ORNL-TM-4179.txt

ORNL-TM-4179

 

IMPROVED REPRESENTATION OF SOME
ASPECTS OF
CIRCULATING-FUEL REACTOR KINETICS

B. E. Prince

 

 

 

e 92

OAK RIDGE NATIONAL LABORATORY

-~ OPERATED BY UNION CARBIDE CORPORATION e FOR THE U.S. ATOMIC ENERGY C(OMMISSION

 
 

 

This report was prepared as an account of work sponsored by the United
States Government. Neither the United States nor the United States Atomic
Energy Commission, nor any of their employees, nor any of their contractors,
subcontractors, or their employees, makes any warranty, express or implied, or
assumes any legal liability or responsibility for the accuracy, completeness or
usefulness of any information, apparatus, product or process disclosed, or
represents that its use would not infringe privately owned rights,

 

 
ORNL-TM=-4179

Contract No. W-7405-eng—~26

REACTOR DIVISION

IMPROVED REPRESENTATION OF SOME ASPECTS
OF CIRCULATING-FUEL REACTOR KINETICS

B. E. Prince

MAY 1973

NOTICE

This report was prepared as an account of work
sponsored by the United States Government. Neither
the United States nor the United States Atomic Energy
Commission, nor any of their employees, nor any of
their contractors, subcontractors, or their employvees,
makes any warranty, express or implied, or assumes any
legat lability or responsibility for the accuracy, com-
pleteness or usefulness of any information, apparatus,
product or process disclosed, or represents that its use
would not infringe privatety owned rights.

      

 

OAK RIDGE NATIONAL LABORATORY
ODak Ridge, Tennessee 37330
operated by
UNION CARBIDE CORPORATION
for the
U.S. ATOMIC ENERGY COMMISSI@N

DISTRIBUTION OF THIS DOCUMENT 18 UN‘].\MTFER )

R%

 
iii

CONTENTS

Preface . « ¢ ¢« ¢ o o o o ¢ o &
AbStract,. o« o o « o o o o o o o
Introduction: + « + o o &+ « ¢ o o
Background. + + « « « o o ¢ o+ o
Mathematical Description. . . . .

Example of Delayed Neutron Kernal

. . . . » L] e

Calculations.,

Discussion of Results and Future Extensions . .

Appendix . . s o v ¢ 0 s e 0 s e

References. + ¢« s s s o o & s & »

Page

v NN g

18
22
26
32
PREFACE

P. N. Haubenreich

The formulations that are presented here were worked out by Blynn
Prince in 1968 in connection with his analysis of the kinetics of the
Molten~Salt Reactor Experiment with 222U fuel, Although he made some
significant progress toward an improved mathematical description of
circulating—fuel reactor kinetics, the work was suspended and these re-
sults were not previously reported because of a contraction of reactor
analysis effort in the Molten~Salt Reactor Program that involved the
assignment of the author to a different program, Whether or not molten
salt reactor development work is continued in the future, the results
contained here may be of interest from the standpoint of theoretical
reactor kinetics analyses. They also indicate a starting point that
could lead to improved, practical computations of molten-salt reactor

kinetics. As such they are recorded here for possible future use.
IMPROVED REPRESENTATION OF SOME ASPECTS
OF CIRCULATING-FUEL REACTOR KINETICS

B. E. Prince

Abstract

The general space-energy dependent reactor kinetic equations for a
circulating—fuel reactor were studied to help determine the type of mathe-
matical representation most appropriate for analysis and computation of
reactor transient behavior. It is shown that, with inclusion of fluid
transport terms in these equations, the application of the usual adjoint-
weighting and integration techniques used to derive '"global" kinetic equa-
tions from the general equations do not result in the usual set of time-
dependent ordinary differential equations associated with stationary-fuel
reactor-kinetics. However, a time-dependent integro-differential equation
describing the kinetics of the neutron population can still be obtained.
General formulas for calculating the weighted delayed-neutron precursor
kernels in this equation are given, and a numerical example is included
which illustrates the nature of the solution. Directions are also sug-
gested for calculating the analogous weighted temperature-distribution
kernels for analysis of power-temperature kinetics. The qualitative in-
fluence of fluid mixing on the kernels is described, and the connections
between the distributed parameter and lumped-parameter representations
of the system kinetics are also discussed,
INTRODUCTION

A complete mathematical description of the nuclear fission chain re-
action in any power reactor is a formidable task, which is further compli-
cated by circulation of the fuel. Fortunately, for many purposes greatly
simplified descriptions are sufficient — as Weinberg and Wigner point out,
the first full-scale reactors (Hanford) were designed with desk calcu-
lators and slide rules.? More detailed analyses are increasingly desira-
ble, however, as reactor designs are refined to obtain higher performance
without compromising reliability and safety. As part of the vast growth
in reactor technology, analysis of stationary-fuel reactors has evolved
to a high level. Representation of the unique aspects of the kinetics of
circulating-fuel reactors has naturally received much less attention and
so has advanced to a lesser degree. Methods were developed for repre-
senting the latest circulating-fuel reactor, the Molten-Salt Reactor Ex-
periment, that proved to be quite adequate for that purpose. But design
of large-scale, high-performance MSR power plants would undoubtedly lead
to demands for improved kinetics calculations. The work described in this

report is intended to help lay the groundwork for these calculations.

BACKGROUND

In the analysis of reactor dynamics, wide use has always been made
of the so-called "point" kinetics model, The great utility of this space~-
independent model is largely a result of the ability to decompose the prob-
lem of calculating the gross details of the time dependence of a system
from the multi-dimensional problem of calculating the neutron distribution.
Although the early reactor physics literature contains some discussion of
the relation between the point kinetics model and the complete mathematical

?® to the writer's

description of the time-~dependent neutron population,z’
knowledge, the first rigorous exposition of the relation, showing its deri-
vation from the time-dependent Boltzman equation, and describing the cri-
teria for the point-kinetics equations to provide a precise description of

the system motion, was given in 1958 by A. F. Henry.“
The derivation of the point kinetics equations is ordinarily carried
out for the case of a stationary-fueled reactor. Although the point ki-
netics approximation has been applied to circulating-fuel reactors, if one
begins at the most basic level to describe a circulating fluid-fueled re-
actor, it is somewhat more natural to consider an Eulerian type of descrip-
tion of the basic mathematical relations between the important variables
such as flux, precursor densities, and temperatures. One is then led to
inquire what differences in mathematical formalism from the standard point-
kinetics equations are suggested for the practical analysis of circulating-
fuel-reactor kinetics problems.

