g y m e s h p o i n t directly f r o m the p r e v i o u s o n e o r t w o m e s h p o i n t s , so t h a t o n l y a small c o m p u t e r m e m o r y is r e q u i r e d a n d iterative m e t h o d s a r e a v o i d e d . T h e r e a r e o t h e r m o d e l s of n e u t r o n t h e r m a l i z a t i o n t h a t exploit t h e t e c h n i q u e s of c o n v e r t i n g t h e i n t e g r a l e q u a t i o n (9-22) i n t o a d i f f e r e n t i a l e q u a t i o n m o r e s u i t a b l e f o r m a c h i n e c a l c u l a t i o n s . W e will n e x t e x a m i n e a m o d e l t h a t c a n b e utilized f o r s u r v e y e s t i m a t e s of t h e t h e r m a l s p e c t r u m w h e n t h e m o d e r a t o r m a s s n u m b e r A is large (e.g., I 2 C).
B. The Heavy Gas Model14 O b v i o u s l y t h e p r o t o n gas m o d e l is of l i m i t e d utility in a n a l y z i n g n e u t r o n t h e r m a l i z a t i o n in n o n h y d r o g e n e o u s m o d e r a t o r s . A l t e r n a t i v e m o d e l s a r e n e e d e d f o r h e a v i e r m a s s m o d e r a t o r s s u c h as g r a p h i t e , w h i c h is u s e d in g a s - c o o l e d r e a c t o r s . O n e of t h e simplest t h e r m a l i z a t i o n m o d e l s is b a s e d o n e x p a n d i n g t h e c r o s s section f o r a f r e e g a s of a r b i t r a r y m a s s n u m b e r A in p o w e r s of 1 / A , h e n c e a r r i v i n g a t a n a p p r o x i m a t i o n w h i c h s h o u l d b e valid f o r large m a s s n u m b e r s . T h e d i f f e r e n t i a l s c a t t e r i n g cross section f o r a f r e e gas c a n b e d e r i v e d in a s t r a i g h t f o r w a r d , if s o m e w h a t c u m b e r s o m e , c o n s i d e r a t i o n of t w o - b o d y k i n e m a t i c s (similar to t h a t of Section 2 - I - D ) . T h i s cross section c a n b e w r i t t e n as
e)
2 Ef
| |^erf(0V€ ' — p V e ) ± e r f ( # V c ' H - p V e )
+ erf(0V€ — pVc
Terf(#Ve + p V 7 ) } ,
(9-38)
where (.4 + 1) v=
(A-1) ,
2VA
p=
E
, 2 VA
If w e n o w e x p a n d in inverse m a s s n u m b e r A
e = -p=
(9-39)
k T
a n d r e t a i n o n l y lowest o r d e r terms,
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
/
389
w e arrive at the heavy gas model of the s c a t t e r i n g k e r n e l (E+E') [8'(E-
E') + kT8"(E-
E')] (9-40)
H e r e 5 ' a n d S " a r e t h e first a n d s e c o n d derivatives of t h e D i r a c 8 - f u n c t i o n a n d a r e d e f i n e d in A p p e n d i x C. T h e c o r r e s p o n d i n g t o t a l s c a t t e r i n g cross section f o r this m o d e l is (9-41) If w e n o w s u b s t i t u t e this r a t h e r singular s c a t t e r i n g kernel b a c k i n t o o u r integral e q u a t i o n a n d n o t e t h a t f o r large A 9 / A , such t h a t
d2
<}>(£) .
(9-43)
T h i s simple s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n is k n o w n as t h e heavy gas equation. I n m a n y w a y s t h e h e a v y gas a p p r o x i m a t i o n is similar to the age a p p r o x i m a t i o n w e d e v e l o p e d to d e s c r i b e n e u t r o n slowing d o w n in h e a v y m a s s m o d e r a t o r s , a n d in the h i g h - e n e r g y limit t h e h e a v y gas m o d e l j u s t r e d u c e s to age t h e o r y . U s i n g the u s u a l t e c h n i q u e s f r o m the t h e o r y of s e c o n d - o r d e r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , o n e c a n l a b o r i o u s l y g e n e r a t e a s y m p t o t i c solutions to this e q u a t i o n f o r different a b s o r p t i o n cross section energy d e p e n d e n c e s — f o r example, 2a(Zs) — c o n s t a n t = 2 a , o r 2 a ( i s ) = y/v. It c a n also b e i n t e g r a t e d directly to g e n e r a t e t h e r m a l n e u t r o n s p e c t r a f o r m o d e r a t o r s s u c h as g r a p h i t e . F o r e x a m p l e , the n u m e r i cal solution of t h e h e a v y gas e q u a t i o n ( a s s u m i n g 1 / v a b s o r p t i o n ) is s h o w n in F i g u r e 9-4 f o r several d i f f e r e n t values of t h e p a r a m e t e r r = 2 a ( / c r ) / £ 2 f r . It s h o u l d b e n o t e d t h a t as t h e m o d e r a t o r m a s s b e c o m e s increasingly heavier (i.e., £ smaller), the m o d e r a t o r is increasingly less c a p a b l e of slowing d o w n n e u t r o n s a n d h e n c e the s p e c t r u m h a r d e n s . I n this m o d e l , then, a heavier gas is e q u i v a l e n t to the a d d i t i o n of a b s o r p t i o n . U n f o r t u n a t e l y t h e h e a v y gas (or even f r e e gas) p r e d i c t i n g t h e r m a l s p e c t r a in g r a p h i t e m o d e r a t e d t h a n is the p r o t o n gas m o d e l f o r t h e L W R 1 5 b e c a u s e b i n d i n g effects. T h i s h a s led to a r a t h e r limited use
m o d e l is f a r less effective at r e a c t o r s (such as t h e H T G R ) of the significance of c h e m i c a l of this m o d e l in p r a c t i c e .
C. Synthetic Scattering Kernel Models I n o u r t w o p r e v i o u s m o d e l s of n e u t r o n t h e r m a l i z a t i o n w e f o u n d t h a t the s c a t t e r i n g k e r n e l 2 g ( i s ' - > i i ) w a s of s u c h a f o r m t h a t w e c o u l d t r a n s f o r m the
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THE MULTIGROUP DIFFUSION METHOD
FIGURE 9-4
The neutron spectrum predicted by the heavy gas model.
i n t e g r a l f o r m of t h e i n f i n i t e m e d i u m s p e c t r u m e q u a t i o n (9-1) i n t o a d i f f e r e n t i a l equation that was far better suited to m a c h i n e computation. It is n a t u r a l t o seek o t h e r m o d e l s of n e u t r o n t h e r m a l i z a t i o n w h i c h a l s o l e a d t o d i f f e r e n t i a l e q u a t i o n s . O n e s u c h m o d e l is t h e generalized heavy gas o r primary model16 p r o p o s e d b y H o r o w i t z in w h i c h t h e s c a t t e r i n g t e r m s i n E q . (9-1) a r e m o d e l e d b y a d i f f e r e n t i a l e q u a t i o n very similar t o t h e h e a v y gas m o d e l (9-42):
EkT-^+(E-kT)4> (9-44) H e r e f ( E ) is a n a r b i t r a r y f u n c t i o n t h a t m u s t b e d e t e r m i n e d either b y f i t t i n g t o e x p e r i m e n t a l s p e c t r u m m e a s u r e m e n t s or b y a fit t o a n i n t e g r a l of t h e s c a t t e r i n g k e r n e l itself H,s(Ef->E). T h e p r i m a r y m o d e l satisfies t h e d e t a i l e d b a l a n c e c o n d i t i o n a n d i n c l u d e s s o m e a c c o u n t i n g f o r c h e m i c a l b i n d i n g effects. M o r e o v e r t h e s t o r a g e r e q u i r e m e n t s a n d m a c h i n e t i m e r e q u i r e d t o solve t h e d i f f e r e n t i a l e q u a t i o n g e n e r a t e d b y E q . (9-44) a r e q u i t e small w h e n c o m p a r e d t o t h e m a c h i n e l a b o r i n v o l v e d in solving t h e i n t e g r a l e q u a t i o n (9-1) directly. U n f o r t u n a t e l y t h e p r i m a r y m o d e l d u e t o H o r o w i t z fails to yield s a t i s f a c t o r y results w h e n s t r o n g a b s o r p t i o n is p r e s e n t — p a r t i c u l a r l y in t h e vicinity of t h e r m a l r e s o n a n c e s . T o c i r c u m v e n t this, C a d i l h a c 1 7 d e v e l o p e d a slightly m o r e e l a b o r a t e
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
/
391
a p p r o x i m a t i o n k n o w n as the secondary model c o n t a i n i n g t w o ( r a t h e r t h a n o n e ) f r e e f u n c t i o n s . O n e first writes t h e s c a t t e r i n g k e r n e l as the p r o d u c t of these t w o a r b i t r a r y f u n c t i o n s u(E) a n d v(E) s u c h t h a t «(£>(£"),
E>E'
u(E')v(E)9
E<Ef.