The reactor physics literature describes many different investigations
of the unusual aspects of circulating-fuel-reactor-kinetics, of which ref-
erences 5—10 are significant examples. These unusual aspects are especi-
ally well identified in a 1962 article by B. Wolfe® in which he considers,
inter alia, the direct effects of motion imparted to the entire neutron
population by the moving fluid. Wolfe concludes that, except in very se-
vere reactor accident conditions, the special reactivity effects so intro-
duced are quite small. On the other hand, in the calculation of the de-
layed neutron precursor distributions and effectiveness in a circulating-
fuel reactor, he reemphasizes the importance of an accurate mathematical
description of the fluid motion effects in kinetics analysis.

Of the variety of mathematical models which have been used in studies
of the kinetic behavior of circulating fuel reactors, most can be desig-
nated as ''special purpose approximations,' useful for the analysis of par-
ticular characteristics or regimes of the system motion, but each neglecting
certain features of the physical system which would be required for other
applications. For example, analyses focusing mainly on determining the
conditions for ultimate dynamic stability of the reactor core will often
neglect the effects of the delayed neutrons. In another case, studies of
reactor transients under abnormal, or accident conditions, which occur on
a time scale less than, or comparable to, the transit time of a fluid par-
ticle through the core, can often neglect the description of the system
external to the core, together with any transients in the temperature or
precursor concentrations in the fluid re-entering the core. As an example,

the ZORCH program, developed for studies of the nuclear safety of the
MSRE,'* is based on this approach. ZORCH uses a simplified treatment of
the delayed neutron precursor dynamics based on an ''effective'" delayed
fraction, which gives the correct initial normalization for the reactivity
margin between delayed and prompt critical. The main effort is then given
to a numerical treatment of the distributed parameter problem of heat con-
vection and temperature feedback during the transient.

Other investigations connected with the MSRE were aimed at describing
the reactor dynamic characteristics appropriate to a time scale comparable
to, or larger than, the core transit time.'? Here the entire circulating
system, including the heat exchanger, must be included in the description.
The general approach has been to develop a "lumped parameter" model for the
system, which provides an adequate description of the dynamics of the power,
precursor concentrations, and temperatures, for the purposes intended.

The various investigations of the kinetics of the MSRE and subsequent
MSR designs are briefly described in a recent memorandum by Haubenreich.'?®
All involve approximations of one kind or another that limit the general
applicability of the methods. If further development of molten-salt re-
actors takes place, it seems likely that kinetics-—computational models
which are of greater generality and flexibility would ultimately be re-
quired for the analysis of routine nuclear operations, kinetics experi-
ments, and unusual occurrences. The investigations reported here were
initiated with this general philosophy in mind. They are aimed at ana-
lyzing some of the most important consequences of the fuel motion in prac-
tical kinetics computations and the interpretation of kinetics experiments
for circulating-fuel reactors. Although differing in emphasis, the ap-
proach has much in common with some of the past investigations mentioned
above. However, we wish to focus on certain aspects of the differences
in mathematical formulation and practical computation with the kinetics
equations which, in our opinion, previous studies have not sufficiently
developed and clarified. In this writing, we shall consider in detail
only the simplest case of interest, the case of negligible temperature
feedback effects, or the '"zero-power' case. However, following the dis~
cussion of this case, we will indicate some connections to the case of

temperature~dependent kinetics.
MATHEMATICAL DESCRIPTION

In the case where one is able to neglect the direct effects of fluid
motion on the neutron population, as described in the preceding section,
one can show that the main line of Henry's derivation can be carried over
to the circulating-fuel reactor, and that the form obtained for the re-
sulting '"global," or space-lethargy-integrated kinetic equation governing
the magnitude of the neutron population is the same as in the stationary
fuel case. This is demonstrated mathematically in the Appendix of this
report. In each case, the resulting kinetic equation for the neutron popu-

lation magnitude is,

 

dT _ p - B
e - Tl oMo (1)

where T(t) is a time-dependent amplitude function, obtained by factoring
the general transient flux distribution, ¢(r, u, t)), into a product of
T(t) and a normalized "shape" function, ¢(xr, u, t). In Eq. 1, the source
terms,,lici, associated with decay of delayed neutron precursors have the

form,

e () = = [ [ C, ) £, @ 6, (z, w) dr du . (2)
R u
Here, Ci (r, t) is the local density of precursors for the :'Lth delayed
group, fdi(u) are the l:thargy spectra of delayed neutron emission (each
normalized to unity), ¢, (r, u) is the solution of the adjoint equation
for a reference reactor condition, and R is the reactor volume. As Henry's
derivation shows (see Appendix), the parameters p(t), A(t), and B(t) are
defined quantities which intrinsically require knowledge of the time-
dependent neutron distribution for their exact calculation, but which are
useful because they can be closely approximated by simpler indirect calcu-
lations, in many practical cases., The parameter, p(t), is the reactivity
change, relative to a reference, stationary state of the reactor, where
there is no circulation of the fuel. The parameter, A(t), is the prompt-

neutron generation-time, and B(t) is the effective delayed neutron fraction,
weighted according to the lethargy spectra of delayed neutron emissions.
Mathematical definitions for all these quantities are given in the Appen-
dix. The factor F(t), is a normalized rate-of-production (of prompt neu-
trons plus precursors). This factor is included in the definition of p,
A, and E, but in such a way that the ratio (p - B)/A in Eq. (1), and the
product A F in Eq. (2) are independent of its magnitude.
The important difference introduced in the case when the fuel is cir-

culating is in the equation governing Ci (r, t). The latter now has the

form of a continuity equation,

3C,
1
== = B, PO -1 C -VovVC (3)

where P is a time-dependent linear operator on the flux distribution, such
that Bi P ¢ (r,t) is the total production rate of ith group precursors at
position r and time t. (Here, P can be regarded a linear integral operator
in the lethargy, which may also depend on position.) The last term on the
right<hand side of Eq. 3 represents the spatial transport of precursors

by fluid motion, with V as the velocity of the fluid.

In applications to a circulating~fuel reactor such as the MSRE, where
the fuel motion was in channels parallel to the core axis, it is sufficient
to consider the one-dimensional version of the transport term, VBCifaz in
Eq. 3. Here, the velocity within an individual channel is assumed to be
uniform across the channel, equal to the average axial velocity of the
fluid; the velocity may, however, vary according to the radial position
of the channel within the reactor. For practical purposes, therefore, our
problem is one of including an adequate mathematical treatment of the re-
sulting partial differential equation into the calculation of the global
quantities, ci(t), defined by (2).

As a starting point for mathematical treatment of Eq. (3), the general
time~dependent flux distribution, &(r, u, t) may, as in the derivation of
Eq. (1), be written in form of a product, T(t)¢(x, u, t). Then, the source

term in Eq. (3) becomes,
= Bi T(t) G(_I_'_st) ’ (4)

where G(r,t) is a normalized, time-dependent distribution of fissions in
the reactor.

Now, in the analysis of a number of reactor kinetic experiments, we
are interested in describing situations where the core properties do not
vary markedly during the transient. For these situations, we can approximate
the production operator by its time-average values, P, during the transient.
It is then conceptually useful to represent the time~dependent normalized
flux distribution, ¢(r, u, t) by an expansion in a set of basis functions,
appropriate to the boundary conditions on the reactor. Although there is
some flexibility in the choice of these basis functions, one possible choice
is that of the eigenfunctions of the time-independent problem (i.e., the
neutron flux equation with the neutron multiplication parameters adjusted
to obtain a stationary solution), corresponding to the average material
properties during the transient, In this approach, the lead term in the
expansion can be chosen to approximate the asymptotic, or persisting neu-
tron distribution which would be associated with this material configura-

tion,** Thus, if we write

0z, u, €)= ) A(E) ¢ (x, W), (5)
k=0
Pé=P¢ = ) A(t)P¢ (r, u,
k=0 k k
= ¥ A(t) ¢ (), (6)
k=0 © Kk

then a useful approximation for the treatment of many time-dependent prob-
lems may be obtained by dropping all but the lead terms in the above ex-
pansions. In the physically time-separable case (i.e., the case where the

reactor flux is changing on a stable period), the single-term approximation
® Although this approximation also implies

becomes an exact description.’
that we limit consideration to problems where the initial and asymptotic
flux distributions do not differ markedly, as indicated above, many sig-
nificant kinetics problems are subsumed under this category. For these
cases where a single term approximation is sufficient, Ao (t) may be chosen
equal to unity by appropriate normalization of the fission distribution,
Go(r), and the time-dependence of the precursor source term, Eq. (4), is
entirely contained in T{t). 4

Alternatively, it may be necessary in some instances to include more
than one term in the expansion representation of the flux. For example,
another possible approximate procedure would represent the flux as a linear
combination of two flux functions, appropriate to the “initial" and “final"
configurations of the reactor. The retention of more than one term, or
“mode™ in the flux expansion generally leads to a system of neutron popu-
lation amplitude equations, as opposed to the single kinetic equation of
the form (1). In this type of description, however, note that Eq. (3) is
linear, and superposition of solutions corresponding to individual source
terms of the form Bi T(t) G(r) can always be applied.

Stemming from the arguments given above, we will consider the mathe-
matical treatment of Eq. {3) for kinetics problems where space-time separa-
bility of the source term can be assumed to hold. To simplify our nota-
tion, henceforth, we drop further consideration of the expansion subscript,

and rewrite Eq. (3) as,

3¢, 3C,
—_— - —_— = {
Tt A C TV B, T(t) G(r) . (7

To complete the mathematical description of the problem, we require
the boundary conditions on Eq. (7). At a given instant of time, the de-
layed precursor concentration in the fluid must be continuous around the
circulation path; moreover, in that part of the circulating system which
is "out'" of the neutron flux (i.e. beyond the boundary where the neutron
flux distribution specified in Eq. (5) vanishes), the delayed emitter con-

centrations are governed by the homogeneous form of Eq. (7), where the
right~hand side is set equal to zero and z is considered to be a more
general position variable, parallel to the direction of average flow. To

a close approximation, the flow velocity V can be assumed constant in vari-
ous subregions of the circulation path, and to undergo rapid transitions
between these regions (e.g. between the core and the external piping).

As determined by Eq. 7, the delayed emitter concentrations Ci are de-
pendent on three space dimension variables, through the source distribution
G(r). Our present interest is in applications where channelled flow in
cylindrical geometry with near—azimuthal symmetry is appropriate such as
was the case for the MSRE. For this case, in addition to the time depen-
dence, the delayed emitter concentrations will vary with axial location
along a channel and with the radial position of the channel within the
core. With the MSRE as an example, it is clear that the hydraulic design
of the circulating system has the effect of radially smoothing and aver-~
aging of the concentrations exiting at a given instant from the channelled
region, and of providing essentially a radially uniform concentration of
emitters entering the channels. Because of this feature, it is necessary
to carry out the integration of Eq. 7 along specified channels, radially
average the concentrations exiting from the channels, and then continue
the integration along the remainder of the path of circulation. The treat-
ment of the radial dependences presents no problem in principle, although
the mechanics of the computation become more involved. Because we wish to
focus attention on certain other aspects of the mathematical treatment of
Eq. 7, we limit consideration here to the case where the flux and fission
distributions depends on only one space variable, corresponding to the
axial direction of flow.