(9-45)
( N o t e a g a i n the d i s c o n t i n u o u s derivative at E' = E.) T h e n o n e f i n d s that the i n f i n i t e m e d i u m s p e c t r u m e q u a t i o n (9-1) c a n b e w r i t t e n as the c o u p l e d set of differential equations
Za(E)
w h e r e j ( E ) a n d k(E)
S(E),
dq
HE) dE
+
M(E)
=j(E)q(E)-
a r e given in t e r m s of t h e " f r e e " f u n c t i o n s , u(E)
-jL
(9-46)
dE
[v(E)k{E)l
a n d v(E),
as
1h(E)=fJdE'v(E'),
k(E)=[M(E)2t(E)]~\
(9-47)
/JL
\kT
FIGURE 9-5. Comparison of the primary and secondary synthetic kernel model predictions for the flux in the neighborhood of the 2 4 0 Pu resonance (moderator is graphite).
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THE MULTIGROUP DIFFUSION METHOD
T h i s m o d e l is p a r t i c u l a r l y interesting since f o r p r o p e r choices of u(E) a n d v(E) it will r e d u c e to several of o u r earlier m o d e l s i n c l u d i n g the p r o t o n gas m o d e l , the h e a v y gas (or p r i m a r y ) m o d e l , the F e r m i age m o d e l , or t h e G o e r t z e l - G r e u l i n g m o d e l . H e n c e the s e c o n d a r y m o d e l is e v i d e n t l y c a p a b l e of b r a c k e t i n g the therm a l i z a t i o n p r o p e r t i e s of t h e t r u e s c a t t e r i n g k e r n e l 2 S ( £ " ^ £ ) . O n e c a n c h o o s e the f r e e f u n c t i o n s u(E) a n d v(E) in a n y of a n u m b e r of ways. F o r e x a m p l e , o n e c a n c h o o s e these f u n c t i o n s b y d e m a n d i n g t h a t the first t w o e n e r g y m o m e n t s of t h e true s c a t t e r i n g k e r n e l 2 S ( £ " — > £ ) b e p r e s e r v e d b y t h e a p p r o x i m a t e m o d e l [Eq. (9-45)]. T h e s p e c t r a p r e d i c t e d b y t h e s e c o n d a r y m o d e l are f r e q u e n t l y in excellent a g r e e m e n t w i t h t h e results of a b r u t e f o r c e n u m e r i c a l solution of E q . (9-1) u s i n g t h e t r u e s c a t t e r i n g k e r n e l T h i s is t r u e even in the vicinity of t h e r m a l r e s o n a n c e s , as illustrated in F i g u r e 9-5. A l t h o u g h t h e s e c o n d a r y m o d e l is u s u a l l y n o t u s e d f o r d e t a i l e d t h e r m a l s p e c t r u m studies, it is f r e q u e n t l y u s e d to g e n e r a t e the initial guesses of the s p e c t r u m u s e d in m o r e e l a b o r a t e iterative c a l c u l a t i o n s n e c e s s a r y to solve t h e i n f i n i t e m e d i u m s p e c t r u m e q u a t i o n (9-1) directly.
III. GENERAL CALCULATIONS OF THERMAL NEUTRON SPECTRA A. Generation of Thermal Cross Section Data A s we h a v e m e n t i o n e d , t h e r m a l n e u t r o n cross sections a r e c o m p l i c a t e d b y a sensitive d e p e n d e n c e o n the t e m p e r a t u r e a n d c h e m i c a l s t a t e of t h e s c a t t e r i n g m a t e r i a l . A l t h o u g h such cross section i n f o r m a t i o n is t a b u l a t e d a n d s t o r e d in n u c l e a r d a t a sets such as E N D F / B , it is u s e f u l to discuss briefly h o w these b a s i c d a t a a r e c a l c u l a t e d . It is c u s t o m a r y to express the d i f f e r e n t i a l s c a t t e r i n g cross section < J s ( i T ' — i n t e r m s of a q u a n t i t y , ^ ( a , / ? ) 1 " 6 , 1 5 k n o w n as t h e scattering law f o r t h e m a t e r i a l of i n t e r e s t :
S2'->Q) =
( -§T ) :2 e x p ( - |
)abS
).
(9-48)
H e r e , a b is t h e b o u n d a t o m cross section, while a a n d fi a r e r e l a t e d to the n e u t r o n m o m e n t u m a n d e n e r g y e x c h a n g e in a s c a t t e r i n g collision: a=(E'
+ E-2VWE
S2•
p~(Ef~E)/kT.
S2')/,kT, (9-49)
S ( a , / ? ) d e p e n d s in a very c o m p l i c a t e d m a n n e r o n t h e d e t a i l e d s t r u c t u r e a n d d y n a m i c s of t h e s c a t t e r i n g m a t e r i a l . M o s t c o m m o n l y , h o w e v e r , o n e i n t r o d u c e s several h o r r i f y i n g l y b r u t a l a p p r o x i m a t i o n s (at least to t h e solid-state physicist, a l t h o u g h n o t to t h e n u c l e a r e n g i n e e r ) in o r d e r to allow the c a l c u l a t i o n of S ( a , f i ) . F i r s t o n e a c c o u n t s f o r only i n c o h e r e n t s c a t t e r i n g (i.e., n o d i f f r a c t i o n ) such t h a t S ( a , / $ ) is r e p l a c e d b y t h e i n c o h e r e n t p a r t of t h e s c a t t e r i n g law, Ss(a,P). T h e n this
THERMAL SPECTRUM CALCULATIONS AND THERMAL GROUP CONSTANTS
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393
q u a n t i t y is c a l c u l a t e d in t h e so-called G a u s s i a n a p p r o x i m a t i o n 1 8 in w h i c h
= -L f ^ J r e ^ e x p [ - aw2(r)
Ss(a,p)
p
2/ W
T
< >= I
dHP{H)
],
r
/ f i ]
(9-50)
/3sinh(/3/2)
T h e entire d e p e n d e n c e of the n e u t r o n s c a t t e r i n g o n the m o t i o n of t h e m o d e r a t o r a t o m s is c o n t a i n e d in t h e f u n c t i o n p(j8) o r equivalently, p(co), a>= kTfi/h. For a h a r m o n i c solid, o n e w o u l d i d e n t i f y p(
B. Thermal Spectrum Calculations F o r a c c u r a t e d e t e r m i n a t i o n of t h e r m a l n e u t r o n spectra a n d the g e n e r a t i o n of t h e r m a l g r o u p c o n s t a n t s , o n e m u s t p e r f o r m a direct solution of the i n f i n i t e m e d i u m s p e c t r u m e q u a t i o n (9-1) b y w r i t i n g it in a discrete m u l t i g r o u p f o r m
2
2v„
(9-51)
n'= 1
w h e r e n is t h e f i n e g r o u p i n d e x . Since the t h e r m a l energy r a n g e is r a t h e r n a r r o w ( 0 < £ < 1 eV), a n d typically f r o m 50 to 100 e n e r g y g r o u p s a r e used, o n e c a n usually j u s t use i n p u t cross section d a t a o n a pointwise basis ( e v a l u a t e d a t the c e n t e r of a g r o u p , f o r e x a m p l e ) . A c t u a l l y E q . (9-1) is a bit t o o simplified, since o n e usually a d d s in a c r u d e spatial t r e a t m e n t u s i n g t h e Px o r Bx m e t h o d in a m a n n e r similar to t h a t u s e d to c a l c u l a t e f a s t spectra. T h e n the t h e r m a l l e a k a g e is d e t e r m i n e d b y s p e c i f y i n g a n e q u i v a l e n t geometric buckling Such i n f i n i t e m e d i u m s p e c t r u m c a l c u l a t i o n s m u s t also b e c o r r e c t e d for core heterogeneities. I n the n e x t c h a p t e r w e will d e v e l o p several p r e s c r i p t i o n s f o r
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m o d i f y i n g t h e t h e r m a l g r o u p c o n s t a n t s to a c c o u n t f o r f l u x n o n u n i f o r m i t i e s w h i c h arise in a f u e l lattice cell.
C. Coupling between Fast and Thermal Spectrum Calculations I n a r i g o r o u s d e t e r m i n a t i o n of the f a s t a n d t h e r m a l n e u t r o n e n e r g y s p e c t r u m , t h e f a s t a n d t h e r m a l e n e r g y r e g i o n s will b e c o u p l e d t o g e t h e r . D o w n s c a t t e r i n g f r o m h i g h e r energies p r o v i d e s t h e s o u r c e t e r m in t h e t h e r m a l range, while t h e r m a l n e u t r o n i n d u c e d fission r e a c t i o n s will give rise to the fission s o u r c e in the fast n e u t r o n region. I n a d d i t i o n t h e r e m a y b e s o m e m i l d i n f l u e n c e o n t h e l o w e r energy b e h a v i o r of t h e f a s t s p e c t r u m d u e to u p s c a t t e r i n g of t h e r m a l n e u t r o n s . F o r t u n a t e l y t h e c o u p l i n g b e t w e e n f a s t a n d t h e r m a l s p e c t r u m r e g i o n s is sufficiently w e a k t h a t t h e c a l c u l a t i o n of f a s t a n d t h e r m a l n e u t r o n s p e c t r a are usually p e r f o r m e d separately. H o w e v e r m a n y f a s t s p e c t r u m c o d e s will g e n e r a t e the P0 a n d Px slowing d o w n s o u r c e s to b e i n s e r t e d i n t o t h e r m a l s p e c t r u m c a l c u l a t i o n s . U s u a l l y all d o w n s c a t t e r e d n e u t r o n s f r o m t h e f a s t r e g i o n a r e a s s u m e d to go i n t o the highest e n e r g y t h e r m a l g r o u p , e x c e p t f o r h y d r o g e n e o u s m o d e r a t o r s in w h i c h t h e slowing d o w n n e u t r o n s a r e d i s t r i b u t e d o v e r all t h e r m a l g r o u p s . If u p s c a t t e r i n g o u t of t h e t h e r m a l r e g i o n is allowed, these n e u t r o n s a r e u s u a l l y a s s u m e d to g o i n t o the lowest e n e r g y f a s t g r o u p .
REFERENCES
1. M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam (1966). 2. G. I. Bell and S. Glasstone, Nuclear Reactor Theory, Van Nostrand, New York (1970), Chapter 7. 3. K. H. Beckurts and K. Wirtz, Neutron Physics, Springer, Berlin (1964). 4. Neutron Thermalization and Reactor Spectra, International Atomic Energy Agency, Vienna (1968), Vols. I—II. 5. I. I. Gurevich and L. V. Tarasov, Low Energy Neutron Physics, Wiley, New York (1968). 6. D. E. Parks, M. S. Nelkin, N . F. Wikner, and J. R. Beyster, Slow Neutron Scattering and Thermalization with Reactor Applications, Benjamin, New York (1970). 7. A. C. Zemach and R. L. Glauber, Phys. Rev. 101, 118 (1956). 8. R. F. Coveyou, R. R. Bate, and R. K. Osborn, ORNL-1958 (1955). 9. E. R. Cohen, Nucl. Sci. Eng. 2, 227 (1957). 10. C. H. Westcott, Effective cross section values for well-moderated thermal reactor spectra, AECL-1101, 1962 (3rd Edition). 11. E. P. Wigner and J. E. Wilkins, AECD-2275 (1944); M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam (1966), p. 75. 12. H. Amster and R. Suarez, The calculation of thermal constants averaged over a Wigner Wilkins flux spectrum; SOFOCATE, WAPD-TM-39 (1957). 13. B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods, Wiley, New York (1969) p. 386. 14. J. E. Wilkins, USAEC Document CP-2481 (1944); Ann. Math. 49, 189 (1948). 15. J. R. Beyster, N . Corngold, H. C. Honeck, G. D. Joanou and D. E. Parks, P / 2 5 8 in Third U. N. Conference on Peaceful Uses of Atomic Energy (1964). 16. M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam (1966), p. 273. 17. M. Cadilhac, J. Horowitz, J. L. Soule, and O. Tretiakoff, Proceedings BNL Conference on Thermalization 2, 1962; M. Cadilhac, et. al., Third U . N . Conference on Peaceful Uses
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of Atomic Energy (1964); M. M. R. Williams, The Slowing Down and Thermalization of Neutrons, North-Holland, Amsterdam (1966), p. 292. 18. P. A. Egelstaff and P. Schofield, Nucl Sci. Eng. 12, 260 (1962). 19. J. U. Koppel, J. R. Triplett and Y. D. Naliboff, GASKET, a unified code for thermal neutron scattering, General Atomic Report GA-7417 (1966). 20. H. C. Honeck and D. R. Finch, F L A N G I I (Version 71-1), a code to process thermal neutron data from an E N D F / B tape, Savannah River Laboratory Report DP-1278 (1971).
PROBLEMS 9-1
Show that the scattering cross characterizing a perfectly bound doh onstrate that a b = 4?r —
section a f r of a free nucleus is related to that nucleus o h by a b = (l + ^4 _ 1 ) 2 a f r . (Hint: First dem-
- dOfT 4 <77 dQ COS0- 1
. Then use the expression for
doh/dQ
cos 0 = 1
for s-wave scattering.) 9-2. Generalize the definition of the slowing down density q(r,E) given by Eq. (8-18) to account for upscattering as well as downscattering. 9-3 Using the definition of the slowing down density for energies E > Ec in the infinitemedium spectrum equation (9-1), derive the balance condition Eq. (9-4). 9-4 Check by explicit calculation whether the detailed balance condition holds for the scattering kernels characterizing: (a) a free proton gas, (b) the heavy gas model, and (c) a free gas of arbitrary mass number A. 9-5 Give three reasons why the neutron energy distribution in a thermal homogeneous reactor is not Maxwellian. What specific physical effects cause deviations at high energies and what physical effects give rise to deviations at low energies? 9-6 Calculate the average energy and velocity of neutrons in a Maxwellian distribution at a temperature Tn. 9-7 Plot the behavior of E
396 9-13 9-14
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THE MULTIGROUP DIFFUSION METHOD Verify the expression given for "2S(E) characterizing a free proton gas by explicitly integrating the form given in Eq. (9-23) for 2 S (.E"—•£). The second energy moment of the scattering kernel, M2 = ——z (kT)
9-15
9-16 9-17
9-18
9-19
9-20 9-21
(*>dEM(E')(E'-E)2oXE'-*E)
dE'
J
o
Jo
is a measure of the mean squared energy exchanged between the neutron and the scattering atoms. Demonstrate that for a free proton gas M2 — 2V2 a f r . Derive the Wigner-Wilkins equation (9-31) from the infinite medium spectrum equation (9-22) characterizing a proton gas. [Refer to E. P. Wigner and J. E. Wilkins, A E C D 2275 (1944) or M. Williams, Thermalization and Slowing Down of Neutrons, North Holland, Amsterdam (1967), p. 77 for assistance.] Using the variable transformation defined by Eq. (9-34), derive the Ricatti equation (9-33) from the Wigner-Wilkins equation. Write a computer program that integrates the nonlinear differential equation (9-33) characterizing a free proton gas. Use either a predictor-corrector or Runge-Kutta scheme with an energy mesh size of AE = 0.001 eV. Determine the flux
9-22
Demonstrate that for E^>kT,
9-23
Determine the choice of the functions u(E) and v(E) of the secondary model that will yield the proton gas model and the heavy gas model. A very useful approximation to the thermal neutron scattering kernel 2 S (£" '^>E) is the so-called synthetic or simple degenerate kernel (SDK) model
9-24
2s(E'^E)
the solution to the heavy gas equation is just
=
p2s(E')vM(E)2s(E),
where (™dEvM(E)^(E). Jo [See N. Corngold, P. Michael, and W. Wollman, NucL ScL Eng. 15, 13 (1963).] Demonstrate that this kernel preserves the correct total scattering cross section 2 S (£") and also satisfies the principle of detailed balance.
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397
Solve the infinite medium spectrum equation for the flux resulting from an arbitrary source S(E) using the SDK model. The S D K model can be used to study the time-dependent thermalization of a neutron pulse injected into a moderator at an energy E0. To this end, consider the initial value problem f™dE'2s(E'^E)
9-27
9-28
9-29
9-30
Assuming an absorption 2 a (.E) = 2 ° / t ; and the SDK model, solve this problem for (£,/). (Hint: Use a Laplace transform in time.) Age-diffusion theory can be used to provide the effective slowing down source for the neutron diffusion equation characterizing the thermal group. In this manner, determine the distribution of thermal neutrons resulting from an isotropic point source of monoenergetic fast neutrons located at the origin. In particular, discuss the asymptotic behavior of your solution far from the source for V r s L. Describe possible experimental techniques for measuring: (a) the neutron age to thermal and (b) the thermal neutron diffusion length. Assume that the only available neutron source is a fast (—2 MeV) source. Justify your discussion by simple calculations. Then discuss how your experiments would differ for graphite and for light water. Neutrons of lethargy zero are produced uniformly throughout an infinite medium. Assume that they then slow down by elastic collisions until they reach Eih and there enter a one-speed diffusion process. The medium is characterized by the macroscopic cross sections 2 S , 2 a , 2 f , and 2 t , which are all independent of neutron energy for all energies. Find the fraction of fissions caused by the thermal neutrons. Assume that the age approximation is valid, that is, #(w) = £2t<J>(w). Demonstrate that when age-diffusion theory is used to describe neutron slowing down, and one-speed diffusion theory is used to describe thermal neutron diffusion, then the multiplication factor for a bare, homogeneous reactor is exp -
9-31
B^t
Compute and plot the critical mass and the fast and slow nonleakage probabilities for a spherical assembly of 2 3 5 U and moderator as a function of the radius of the assembly for the following moderators: (a) H 2 0 , (b) D 2 0 , (c) Be, and (d) graphite. Determine the minimum critical mass in each case. (Use the parameters listed in Table 8-7.)