Laplace transform theory provides a convenient and general approach
to the treatment of Eq. 7. To apply this approach, it is useful to first
separate the problem of obtaining the initial conditions in time, the dis-
tributions Ci (z,t=0) = Cio(z), from that of solving the time-dependent
equation. In many cases of physical interest, steady-state conditilons
will prevail at t=o, and the Cio(z) are determined by,

BCio

e : = a £
V=7t G By T/0) G(z) . (8)
10

Subtracting Eq. 8 from Eq. 7, we obtain a similar partial differential
equation for the change in emitter concentrations, Ei = Ci - C, , which
io

has zero initial conditions on the dependent variable,

oE,

JE, i
+?\iEi+V"a—-z-— =

1

9t (9)

B, (T{t) - T(0)] c(2)

Although the Laplace transform technique can be applied directly to
the solution of Eq. 9, one should first observe the T(t) is not an arbi-
trarily specified function of time; rather, as indicated previously, it
is determined by the "global" time-dependent equation for the neutron
population, required to complete the description of the system dynamics.
This latter equation and the delayed emitter equations are coupled through
terms of the form (2). Eliminating the dependent wvariables Ci by solving

Eqs. 7 in terms of T(t) is equivalent to replacing the space-lethargy-

integrated, system-kinetics equations
ential equation in the time variable.

obtain the solution of Eq. 9 in terms

by a single Volterra integro-differ-
A direct route to this end is to

of the "impulse' response,'® the

response when the source term in Eq. 9 is concentrated as a delta function

 

at t = §. Thus, if we denote Ei by Ki for this special case,
BKi BKi
o + Ai Ki + Ifsz— Bi & (t-£) G(z) 1in core region (10)
= 0 in external piping.
The same initial and boundary conditions apply to Eqs. 9 and 10. Once the

impulse response of the system is obtained, by using linear superposition
and the properties of the delta function, we may set
t
E, (z,8) = [K (z,t-8) [T(§) - T(o)] dg (11)
0

and from the above definition of Ei’ together with the solution of Eq. 8,
11

t
c, (z,t) = ¢, (z) + £ K, (z,t-8) [T(E) - T(o)] dt . (12)

Finally we may obtain integral expressions for the central quantities of
interest by substituting (12) in the "global" delayed neutron source terms
defined by Eq. 2. Upon interchanging the order of the time integration

and the space-lethargy integrations, these may be written in the form,

) t

- ido 1 * (e - <

re, = Rt IT £ Ay Ky (£-8) [T(E) - T(o)] d& (13)
where we define,
* _ H %
Cio = [ [, (@ £, ¢ (z,u) dz du , (14)
ou
* H *

K, (¢-8) = £ £Ki<z,t—a> £y, ¢, (z,0) dz du, (15)

and 0 <z<H represents the region of the flow path in which the neutron fiux
and adjoint functions are non-zero.

The kinetic equation (1) for the flux magnitude now becomes

AL

_p -8 %10 AR w
dr/dt = — T+ ) R Y - f K, (t-£) [T(€) - T(o)] dg , (16)
i i o

or if we define the effective fraction, Eic of itthroup delayed neutrons

emitted under conditions of steady-state circulation and stationary neutron

population,

= _ TiTio
Bic ~ Fr(o) °’ (17)
12

then the modified form of the neutron population kinetic equation is,

- B By T M *
dT/dt = ET—T + ) — 5t ) 17 J K (e-8) [T(€) - T(e)] d& . (18)
i i

QO ‘*—rt

In this form, it may readily be seen that stationary conditions of the neu-
tron population will prevail when the reactivity has a small positive mag-

nitude equal to the net "loss" of B due to circulation, i.e.,
Q = B - z E- . (19)
i

As a result of reformulating the kinetic equation as a single time-
dependent integro-differential equation, it is possible to regard the cal-
lation of the adjoint-weighted impulse responses, or kernel function, K:,
as a fundamental element of circulating-fuel reactor-kinetics analysis.
The remainder of this section, therefore, is devoted to obtaining explicit
mathematical expressions for these functions.

Since £ is to be regarded as a fixed time in Eq. 10, one may simplify
(10) by shifting the origin of time to this point. This is equivalent to
replacing the variable t by u = t-£, where p is the "'age' between the appli-
cation of the impulse and the evaluation of the response. Denoting the
unilateral Laplace transform of Ki with respect to the variables u by,

fii (z;8) = e °F Ki(z,u) du , (20)

0“8

we may obtain the transform of Eq. 10 as,

_ _ afii
s K, + X, K, +V—
i i 0Z

By G(z) in core region (21a)

= 0 in external piping. (21b)

Since the transformed equation can be treated as an ordinary differential
equation, its solution, obeying the conditions of continuity along the cir-
culation path, is easily obtained. The integration of (2la) along the path

through the core yields,
13

_ _ S 0Dz —(sbh) (RS2 .
K, (z38) = K, (03;8) e Ve 4 fBi e L G(z') %— , (22a)
O c

Where VC is the fluid velocity in the core region. Similarly, integration
of (210) between z=H and z=0 (the entrance and exit of the external piping)
results in,

L

— — "(S+Ai) v — _(S"'A.)T
K, (038) = K, (i3s) e P o= K, (H;s) e P (22b)

where L, Vé, and Tp are the effective length, fluid velocity, and residence
time in the region designated as external piping.

Equations 22g and 22b constitute the complete system of relations
necessary to solve for the transforms of the kernel functions E; (z38).
For example, setting z = H in (22a) and then substituting (220) into (22a)

results in the following relation for Efi (H;s),

H-z'
~(s+r.) (T +1 ) H -(s+1.) (— .
e afee Vel e L, @
C

Ki (H;s) = Ki(H;s)e s 8

where T, = H/Vé is the fluid residence time in the core region.

We next consider the inversion of the transforms in Eqs. 22 and 23,
in order to obtain the explicit relations for Ki in the time domain. To
accomplish this, while still retaining the general form for the functional
dependence of the fission distribution, G(z), it is necessary to invoke
some formal mathematical manipulations involving delta functions, whose
rigorous justification requires the theory of generalized functions, or
"distributions" (see Ref. 16, Appendix A). We will not attempt to present
rigorous proofs here. Instead, after indicating these manipulations and

the resulting formulas, we will discuss the results in terms of a specific
14

example which does not require these formal manipulations. In the de-
velopment given below, use is made of the following important property of
the Laplace transform:

-as
Translation Property: The inverse transform of the product e f(s)

is f(t-a), where f(t) = 0 when t<O0.

To carry out the formal inversion of the transforms in Eq. 23, we
interchange the order of the spatial integration and inversion, and employ
the translation property in both terms on the right-hand side of the equa-~

tion. Thus

H-z'

 

-\, T H -\, G5
_ i'T H-z' it v vy dz’
K, Bw) = K. (H,u-r1) e + | B, &(u 7 o)e c G(z") 57—,
o c C
(24)
where Tp = T, + Tp is the total circuit time. Next, performing the spatial

integration in the second term and again making use of the formal properties

of the delta function, we obtain the basic recurrence relation,

Ki (H,u) = 0 if u<0 (25q)
—AiTT —Aiu
Ki (B,u) = Ki(H,p—TT)e + Bi e G(H*ch) if ngsxc (250)
—AiTT
Ki (H,u) = Ki(H,u—TT)e if u>Tc (25¢)

Note that we have formally included the "time-lagged" first term on the
right hand side of (25b), although, by use of (25a), this term is identi-
cally zero for Ofipg;c

Application of a similar procedure to Eq. 22 results in,

Ki (z,u) = 0 if u<0 (26a)
15

Z
z —Ai(Tp +.57— "Aiu Z
Ki (z,u) = Ki (H,U—Tp - V;fi e c + Bi e G(z—ch) if Osuggfiz ,
(26D)
2 —)\i(Tp + Tz/_) 2
K, (z,n) = Ki(H,u—Tp— T—;C-) e c 1if > 7; , (26¢)

where the functions Ki(H,x) in the first term on the right hand of (26)
are to be determined from the recurrence relations (25). By making use of
Eq. 26 and the defining equation (15), we may also write a formal recur-

%
rence relation for the space-lethargy integrated kernel function, Ki (W)

*
K, (W) = 0 if u<0 (27a)
Z
H -A.(t + =)
%* v %
R, G = [[R Gt =F) e TP Te £ (0 ¢ (2,0) dz du
o u C
~hgH H % .
+ 8, e J Je (v £, ¢_(z,u) dz du if 0 < w <t (27D)
V uu
C
Z
H 2. (t + =
% ‘ v * )
Ki (W) = f f Ki(H,u-Tp— %fii e = P c fdi(u) ¢0 (z,u) dz du if u>TC
o u c

(27¢2)
16

Finally, by combining Eqs. 25 with Eq. 27¢, we may also obtain a recurrence
relation which is based on the total circuit time, T and applies for arbi-
trary values of U>TT;
-A,T
* * iT
Although this relation is reasonably obvious from an intuitive standpoint,
its formal proof may be carried out as follows. There are two cases to

distinguish:

(a)
TT§y§;T+TC . Rewrite Eq. 27¢ in the form,
V (u~t.,) -, (1 + 29
® _fc T z it'p ¥V *
K,(w) = [k, @, b=ty = e c £4,(u) ¢_ (z,u) dz du
0 u c
(29)
z
H ) A (1 + =) %
+ f f K.(H,uy-t - E—fie P Vc: f..(u) ¢ (z,u) dz du .
i P V di o
Vc(u—TT) u c

The first term on the right hand side of (29) may be transformed by using
Eq. 25¢, i.e.,

Z —AiTT
Ki(H,u~Tp - 320 = Ki(H,u—Tp —-V: - TT) e

if 0 < 2 S.Vc(u—TT) . (30)

Similarly, the second term in (29) can be transformed using (25b),
17

P (1 —_— o 2 : -
Bie c G[H Vc(p TP )] if VC (u TT) < z < H,. (31)

Putting (30) and (31) into Eq. 29 results in

x —KiTT H z —Ai(Tp + £ %
K, (W) = e £ £ Ky (yu-t VT ) e ¢ £,,(W) ¢_ (z,u) dz du
—hiu H . %
+ B,e f , f G(H-V (y-17_ - ——0] f..(u) ¢ (z,u) dz du
i c P v di o
V (u-1..) u c
C T
-ALT,
* i T
= K, (u-'fT)e , (32)

where the final result follows by applying Eq. 27b, with u replaced by

H=Tops and by using the simple algebraic rearrangement,

z
H - VC (u—rp - Vc)

z - Vc(u—TT) . (33)

(b)

T + rcgyggTT . The required result follows immediately by
putting Eq. 25¢ into Eq. 27¢, with u replaced by H=Top » Thus,
AT, H -?\(T +_Z._.
* o TMT z_ i v -k
K,(W = e [ K, (H,u-1 = Tg)e ¢ f£,(0) ¢ (z,u) dz du
o u c
-A,T
i’ T %
= g Ki (u—TT) . (34)

Finally, the complete proof of the recurrence relation (28) for arbitrary

values of u follows from inductive application of the preceding results.
18

The system of relations expressed by Eqs. 25, 27, and 28 form a basis
%*
for the calculation of the kernels, Ki(u). Explicitly, the problem of com-

%
puting Ki over the interval O < u < 1., is reduced to numerical integration

T
of expressions involving the fission neutron source function, G(z), and the

% *
importance function ¢O(z,u). The function Ki(u) can then be extended to
the interval u>t

T
a procedure is readily adaptable to development of a digital algorithm for

by application of the recurrence relation, Eq. 28. Such

numerical calculation of these functions.

Once the kernels, K:(u), are obtained, they can be applied in the so-
lution of the integro-differential equation for the neutron population,
Eq. 18, when an arbitrary variation of the reactivity is imposed. Because
this part of the analysis, in a sense, subsidiary to the main theme of this
memo (i.e., that of obtaining and interpreting expressions for the kernels),
we will not pursue it in any detail here. Use of an integro-differential
form of the neutron kinetic equation is common to some investigations of
stationary-fuel reactor kinetics, and several approaches are possible for
using the equation for numerical calculation of transients. Instead, we
will attempt to gain further insight into the preceding mathematical de-
scription by considering a special case which illustrates the nature of

the solution.