10 Cell Calculations for Heterogeneous Core Lattices
T h u s f a r w e h a v e restricted ourselves to t h e s t u d y of r e a c t o r cores in w h i c h fuel, m o d e r a t o r , c o o l a n t , a n d s t r u c t u r a l m a t e r i a l s w e r e a s s u m e d to b e i n t i m a t e l y a n d h o m o g e n e o u s l y m i x e d . H o w e v e r , n u c l e a r r e a c t o r cores a r e of c o u r s e c o n s t r u c t e d in a highly h e t e r o g e n e o u s c o n f i g u r a t i o n to facilitate t h e r m a l design ( c o o l a n t c h a n n e l s , h e a t - t r a n s f e r surfaces), m e c h a n i c a l design ( s t r u c t u r a l integrity, f u e l f a b r i c a t i o n a n d h a n d l i n g ) , a n d reactivity c o n t r o l ( c o n t r o l r o d s , b u r n a b l e poisons, i n s t r u m e n t a t i o n ) . F o r e x a m p l e , t h e r e a d e r s h o u l d recall t h e r a t h e r detailed s t r u c t u r e of t h e typical P W R c o r e illustrated in F i g u r e s 3-6 a n d 3-7. Such heterogeneities in t h e r e a c t o r f u e l a r r a y o r lattice m u s t b e t a k e n i n t o a c c o u n t in n u c l e a r design since t h e y will c a u s e a local spatial v a r i a t i o n in t h e n e u t r o n flux w h i c h m a y strongly i n f l u e n c e core multiplication. T h e d e g r e e to w h i c h c o r e lattice effects m u s t b e t a k e n i n t o a c c o u n t in r e a c t o r design d e p e n d s o n t h e c h a r a c t e r i s t i c d i m e n s i o n s of t h e lattice s t r u c t u r e , f o r exa m p l e , t h e d i a m e t e r of a f u e l p i n or t h e s p a c i n g b e t w e e n f u e l elements, c o m p a r e d to t h e m e a n f r e e p a t h of n e u t r o n s in t h e core. F o r e x a m p l e , in t h e L W R the t h e r m a l n e u t r o n m e a n f r e e p a t h is typically o n t h e o r d e r of o n e cm, c o m p a r a b l e to the f u e l p i n d i a m e t e r . H e n c e the flux d i s t r i b u t i o n in t h e f u e l m i g h t b e e x p e c t e d to b e quite d i f f e r e n t f r o m t h a t in t h e m o d e r a t o r o r c o o l a n t c h a n n e l , t h e r e b y necessit a t i n g a d e t a i l e d t r e a t m e n t of t h e h e t e r o g e n e i t y . By w a y of c o n t r a s t , t h e m u c h longer m e a n f r e e p a t h c h a r a c t e r i s t i c of t h e n e u t r o n s in a f a s t r e a c t o r (typically tens of c e n t i m e t e r s ) allows a m u c h grosser t r e a t m e n t of lattice effects. Of course, a detailed t r e a t m e n t of t h e c o r e lattice o n a scale sufficiently f i n e to a c c o u n t f o r t h e spatial v a r i a t i o n of t h e f l u x in t h e n e i g h b o r h o o d of a given f u e l e l e m e n t is clearly o u t of t h e q u e s t i o n , since it w o u l d require a n u n m a n a g e a b l y large a r r a y of m e s h p o i n t s in a m u l t i g r o u p d i f f u s i o n c a l c u l a t i o n (typical L W R s h a v e over 50,000 f u e l elements). I n d e e d d u e to their strongly a b s o r b i n g n a t u r e , f u e l a n d 398
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c o n t r o l e l e m e n t s in t h e c o r e f r e q u e n t l y r e q u i r e a m o r e a c c u r a t e t r e a t m e n t of n e u t r o n transport than that provided by diffusion theory. H e n c e one must adopt a m o r e p i e c e m e a l a p p r o a c h b y seeking to selectively " h o m o g e n i z e " the analysis of t h e core, u s u a l l y b y p r o v i d i n g p r e s c r i p t i o n s f o r i n c l u d i n g lattice e f f e c t s into existing i n f i n i t e m e d i u m n e u t r o n e n e r g y - s p e c t r u m c a l c u l a t i o n s or b y c a l c u l a t i n g f e w - g r o u p c o n s t a n t s t h a t h a v e b e e n spatially a v e r a g e d over t h e f i n e r details of the flux d i s t r i b u t i o n in t h e lattice. Of course, t h e type of t r e a t m e n t o n e c h o o s e s will d e p e n d o n the p u r p o s e of the c a l c u l a t i o n . F o r e x a m p l e , o n e c a n c o n t r a s t a h a n d c a l c u l a t i o n b a s e d o n the s i x - f a c t o r f o r m u l a suitable f o r a c r u d e survey e s t i m a t e with a n extremely detailed t r a n s p o r t c a l c u l a t i o n t h a t m i g h t b e u s e d in a c o m p a r i s o n with a critical e x p e r i m e n t o r p e r h a p s as a b e n c h m a r k f o r the testing of o t h e r c a l c u l a t i o n a l schemes. O u r c o n c e r n in this c h a p t e r is with m o r e r o u t i n e design c a l c u l a t i o n s t h a t m u s t b e p e r f o r m e d very f r e q u e n t l y a n d h e n c e p l a c e a p r e m i u m o n c a l c u l a t i o n a l ease. F o r s u c h s c h e m e s t o yield s u f f i c i e n t a c c u r a c y , o n e is f o r c e d to rely o n f r e q u e n t cross c a l i b r a t i o n w i t h e x p e r i m e n t — t h a t is, to a c c e p t a c e r t a i n a m o u n t of e m p i r i c i s m ( f u d g i n g ) in t h e t e c h n i q u e . A n d of c o u r s e as in m o s t f a s t y e t a c c u r a t e m e t h o d s , o n e relies heavily o n a c a n c e l l a t i o n of errors. F o r e x a m p l e , in a L W R c o r e study, o n e m i g h t a t t e m p t to g e n e r a t e g r o u p c o n s t a n t s f o r a f e w - g r o u p d i f f u s i o n analysis u s i n g a S O F O C A T E - M U F T s c h e m e similar to t h a t discussed in C h a p t e r s 8 a n d 9. Y e t of c o u r s e t h e s e p r o c e d u r e s p e r f o r m c a l c u l a t i o n s f o r a h o m o g e n e o u s m e d i u m in w h i c h only gross l e a k a g e e f f e c t s a r e a c c o u n t e d f o r via t h e Px or Bx a p p r o x i m a t i o n . H e n c e o u r objective h e r e is to p r o v i d e a p r e s c r i p t i o n f o r m o d i f y i n g these h o m o g e n e o u s results to a c c o u n t f o r h e t e r o g e n e o u s lattice effects. T h e g e n e r a l a p p r o a c h is to divide t h e p e r i o d i c a r r a y of t h e r e a c t o r lattice i n t o a n u m b e r of i d e n t i c a l unit cells a n d t h e n calculate the e f f e c t i v e g r o u p c o n s t a n t s c h a r a c t e r i z i n g o n e s u c h cell. T h e d e t a i l e d analysis of a u n i t cell m u s t a c c o u n t f o r t h e s t r o n g spatial v a r i a t i o n of t h e n e u t r o n energy s p e c t r u m w i t h i n the cell. T h e cell is h o m o g e n i z e d b y a v e r a g i n g the g r o u p c o n s t a n t s c h a r a c t e r i z i n g m a t e r i a l s in the cell over t h e spatial f l u x d i s t r i b u t i o n w i t h i n t h e cell to p r o d u c e cell-averaged or so-called self-shielded g r o u p c o n s t a n t s t h a t c a n t h e n be u s e d in a m u l t i g r o u p d i f f u s i o n analysis of t h e e n t i r e core. I n this c h a p t e r w e d e v e l o p several p r o c e d u r e s a p p l i c a b l e f o r p e r f o r m i n g such cell c a l c u l a t i o n s a n d t h e r e b y g e n e r a t i n g cell-averaged g r o u p c o n s t a n t s . T h e highly a b s o r b i n g n a t u r e of t h e f u e l r e g i o n in a r e a c t o r lattice cell will necessitate a t r a n s p o r t t h e o r y d e s c r i p t i o n of t h e n e u t r o n flux, b a s e d o n either so-called collision p r o b a b i l i t y m e t h o d s o r a direct solution of t h e t r a n s p o r t e q u a t i o n itself (or p e r h a p s a M o n t e C a r l o calculation). Since the t r e n d in r e c e n t y e a r s h a s b e e n t o w a r d m o r e d e t a i l e d t r e a t m e n t s of t h e h e t e r o g e n e o u s e f f e c t s in c o r e lattices, w e will i n c l u d e a brief d i s c u s s i o n of b o t h a n a l y t i c a l a n d direct n u m e r i c a l t e c h n i q u e s for s t u d y i n g n e u t r o n t r a n s p o r t w i t h i n lattice cells.