EXAMPLE OF DELAYED NEUTRON KERNEL CALCULATIONS

One specific instance where analytical evaluation of the integrals
implied in the preceding formulas is possible is that of a homogeneous
slab reactor, through which fuel circulates in the direction of variation
of the neutron flux. In fact, the specialization of the preceding formulas
to this case reproduces results of some of the early studies in circulating-
fuel reactor kinetics,® In addition to lending to simple interpretation,
the results for this special case are of interest as a reference in evalu-
ating various quadrature techniques of potential use in treating the more
general inhomogeneous reactor problem (i.e., the case where the spatial de-
pendences of the neutron flux and adjoint functions cannot be specified

analytically, and complete numerical treatment of the problem is necessary).
19

In the special case, the flux and adjoint functions are proportional to
sin mz/H, and for the purposes of the example, we can drop further con-
sideration of the lethargy dependence. It is then possible to calculate
the kernel functions in a more direct manner than used in the preceding
derivations, by first performing the spatial integrations and then in-
verting the Laplace transforms. The resulting expressions can be shown

to be identical with those obtained by application of the preceding formu-

%
las. The expressions for Ki(H,u) and Ki(u) which result in this case are,

 

 

 

Ki(H,u) = 0 if p <0 (35a)
—Aiu T
= Bie sin ?;- if 0 < n < T, (35b)
—AiTT :
= Ki (H,U—TT) e if 1 <u, (35¢)
*
Ki(u) = 0 if w < 0 (36a)
(1 - 29 cos oL gin B 4f0<u <t
T T T T - " = c
—Aiu c c c
= B.e +
0 if TC <y f_’cT.
0 ifO0<p<Tt. ]
+ B.e
m(u=-t_)  (u=-1.) T(u=-1_) L
j = sin L TP cos -—7?—31- (36D)
m Te c c
\ if Tp < U 5-TT )
-\, T
+ Kty e T 1f 1, < 1 (36¢)
it € T2
20

Inspection of Eqs. 35 and 36 reveals several qualitative features of the
impulse response functions for the precursor dynamics. As a result of im-
pulse in the fission rate, occurring at u = 0, additional precursors are
produced in the distribution of the fundamental mode, throughout the core.
The exit concentration, Ki(H,u) begins to rise as the fluid containing
these precursors leaves the core region, its time dependence corresponding
to a superposition of the translated '"modal" concentration, with a damping
factor due to radiocactive decay of the precursors. Following the comple-
tion of one core transit time, the exit concentration change remains zero
until the completion of the first circulation cycle. These concentration
changes then have a periodic recurrence during further circulation cycles,
with the changes always damped by the decay factor, e_Aiu.

The same general features describe the changes in the adjoint-weighted
precursor concentrations, K: (u). In this case, however, the variation in
the response function is smoothed by the integral weighting over the entire
neutron flux region, at each instant of the circulation cycle. The time
dependence of this function during one complete circulation cycle, with
the radioactive decay factor suppressed, (i.e., the sum of the bracketed
terms in Eq, 36b) is plotted in Fig. 1. Note that the function is sym-
metric about p = TT/Z, regardless of the relative values of T, and Tp.

Perhaps the most basic characteristics of the impulse response func-
tions, exhibited by this example, is the (damped) periodicity in concen-
trations corresponding to the circulation period, and the fact that the
only damping introduced into the response functions is that due to radio-
active decay. Put in other terms, the precursor impulse response function
tends to exhibit a “memory' of the modal shape dependence of production by
fission, which is diminished only by radiocactive decay. The origin of
this characteristic lies in the assumption, contained in our basic mathe-
matical description, of a slug-flow regime throughout the circulation path.
This description does not account for any effects of mixing, and therefore
it gives rise to the basic characteristic of our solution — a translation
of the precursor distributions in the direction of flow, superimposed on
their radiocactive decay. Now, it is apparent that the hydralic charac-
teristics of any real circulating fuel system will give rise to non-

negligible fluid mixing effects., One type of effect, that of radial mixing
Fig.

1.

Delayed

Neutron Kernel Function for a Homogeneous Slab

Reactor

 

1Z
22

which occurs as new precursors exit from the core and enter the external
piping system, has already been mentioned. This gives rise to the neces-
sity of averaging the channel exit concentrations over the radial direction
before computing the concentrations subsequently reentering the core. How-
ever; there will also be some degree of mixing along the direction of flow,
before the precursors produced by the "primary" impulse re-enter the core
for subsequent circulation cycles. Hence, mixing will likely have the ef-
fect of attenuating the secondary peak concentrations additionally to that
resulting from radicactive decay, and also of broadening the response inter-
val during these secondary cycles. Ultimately, the precursors would tend
to become uniformly mixed throughout the circulating system. The important
point to emphasize 1s that, as long as the flow in the core is channeled,
that portion of the weighted primary impulse response between the time of
precursor production and the first exit from the core should have a magni-
tude much closer to that derived through the procedure we have described,
rather than one which assumes complete and instantaneous mixing in the
core. The specific effects of mixing external to the core might be investi-
gated by introducing idealized "mixing chambers'" along the path of flow ex-
ternal to the core, or otherwise modeling the real system hydraulies. In
this way, the influence of mixing on the concentrations of the original im-
pulse of precursors on successive re-entries to the core could be syste-

matically studied.
DISCUSSION OF THE RESULTS AND FUTURE EXTENSIONS

As was specified at the outset, the mathematical description in the
preceding sections is limited in scope, and by no means constitutes a com—
plete foundation for analysis of all types of circulating-fuel-reactor-
kinetics problems. Perhaps the most important of these limitations was
the assumption of negligible temperature feedback conditions. Even with
these restrictions, however, several important features emerge from an
examination of the kinetic description for this case. Their significance
is best seen in reference to the analogous description for the stationary-
fuel reactor. As Henry's derivation of the conventional global kinetic

equations for this latter case shows,” the resulting equations governing
23

the changes in delayed neutron precursors have the form of ordinary dif-
ferential equations, and the complete mathematical system, including the
global equation for the amplitude of the neutron population, is a system
of coupled ordinary differential equations in the time variable.