I. LATTICE EFFECTS IN,NUCLEAR REACTOR ANALYSIS A. A Qualitative Discussion of Heterogeneous Effects on Core Multiplication in Thermal Reactors B e f o r e w e l a u n c h i n t o a d e t a i l e d s t u d y of h o w lattice e f f e c t s a r e i n c l u d e d in r e a c t o r c o r e analysis m e t h o d s , it is u s e f u l to give a simple q u a l i t a t i v e discussion of t h e e f f e c t s of f u e l l u m p i n g o n core m u l t i p l i c a t i o n in t h e r m a l r e a c t o r s , since this
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THE MULTIGROUP DIFFUSION METHOD
reveals m o s t of t h e r e l e v a n t physics. T o this e n d w e will s i m p l y e x a m i n e h o w e a c h of t h e v a r i o u s t e r m s in t h e six-factor f o r m u l a a r e m o d i f i e d in p a s s i n g f r o m a h o m o g e n e o u s r e a c t o r core, in w h i c h t h e f u e l a n d m o d e r a t o r a r e i n t i m a t e l y m i x e d , t o a h e t e r o g e n e o u s lattice, i n w h i c h t h e f u e l is l u m p e d s e p a r a t e l y f r o m t h e m o d e r a t o r . T h i s d i s c u s s i o n a c t u a l l y h a s a r a t h e r i n t e r e s t i n g historical significance, since w i t h o u t f u e l l u m p i n g it w o u l d h a v e b e e n i m p o s s i b l e t o a c h i e v e a critical a s s e m b l y u s i n g n a t u r a l u r a n i u m a n d g r a p h i t e in F e r m i ' s " p i l e " a t t h e U n i v e r s i t y of C h i c a g o in 1942. M o r e specifically, f o r a n a t u r a l u r a n i u m system, 77 = 1.33 w h i l e € = 1.05. If o n e studies / a n d p f o r v a r i o u s h o m o g e n e o u s m i x t u r e s of n a t u r a l u r a n i u m a n d g r a p h i t e , t h e n , a t b e s t o n e f i n d s f p = 0.59. 1 H e n c e , f o r a h o m o g e n e o u s r e a c t o r c o r e c o m p o s e d of these m a t e r i a l s , k ^ < ( 1 . 3 3 ) (1.05) (0.59) = 0.85. O b v i o u s l y s u c h a s y s t e m c o u l d n e v e r b e m a d e critical. F e r m i a n d Szilard 2 n o t e d t h a t if t h e f u e l w e r e l u m p e d i n t o a h e t e r o g e n e o u s lattice, t h e n t h e r e s o n a n c e e s c a p e p r o b a b i l i t y p w o u l d i n c r e a s e d r a m a t i c a l l y . T h i s o c c u r s b e c a u s e n e u t r o n s t h a t a r e slowed d o w n t o r e s o n a n c e e n e r g i e s in t h e m o d e r a t o r a r e p r i m a r i l y a b s o r b e d in t h e o u t e r r e g i o n s of t h e f u e l e l e m e n t — h e n c e l e a d i n g t o a d e p r e s s i o n in t h e n e u t r o n f l u x w i t h i n t h e f u e l at r e s o n a n c e energies (see F i g u r e 10-1). T h a t is, t h e o u t e r layers of t h e f u e l t e n d t o shield its i n t e r i o r f r o m resonance energy neutrons, thereby decreasing the net resonance absorption a n d
'fission
FIGURE 10-1.
Flux behavior in the neighborhood of a fuel pin.
CELL CALCULATIONS FOR HETEROGENEOUS CORE LATTICES
/
401
h e n c e i n c r e a s i n g t h e r e s o n a n c e e s c a p e p r o b a b i l i t y p. T h i s " s e l f - s h i e l d i n g " e f f e c t is sufficiently s t r o n g t h a t k ^ i n c r e a s e s to a v a l u e of 1.08 in a g r a p h i t e - n a t u r a l u r a n i u m lattice. (It s h o u l d b e n o t e d t h a t this spatial self-shielding is in m a n y w a y s a n a l o g o u s to t h e energy self-shielding w e e n c o u n t e r e d in o u r d i s c u s s i o n of the D o p p l e r e f f e c t in r e s o n a n c e a b s o r p t i o n . B o t h e f f e c t s t e n d to d e c r e a s e r e s o n a n c e absorption, thereby increasing the resonance escape probability.) T h e r e a r e o t h e r e f f e c t s d u e to f u e l l u m p i n g , h o w e v e r . O n the positive side, the f a s t fission f a c t o r € will i n c r e a s e s o m e w h a t in a h e t e r o g e n e o u s a s s e m b l y b e c a u s e t h e p r o b a b i l i t y of a f a s t n e u t r o n s u f f e r i n g a collision with a f u e l n u c l e u s while its e n e r g y is still a b o v e t h e f a s t fission t h r e s h o l d will increase. O n the n e g a t i v e side, the t h e r m a l utilization / will d e c r e a s e s o m e w h a t b e c a u s e the t h e r m a l flux t e n d s to b e d e p r e s s e d in t h e fuel, h e n c e y i e l d i n g less a b s o r p t i o n in the f u e l at t h e r m a l energies (again d u e to self-shielding). Since t h e r m a l a b s o r p t i o n in t h e f u e l (in c o n t r a s t to e p i t h e r m a l r e s o n a n c e a b s o r p t i o n ) u s u a l l y leads to fission, the n e t result is a d e c r e a s e in c o r e m u l t i p l i c a t i o n d u e to / . F o r t u n a t e l y this d e c r e a s e is f a r o u t w e i g h e d b y the i n c r e a s e in p. T o e x a m i n e t h e s e e f f e c t s in s o m e w h a t m o r e detail, we will n o w c o n s i d e r the i n f l u e n c e of f u e l l u m p i n g o n e a c h t e r m in t h e six-factor f o r m u l a . T h e n in later sections of this c h a p t e r , w e will t u r n to t h e m o r e p r a c t i c a l p r o b l e m of j u s t h o w s u c h e f f e c t s a r e i n c l u d e d in c o r e n e u t r o n i c s analysis. F i r s t recall t h a t 17 d e p e n d s o n l y o n the m a c r o s c o p i c cross sections c h a r a c t e r i z i n g the fuel vof F
j (10-1) j
( w h e r e the latter e x p r e s s i o n h o l d s f o r a m i x t u r e of f u e l isotopes). H e n c e o n e w o u l d n o t expect f u e l l u m p i n g to a p p r e c i a b l y a f f e c t this ratio. In a c t u a l i t y , h o w e v e r , the cross sections t h a t a p p e a r in 17 are g r o u p c o n s t a n t s c h a r a c t e r i z i n g the t h e r m a l e n e r g y g r o u p . T h e s e a r e d e p e n d e n t , of c o u r s e , o n the t h e r m a l n e u t r o n e n e r g y s p e c t r u m , a n d this s p e c t r u m d e p e n d s , in t u r n , o n the f u e l - m o d e r a t o r lattice c o n f i g u r a t i o n . H e n c e t h e r e will b e a slight m o d i f i c a t i o n in rj w h e n g o i n g to a h e t e r o g e n e o u s lattice. T h i s c h a n g e is usually i g n o r e d in less s o p h i s t i c a t e d t h e r m a l s p e c t r u m c o d e s t h a t i n c l u d e h e t e r o g e n e i t i e s via t h e r m a l d i s a d v a n t a g e f a c t o r s . M o r e e l a b o r a t e cell c a l c u l a t i o n s a c c o u n t i n g f o r s p a c e - e n e r g y e f f e c t s w i t h i n t h e cell will i n c l u d e this c o r r e c t i o n a u t o m a t i c a l l y . I n o u r earlier t r e a t m e n t of h o m o g e n e o u s s y s t e m s w e d e f i n e d t h e t h e r m a l utilizat i o n / as t h e r a t i o of the r a t e of t h e r m a l n e u t r o n a b s o r p t i o n in the f u e l to the total r a t e of t h e r m a l n e u t r o n a b s o r p t i o n in all m a t e r i a l s . T h i s d e f i n i t i o n c a n b e a p p l i e d a s well to a h e t e r o g e n e o u s c o r e b y w r i t i n g f /=
^2aF(r)
(10-2)
H e r e we a r e c o n s i d e r i n g t h e c o r e to b e m a d e u p of o n l y two types of m a t e r i a l , fuel, d e n o t e d b y t h e s u p e r s c r i p t " F " , a n d m o d e r a t o r , d e n o t e d b y " M " . ( T h e e x t e n s i o n to
402 /
THE MULTIGROUP DIFFUSION METHOD
m o r e t h a n t w o r e g i o n s will b e given later.) Since t h e c o r e is m a d e u p of a n u m b e r of i d e n t i c a l f u e l cells, w e c a n c o n s i d e r t h e a v e r a g e in E q . (10-2) as b e i n g t a k e n only over t h e v o l u m e F c e l l of o n e s u c h cell. N o w if w e r e c o g n i z e t h a t t h e m a c r o s c o p i c cross sections 2 a ( r ) a n d 2 ^ ( r ) a r e a c t u a l l y c o n s t a n t over t h e v o l u m e VF of t h e f u e l a n d VM of t h e m o d e r a t o r respectively, a n d v a n i s h elsewhere, w e c a n limit t h e r a n g e of i n t e g r a t i o n t o e a c h of t h e s e regions a n d pull t h e c r o s s sections o u t of t h e integrals t o w r i t e 2 a FJf
dH(r)
vp
/
:
7 s n
J
7
3
d3r
d r<j>(r) + ^ (
J
Vf
•
(10-3)
VM
N e x t , s u p p o s e w e d e f i n e t h e spatially a v e r a g e d f l u x in e a c h r e g i o n as
=
F JVF
=
d3r
d3r
(10-4)
Vm
T h e n w e c a n r e w r i t e t h e t h e r m a l utilization / in t e r m s of t h e s e a v e r a g e s a s
/=
p
s r ^ F _ M
_
,
(10-5)
or, d i v i d i n g b o t h n u m e r a t o r a n d d e n o m i n a t o r b y
w h e r e w e h a v e d e f i n e d t h e thermal disadvantage f l u x in t h e m o d e r a t o r to t h a t in t h e f u e l
factor
J as t h e r a t i o of t h e a v e r a g e
(10-7)
T h i s t e r m i n o l o g y arises b e c a u s e t h e t h e r m a l flux t e n d s t o b e d e p r e s s e d in t h e h i g h l y a b s o r b i n g f u e l region, l e a d i n g t o a v a l u e of f > 1. H e n c e since t h e a v e r a g e f l u x is s o m e w h a t h i g h e r i n t h e m o d e r a t o r t h a n i n t h e fuel, t h e f u e l n u c l e i a r e at a relative
J
FIGURE 10-2
A two-region unit fuel cell.
CELL CALCULATIONS FOR HETEROGENEOUS CORE LATTICES
/
403
d i s a d v a n t a g e in c o m p e t i n g with m o d e r a t o r nuclei f o r t h e c a p t u r e of t h e r m a l neutrons. T h e d e p r e s s i o n of t h e t h e r m a l flux in t h e f u e l is a g a i n a c o n s e q u e n c e of the self-shielding e f f e c t . T h a t is, n e u t r o n s b o r n in fission events in the f u e l will t e n d to t h e r m a l i z e in t h e m o d e r a t o r a n d t h e n m u s t e v e n t u a l l y d i f f u s e b a c k i n t o t h e f u e l to i n d u c e a f u r t h e r fission. H o w e v e r the highly a b s o r b i n g n u c l e i n e a r the s u r f a c e of t h e f u e l p i n t e n d to a b s o r b t h e t h e r m a l n e u t r o n s d i f f u s i n g b a c k in f r o m the m o d e r a t o r a n d h e n c e in e f f e c t shield t h e f u e l n u c l e i in the interior of t h e p i n . T h i s leads to the observed flux depression. W e c a n c o m p a r e this m o r e g e n e r a l d e f i n i t i o n of t h e r m a l utilization / with o u r earlier e x p r e s s i o n f o r a h o m o g e n e o u s system ^Fhom j*hom _
5
J
Fhom
2&
(10-8) Mhom
+ 2&
if w e c o n s i d e r t h e h o m o g e n e o u s system to consist of u n i t cells of the s a m e v o l u m e ^ceii ~ ^ f + ^ m a s o u r h e t e r o g e n e o u s cell, b u t with t h e f u e l a n d m o d e r a t o r n o w s p r e a d u n i f o r m l y over t h e cell. H e n c e w e w o u l d n o w f i n d t h e f u e l a n d m o d e r a t o r n u m b e r densities in t h e h o m o g e n e o u s cell as N
F°m=
cell
^Mm=
-
cell
'
(10-9)
If w e n o w n o t e t h a t t h e m a c r o s c o p i c cross sections f o r the h o m o g e n e o u s cell are 2
Fhom
= i V
hom
^
^Mhom
=
^hom
0
fhom—
M
W £
find
f r Q m
£ q
(1
Q_9)
2f 2f + 2aM(FM/KF)
.
(10-10)
C o m p a r i n g this w i t h o u r m o r e g e n e r a l d e f i n i t i o n f o r a h e t e r o g e n e o u s s y s t e m in E q . (10-6), w h i c h w e will n o w r e f e r to a s / h e t , w e n o t e t h a t in g e n e r a l / h e t < / h o m , since f > 1 (as t h e f l u x d e p r e s s i o n in t h e f u e l w o u l d imply). T h e r e f o r e l u m p i n g t h e fuel i n t o a h e t e r o g e n e o u s lattice will a c t u a l l y l o w e r t h e r m a l utilization, t h e r e b y d e c r e a s ing c o r e m u l t i p l i c a t i o n . O n e c a n generalize t h e c o n c e p t of t h e r m a l utilization even f u r t h e r to a c c o u n t f o r a m u l t i r e g i o n f u e l cell. C o n s i d e r , f o r e x a m p l e , a t h r e e - r e g i o n f u e l cell c o m p o s e d of fuel, clad, a n d m o d e r a t o r m a t e r i a l . T h e n in a n a l o g y with o u r t w o - r e g i o n e x a m p l e , we would write /=
= ZlV F
— . ^Vc
(10-11)
If w e n o w d i v i d e t h r o u g h b y t h e a v e r a g e f l u x in t h e fuel, w e f i n d t h a t the t h e r m a l utilization / involves t w o t h e r m a l d i s a d v a n t a g e f a c t o r s , o n e f o r t h e f u e l to m o d e r a tor, M/F, a n d o n e f o r the f u e l to clad, c/<j>F. A s y e t w e h a v e said n o t h i n g c o n c e r n i n g j u s t h o w o n e calculates these t h e r m a l d i s a d v a n t a g e f a c t o r s . Of course, it m i g h t b e a r g u e d t h a t the c o n c e p t of t h e r m a l utilization w i t h i n t h e c o n t e x t of t h e six-factor f o r m u l a h a s very limited utility aside f r o m c r u d e survey estimates. W e will see later, h o w e v e r , t h a t the d i s a d v a n t a g e f a c t o r f c a n b e u s e d to spatially a v e r a g e t h e r m a l g r o u p c o n s t a n t s over u n i t fuel cells, a n d h e n c e p l a y s a n e x t r e m e l y i m p o r t a n t role in r e a c t o r design. H e n c e w e will d e v o t e a c o n s i d e r a b l e a m o u n t of a t t e n t i o n t o w a r d its c a l c u l a t i o n in t h e n e x t
404 /
THE MULTIGROUP DIFFUSION METHOD