By contrast, in attempting to obtain an analogous global kinetic
description for the circulating-fuel reactor, one must abandon efforts to
force the description (without introducing ad hoe approximations) into the
conventional ordinary differential equation format; instead, it is neces-
sary to work directly with the integro-differential equation for the neutron
population amplitude. We should include the possibly obvious remark that
this latter mathematical formalism can also be used in the case of the
fixed«fuel reactor. (It is a trivial exercise to show that the kernal
function, K:, for this case are proportional to e_liu.) Because the global
kinetic equations governing the precursor concentrations in a fixed-fuel re-
actor have the form of linear, time-invariant ordinary differential equa-
tions, the Laplace transforms of the kernel functions are simple rational
algebraic expressions. For the circulating fuel reactor, however, we have
seen that the transforms contain transcendental functions, whose inversion
yields relations exhibiting the fluid translation effects.

It may also be noted that multimode generalizations of the preceding
description of the neutron kinetic equations can be developed. This gener-
alization would give rise to coupled sets of integro—-differential equations
for the modal amplitudes of the neutron flux. However, for the analysis of
many kinetics experiments of interest, the single-mode description is quite
adequate, and therefore we have avoided this further complication in this
memorandumn,

Another important feature arising from the preceding mathematical de-
scription is the use of recurrence relations in describing the kernel func-
tions for the precursor concentrations. These relations constitute a nat-
ural means of expressing the particular physical attributes and boundary
conditions of this problem, i.e., the periodicity effects in the delayed
neutron source terms introduced by the steady recirculation of the fuel.
One of the reasons why some of the mathematical descriptions of this same
general problem, given in earlier studies,®*’ appear cumbersome and not

easily amenable to interpretation, is because the attempt is made to develop
24

explicit formulas for the kermel functions by expansion of the functions
in infinite series involving only elementary functions. Contrawise, we
have shown that the preceding description readily admits interpretation.
Moreover, the recurrence relations are readily adaptable to the develop-
ment of a digital computer algorithm for kinetics analysis.

The question arises as to what extent the general approach used in
the preceding description of the precursor kinetics can be extended to the
temperature-dependent kinetics. There are certain formal similarities in
the basic equations governing the fuel temperatures variations and the pre-
cursor concentration variations (Eq. 3) along the circulation path. In
addition, the way in which the temperature field influences the global
kinetic equation for the neutron population amplitude is through a reac-
tivity feedback term approximated by a functional relationship similar to
Eq. 1. In this analogy, C(x,t) would be replaced by the temperature
T(r,t), and the weighting function would contain products of the direct
and adjoint fluxes, rather than only the latter.'’ Aside from these simi-
larities, however, there are also differences. 1In the MSRE, for example,
a small fraction of the energy released in fission was deposited directly
in the graphite moderator. Thus, the temperature equations must include
coupling terms describing the heat transfer lags between fuel and graphite,
Also, in the precursor equations, removal from the system only occurs through
radioactive decay along the circulation path. The analogous treatment of
the temperature response kernels corresponding to secondary passes of fuel
through the reactor after the primary impulse in the power level would re-
quire coupling with the temperature-kinetic equations for the heat exchanger
and secondary coolant system, Hence, several extensions of the preceding
mathematical description would be necessary before the method could be read-
ily transcribed to analysis of power-temperature kinetics. However, in one
interesting example where there was need to determine the reactivity per-
turbations for short periods during observed power variations in the MSRE,
the graphite heat~transfer lags and the secondary heat removal effects
could be ignored. Hence, the integro-differential Eq. 18 was used directly
to calculate these reactivity perturbations.*®

In any extended investigation along these lines, we recommend that the

use of the distributed-parameter approach and its method of decomposition
25

described in this memo be further explored in application to the power-
temperature kinetics of circulating-fuel reactors. This includes the de-
velopment of numerical procedures to calculate the kernels for precursor
concentrations and temperatures, and systematic study of the effects of
fluid mixing on these kernels. This approach should be considered as a
parallel and complementary step to the development of approximate lumped-
parameter models for the system dynamics. We do not imply that the dis-
tributed-parameter approach should necessarily replace the use of the
lumped-parameter models in analysis and design of control systems for these
reactors. For many purposes, the lumped models are quite adequate, and
moreover, are well suited to techniques of digital and analogue computation.
They proved very useful in many of the studies of system kinetics of the
MSRE.'*® However, in certain types of large transients, differences in the
treatment of spatial neutron importances and time-lag effects between these
two approaches can give rise to significant differences in kinetic responses, '’
A combination of the two approaches may ultimately prove essential for studies
of the dynamics and control of this type of reactor. Perhaps the basic mes-
sage of this memo is that, without close attention to the most elemental form
of the mathematical description of the physical system, together with its
"satural' mathematical decomposition, it is often difficult to determine

the precise limitations of the lumped models. Thus, it would not be sur-
prising if these investigations could result in improving the resolution

of the latter models.
26

APPENDIX

Derivation of the Global Kinetic Equation for the

Neutron Population Amplitude

 