section. W e will e x a m i n e b o t h a p p r o x i m a t e ( " q u i c k a n d d i r t y " ) w a y s to e s t i m a t e f a s well a s m o r e e l a b o r a t e s c h e m e s b a s e d o n t r a n s p o r t t h e o r y a n d c o l l i s i o n probability methods. Since the f a s t n e u t r o n m f p is typically large c o m p a r e d to the scale of lattice d i m e n s i o n s , o n e usually n e e d o n l y c o n s i d e r t h e t h e r m a l n o n l e a k a g e p r o b a b i l i t y /* T N L . L a t t i c e e f f e c t s e n t e r p r i m a r i l y t h r o u g h a c h a n g e in t h e t h e r m a l g r o u p c o n s t a n t s , w h i c h a c c o u n t s f o r spatial flux v a r i a t i o n w i t h i n a cell. I n p a r t i c u l a r o n e c a n s h o w t h a t t h e d i f f u s i o n l e n g t h c h a r a c t e r i z i n g a h e t e r o g e n e o u s c o r e is a p p r o x i m a t e l y j u s t t h a t c h a r a c t e r i z i n g a p u r e m o d e r a t i n g m e d i u m d e c r e a s e d b y the t h e r m a l utilization (1 — f ) (see P r o b l e m 10-3). H o w e v e r since l e a k a g e f r o m a large t h e r m a l r e a c t o r c o r e is a relatively m i n o r effect, w e will a v o i d a n explicit d i s c u s s i o n of h e t e r o g e n e o u s e f f e c t s o n PTNL since it will h a v e a small e f f e c t o n core m u l t i p l i c a t i o n ( a n d will b e a c c o u n t e d f o r b y u s i n g cell-averaged g r o u p c o n s t a n t s in a n y event). P e r h a p s t h e m o s t s i g n i f i c a n t e f f e c t d u e to h e t e r o g e n e o u s a r r a n g e m e n t of f u e l in a t h e r m a l r e a c t o r is a s i g n i f i c a n t i n c r e a s e in t h e r e s o n a n c e e s c a p e p r o b a b i l i t y p. T h i s m o d i f i c a t i o n o c c u r s as a c o n s e q u e n c e of t w o p h e n o m e n a . F i r s t t h e r e is a g e o m e t r i cal e f f e c t arising b e c a u s e t h e p h y s i c a l s e p a r a t i o n of the f u e l a n d t h e m o d e r a t o r will a l l o w s o m e n e u t r o n s to slow d o w n w i t h o u t ever e n c o u n t e r i n g the f u e l . T h i s e f f e c t is of s e c o n d a r y i m p o r t a n c e to the p h e n o m e n o n of self-shielding, h o w e v e r . T o u n d e r s t a n d this s e c o n d e f f e c t m o r e clearly, c o n s i d e r the d i a g r a m of the spatial d e p e n d e n c e of t h e n e u t r o n f l u x at several d i f f e r e n t energies c h a r a c t e r i z i n g a r e s o n a n c e given in F i g u r e 10-1. Of c o u r s e at fission energies w e m i g h t e x p e c t the f l u x to p e a k in t h e f u e l since t h e fission sources are c o n f i n e d to the fuel. H o w e v e r o n c e w e h a v e d r o p p e d in e n e r g y m u c h b e l o w t h e fission e n e r g y , w e will b e g i n to see a flux d e p r e s s i o n in the f u e l . T h i s arises b e c a u s e the fission n e u t r o n s m u s t e s c a p e t h e f u e l p i n i n t o t h e m o d e r a t o r in o r d e r to b e a p p r e c i a b l y m o d e r a t e d . ( N u c l e a r inelastic s c a t t e r i n g f r o m f u e l i s o t o p e s as well as elastic s c a t t e r i n g f r o m light i s o t o p e s s u c h a s o x y g e n a d m i x e d i n t o the f u e l c a u s e s o m e m o d e r a t i o n , b u t this is a s e c o n d a r y e f f e c t in t h e r m a l r e a c t o r s . ) H e n c e t h e m o d e r a t o r p r e s e n t s effectively a v o l u m e t r i c s o u r c e of n e u t r o n s a p p e a r i n g at the lower energies. T h e s e n e u t r o n s m u s t t h e n either d o w n s c a t t e r to even lower energies or d i f f u s e i n t o the fuel where they are absorbed. T h e f u e l p r e s e n t s a very highly a b s o r b i n g m e d i u m to t h e n e u t r o n s of r e s o n a n c e e n e r g y d i f f u s i n g in f r o m t h e m o d e r a t o r . T h i s a b s o r p t i o n is s u f f i c i e n t l y s t r o n g t h a t m a n y of the n e u t r o n s i n c i d e n t o n t h e f u e l a r e a b s o r b e d in t h e o u t e r layers of the f u e l p i n . H e n c e t h e f u e l n u c l e i in the p i n i n t e r i o r see a s o m e w h a t d e p r e s s e d f l u x d u e to the e f f e c t i v e shielding p r e s e n t e d b y the f u e l nuclei n e a r the f u e l p i n s u r f a c e . S u c h self-shielding is p r e s e n t to a c e r t a i n d e g r e e at all energies b e l o w fission energies. H o w e v e r it b e c o m e s m u c h m o r e p r o n o u n c e d w h e n the f u e l a b s o r p t i o n cross section is l a r g e — s u c h as at a r e s o n a n c e in the a b s o r p t i o n cross section o r in the t h e r m a l e n e r g y r a n g e . T h i s e f f e c t is q u i t e p r o n o u n c e d . F o r e x a m p l e , t h e r e s o n a n c e i n t e g r a l c h a r a c t e r i s t i c of n a t u r a l u r a n i u m u n i f o r m l y m i x e d w i t h m o d e r a t i n g m a t e r i a l is a b o u t 2 8 0 b ; l u m p i n g t h e u r a n i u m , w e c a n r e d u c e the r e s o n a n c e integral to a v a l u e of 9 b — a 3 0 - f o l d r e d u c t i o n . W e will see later t h a t o n e c a n usually write the e f f e c t i v e r e s o n a n c e integral in t h e f o r m
/-c, + c 2 (^)
(10-12)
w h e r e AF is the s u r f a c e a r e a of t h e f u e l l u m p a n d MF is its m a s s ( p r o p o r t i o n a l to its
CELL CALCULATIONS FOR HETEROGENEOUS CORE LATTICES
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405
v o l u m e ) . A s ( A ¥ / M ¥ ) d e c r e a s e s — c o r r e s p o n d i n g to m o r e highly h e t e r o g e n e o u s lattice c o n f i g u r a t i o n s — t h e r e s o n a n c e integral d e c r e a s e s . T o t r e a t t h e e f f e c t s of h e t e r o g e n e i t i e s in r e s o n a n c e a b s o r p t i o n r e q u i r e s t h e use of several c o n c e p t s f r o m t r a n s p o r t t h e o r y . Since this s u b j e c t is of c o n s i d e r a b l e i m p o r t a n c e in t h e r m a l r e a c t o r design, w e will discuss it in s o m e d e t a i l in Section 10-111. T h e f a s t fission f a c t o r c is also i n c r e a s e d s o m e w h a t b y g o i n g to a h e t e r o g e n e o u s lattice. T o u n d e r s t a n d w h y , o n e n e e d o n l y recall t h a t t h e e n e r g y of a n e u t r o n m u s t b e a b o v e a c e r t a i n t h r e s h o l d in o r d e r to i n d u c e a fast fission r e a c t i o n in a f i s s i o n a b l e i s o t o p e s u c h as 2 3 8 U . By l u m p i n g t h e fuel, o n e effectively i n c r e a s e s the p r o b a b i l i t y t h a t a h i g h - e n e r g y fission n e u t r o n will e n c o u n t e r a f u e l n u c l e u s b e f o r e it is slowed d o w n b e l o w t h e f a s t fission e n e r g y t h r e s h o l d , either b y elastic s c a t t e r i n g collisions w i t h m o d e r a t o r nuclei or inelastic s c a t t e r i n g f r o m f u e l nuclei. I n s u m m a r y , t h e n , l u m p i n g t h e f u e l i n t o a h e t e r o g e n e o u s lattice c a n significantly i n c r e a s e k ^ f o r n a t u r a l a n d slightly e n r i c h e d (^£5%) u r a n i u m cores. T h e d o m i n a n t e f f e c t is c o n t a i n e d in t h e b e h a v i o r of t h e t h e r m a l utilization / a n d t h e r e s o n a n c e e s c a p e p r o b a b i l i t y p. T o illustrate these t r e n d s , w e h a v e i n d i c a t e d in F i g u r e 10-3 the v a r i a t i o n of e a c h of t h e f a c t o r s in t h e f o u r - f a c t o r f o r m u l a w i t h h e t e r o g e n e i t y (in this case m e a s u r e d b y t h e f u e l p i n p i t c h / d i a m e t e r ratio) f o r a typical P W R lattice.
Pitch/diameter
FIGURE 10-3.
Effect of fuel lumping on k^.
406 /
THE MULTIGROUP DIFFUSION METHOD
B. Core Homogenization T o b e m o r e specific, let us o u t l i n e o n e possible a p p r o a c h to t h e t r e a t m e n t of c o r e lattice e f f e c t s . W e b e g i n b y n o t i n g t h a t r e a c t o r cores h a v e a r e g u l a r o r p e r i o d i c l a t t i c e s t r u c t u r e in w h i c h o n e s u b e l e m e n t o r so-called unit cell is r e p e a t e d t h r o u g h o u t the core. F o r e x a m p l e , a f u e l s u b a s s e m b l y or g r o u p of f u e l s u b a s s e m b lies s u c h as t h o s e s k e t c h e d in F i g u r e 10-4 c o u l d b e r e g a r d e d as a u n i t cell. O n a m o r e d e t a i l e d scale, a given f u e l e l e m e n t a n d a d j a c e n t c o o l a n t c h a n n e l m i g h t b e c h o s e n as t h e u n i t cell ( F i g u r e 10-5). I n f a c t , f o r r e a c t o r types u s i n g c o a t e d p a r t i c l e fuels, o n e is f r e q u e n t l y r e q u i r e d to c o n s i d e r a f u e l m i c r o s p h e r e o r " g r a i n " as a u n i t cell. Of c o u r s e the r e a c t o r lattice s t r u c t u r e is n o t precisely r e g u l a r b e c a u s e of c o n t r o l or i n s t r u m e n t a t i o n devices, n o n u n i f o r m f u e l l o a d i n g s a n d c o o l a n t densities, c o r e b o u n d a r i e s , a n d so o n . T h e s e lattice irregularities a r e usually h a n d l e d w i t h i n the g l o b a l m u l t i g r o u p d i f f u s i o n c a l c u l a t i o n of t h e core f l u x d i s t r i b u t i o n . F o r the p u r p o s e s of o u r p r e s e n t analysis, w e will i g n o r e these gross lattice irregularities a n d a s s u m e t h a t t h e c o r e c a n b e r e p r e s e n t e d as a n i n f i n i t e a r r a y of i d e n t i c a l lattice cells. T h e essential s c h e m e is t h e n to p e r f o r m a d e t a i l e d c a l c u l a t i o n of the flux d i s t r i b u t i o n in a given u n i t cell of t h e l a t t i c e — u s u a l l y a s s u m i n g t h a t t h e r e is z e r o n e t n e u t r o n c u r r e n t a c r o s s the b o u n d a r y of t h e cell (using a r g u m e n t s b a s e d o n the s y m m e t r y of t h e lattice). T h e v a r i o u s m u l t i g r o u p cross sections c h a r a c t e r i z i n g m a t e r i a l s in t h e cell a r e t h e n spatially a v e r a g e d over t h e cell, u s i n g the flux d i s t r i b u t i o n as a w e i g h t i n g f u n c t i o n . I n this w a y o n e c a n c h a r a c t e r i z e t h e cell b y e f f e c t i v e g r o u p c o n s t a n t s a c c o u n t i n g f o r t h e i n h o m o g e n e o u s f l u x d i s t r i b u t i o n in t h e cell. T h i s s c h e m e essentially r e p l a c e s t h e a c t u a l u n i t cell b y a n e q u i v a l e n t homogeneous u n i t cell c h a r a c t e r i z e d b y these e f f e c t i v e cross sections. F o r e x a m p l e , o n e usually b e g i n s b y c o n s i d e r i n g a typical f u e l c e l l — n a m e l y , f u e l p l u s c l a d plus c o o l a n t . T h e f u e l cell is first r e d u c e d to a n e q u i v a l e n t cell of simpler g e o m e t r y to e x p e d i t e c a l c u l a t i o n s (see F i g u r e 10-6a) ( t a k i n g c a r e to p r e s e r v e volume fractions). O u r p r i m a r y i n t e r e s t is usually c o n c e r n e d w i t h the g e n e r a t i o n of e f f e c t i v e fast a n d t h e r m a l g r o u p c o n s t a n t s f o r s u c h a cell. I n t h e f a s t r a n g e , t h e h e t e r o g e n e i t i e s e n t e r p r i m a r i l y as m o d i f i c a t i o n s to the r e s o n a n c e e s c a p e p r o b a b i l i t y a n d f a s t fission cross sections. H e n c e it is usually s u f f i c i e n t t o s i m p l y p e r f o r m t h e u s u a l i n f i n i t e m e d i u m f a s t s p e c t r u m c a l c u l a t i o n , t a k i n g care, h o w e v e r , to a c c o u n t f o r h e t e r o g e n e o u s e f f e c t s in r e s o n a n c e a b s o r p t i o n a n d f a s t fission via t e c h n i q u e s t h a t will b e d i s c u s s e d later in this c h a p t e r . The m u c h shorter m e a n free p a t h characterizing thermal neutrons necessitates a s o m e w h a t m o r e d e t a i l e d t r e a t m e n t of h e t e r o g e n e i t i e s in d e t e r m i n i n g t h e t h e r m a l f l u x s p e c t r u m in a f u e l cell. F o r less d e t a i l e d c o r e c a l c u l a t i o n s , o n e c a n f r e q u e n t l y get b y w i t h s i m p l y m o d i f y i n g t h e results of a n i n f i n i t e m e d i u m t h e r m a l s p e c t r u m c a l c u l a t i o n (e.g., S O F O C A T E ) to a c c o u n t f o r t h e v a r i a t i o n s in a v e r a g e f l u x in the fuel,
CELL CALCULATIONS FOR HETEROGENEOUS CORE LATTICES
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407
Fuel assembly w i t h rod—cluster c o n t r o l
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V BWR assemblies
FIGURE 10-4.
Typical fuel assemblies.
validity w h e n r o d n e i g h b o r s i n c l u d e w a t e r holes, p o i s o n shims, c o n t r o l rods, or P u - l o a d e d f u e l pins, since o n e t h e n n e e d s to a c c o u n t f o r cell-to-cell leakage. 3 F o r c o a t e d p a r t i c l e fuels, s u c h a s t h o s e utilized in t h e H T G R , a d d i t i o n a l c o r r e c t i o n f a c t o r s m u s t b e u s e d t o a c c o u n t f o r the h e t e r o g e n e i t y r e p r e s e n t e d b y the m i c r o s c o p i c g r a i n s t r u c t u r e of t h e f u e l in the c a l c u l a t i o n of cell-averaged t h e r m a l spectra and group constants.4 T h e n e x t s t e p in t h e analysis of t h e c o r e is to c o n s i d e r a typical f u e l a s s e m b l y or g r o u p i n g of f u e l assemblies, i n c l u d i n g c o n t r o l o r s h i m e l e m e n t s ( F i g u r e 10-6b). T h e f e w - g r o u p c o n s t a n t s c a l c u l a t e d f o r the f u e l cell c a n b e used to d e s c r i b e m o s t of the a s s e m b l y , w i t h t h e e x c e p t i o n of c o n t r o l m a t e r i a l w h i c h requires r a t h e r specialized
F I G U R E 10-5.
Typical unit fuel assemblies.
(a) Fuel—cell homogenization
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oommoo o#oo#o
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OOOOOO
•
•
(b) Fuel—assembly homogenization
(c) Core homogenization FIGURE 10-6.
408
Reactor-core homogenization.
CELL CALCULATIONS FOR HETEROGENEOUS CORE LATTICES
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409
t e c h n i q u e s . U s u a l l y a d e t a i l e d m u l t i g r o u p t w o - d i m e n s i o n a l d i f f u s i o n or t r a n s p o r t c o d e is u s e d to d e t e r m i n e t h e f l u x in* such a n a s s e m b l y , a n d t h e n o n c e a g a i n these fluxes a r e u s e d to g e n e r a t e a s s e m b l y a v e r a g e d g r o u p c o n s t a n t s . T h e f i n a l step is to use either these a s s e m b l y - a v e r a g e d g r o u p c o n s t a n t s (or, in very d e t a i l e d c a l c u l a t i o n s , the original f u e l cell g r o u p c o n s t a n t s ) to d e t e r m i n e the f l u x a n d p o w e r d i s t r i b u t i o n over t h e e n t i r e core. T h e s y m m e t r y of the core will f r e q u e n t l y allow o n e to c o n s i d e r only o n e q u a d r a n t in detail. It m a y o c c a s i o n a l l y b e d e s i r a b l e to h o m o g e n i z e t h e c o r e still f u r t h e r to facilitate gross survey calculat i o n s (as in F i g u r e 10-6c).
C. Cell-Averaging Techniques T h e g o a l of m o s t of o u r analysis in this c h a p t e r will b e to c a l c u l a t e effective g r o u p c o n s t a n t s t h a t h a v e b e e n spatially a v e r a g e d over t h e f l u x d i s t r i b u t i o n in a lattice cell a n d t h e r e f o r e c a n b e u s e d t o c h a r a c t e r i z e t h e cell in s u b s e q u e n t m u l t i g r o u p d i f f u s i o n c a l c u l a t i o n s in w h i c h t h e cell s t r u c t u r e is i g n o r e d . T o discuss this s u b j e c t of cell a v e r a g i n g in m o r e detail, let us c o n s i d e r t h e c a l c u l a t i o n of the c e l l - a v e r a g e d g r o u p c o n s t a n t c h a r a c t e r i z i n g a t w o - r e g i o n cell in w h i c h m a t e r i a l c o m p o s i t i o n is u n i f o r m in e a c h region. W e will a g a i n d e n o t e these r e g i o n s b y " M " a n d " F " f o r c o n v e n i e n c e (refer to F i g u r e 10-2.) T h e cell-averaged t h e n b e d e f i n e d as
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