To derive Equations 1 and 2 of this report, and hence establish the
point-of~departure for analyzing the special aspects of circulating fuel
systems, we will use a notationally abbreviated version of Henry's origi-
nal derivation.“ This employs a linear operator formulation of the gen-
eral timew-dependent reactor equations, as opposed to the detailed develop-
ment in ref. 4, describing the physical constituents of the transport
equation, The operator form of the equations describing time dependences

of the neutron flux and. precursor population are,

6

PO © o, 00
Lo+ (1-B) £ o + [ Ay £, € = v g (A1)

k i=1

e
aC
Po R
Bi ko - )\i Ci -}L‘Vci - Bt 9 1 l, 2, oooo6o (AZ)
e

Here, ®(r,u,t) and Ci (r,t) represent the local neutron flux and precursor
densities, respectively. (We shall assume a description of the neutron
flux in terms of space {r) and lethargy (u) variables as a starting point;
Henry treats the more general case including directional variables, but
this inclusion is not vital for our discussion.) The operator, L, is a
time-dependent linear operator representing the net of all neutron loss
processes (which includes leakage, absorptions, and energy transfer by
scattering), and P is a time-dependent linear operator representing the
total productions from fission. In Eq. Al, therefore; P¢ is the total rate
at which prompt neutrons plus gelayed precursors are produced at r; multi-

plication of P% by 1-8 = 1 - z Bi gives the total rate of production of
i=1
27

of prompt neutrons at r, and multiplication by Bi gives the rate of pro-
duction of ith—group precursors at En- In Eq. A2, P% is taken to be zero
in that part of the circulating fuel system where the neutron population

is negligible. Of the remaining symbols, A, is the decay constant for the

i
ith precursor group, and the quantities fp and f are lethargy distri-

butions of prompt and delayed neutron production?ii.e., the fractions of
the total production which are emitted in specified lethargy ranges. The
lower case velocity, y(u), is the neutron velocity at lethargy u, and

V is the circulation velocity of the fuel.

Following Henry's derivation, the multiplication factor, kZ’ has been
introduced to provide a correspondence with a reference state of the reac-
tor, known to have stationary (critical) flfix and precursor densities. In
our case, we must further qualify this to mean stationary with no circu-
lation of the fuel. We will denote the physical conditions corresponding
to this stationary state by adding zero subscripts to the operators, flux,
and precursor densities. The equations describing the reference state may
then be obtained from Eqs. Al and A2 by setting the right hand sides equal
to zero and setting the fluid velocity V = 0. Then, the precursor densi-

ties may be eliminated from the equations, and it is easily shown thatJr

P

Lo bo + F~—2>¢ = 0, (A3)
k0
e
where,
_ 6
F = (1-B) fp + _2 B, £qs - (A4)

i=1

To convert the "local' kinetic equation for the neutron distribution
(Eq. Al) to a "global" kinetic equation for a population magnitude, Henry

multiples the former equation by an appropriate weighting function and

 

TWe will distinguish the flux distribution for the stationary case
by use of lower case letters, with zero subscripts.
28

integrates over the independent variables of the neutron distribution.
For the weighting function, it proves convenient to choose the static

%
adjoint flux, ¢°, the solution of the adjoint equation corresponding to

Eq. (A3). With asterisks indicating the adjoint operators of (A3), this is,

- %
% % £p *
Ly 45 + EEel o0 o g, (A5)
K°
e
We will write the integrals using a scalar product notation, e.g.,
% %
(¢o, LP) will represent the product of ¢, and L¢, integrated over the do-
mains of the lethargy and spatial variables for the neutron population.

*
Therefore, by first forming the scalar product of ¢o with Eq. (Al), we

have,
% % P 6 *
~{$o, LO) + (1-B) {40, £ =5 &) + ] A, (00, £, C))
Px i=1
e
3 (a¥ g
+ 5p (9o, V7T 9 (A6)

Secondly, we form a similar scalar product of ¢, the time-dependent so-
Iution of Eq. Al, with the adjoint equation for the reference stationary-

state (Eq. A5),

- *
%
~10, Lo ga) + (o, BBl Ty o0 (47)
k
e

By using the definition (A4), we may rearrange the term associated with

prompt neutron production in Eq. A6 as follows:

% _ P % -
(]-'"B) (¢’09 f — CP) = <¢Oy £
Py

.
k

P
k

_ T *
e e e

Applying this in Eq. A6 and subtracting Eq. A7 from Eq. A6 gives,
29

= - %
% % % % x
~6o, LO) + (0, Lo do) + (b, ~b 8) - (0, [£ Pol 4%
o o
k k
e e
- § B (‘i’iaf.'l?“'@\'i' § A, (9o, £.. C.)
;21 1 di 1.© 12 1 di “i
e
3 * -1
= fi' (Cbo, v ) . (A9)

Henry now partitions the function ¢(r,u,t) into a product of a shape
function, ¢ r,u,t) and a time function T(t), with a normalization re-

quirement that

i
<

3 * -
5 (o, v 9) : (A10)

By doing this, it can be seen that the right hand side of Eq. (A9) may

be written,

-

9 ¥ _ ® -1 ,@E
T,V 9 = o, v ) g (A11)

0

and Eq. A9 becomes, on factoring out the time-dependent amplitude, T(t),

 

— - *
~(b5, L§) = (4, Lo ¢a) + (05, =2 0) = (9, LBl 4By by,
k k
= e
g P Z *
= B (¢o, i _""'d)) T(t) + A, ((bOQ f.. C)
je1 1 kz i=1 * a1

- E v (A12)
30

The group of terms inside the braces of this equation formally represents
the algebraic difference between weighted increments in the production
rate and weighted increments in the neutron loss rates. To put this net
expression, or coefficient, into a form which does not depend explicitly
on the normalization of the flux shapes, it is convenient to rewrite the
coefficient as a fraction of the normalized production rate. The choice
of this normalization factor is somewhat arbitrary; however, as demon-
strated in ref., 4, it is useful to divide both sides of Eq. Al2 by the

factor,

F(t) = (bs, FP ) . (A13)

Thus, by factoring the time-dependent population magnitude, T(t), from
all quantities on the left hand side of Eq. (A9) one obtains the "global"

kinetic equation,

 

A, C, = == (Al4)

8 dr
P S dt

where, by definition,

 

 

o(6) = Ty [—(cbf,w) + (0, Lo o) + (b0, -f-1-<-§ 9
e
- p % %
- ¢, £ qbo)] , (415)
k
e
o = ok e of e, B (416)
F(t) ;2; "1 °° Tdi kZ >
31

AE) = —— (4, v 0) , (A17)
F(t)

1 %*
Ci(t) = m (do, fdi c.) . (A18)
32

REFERENCES

A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain
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H. Hurwitz, Jr., '"Derivation and Integration of the Pile-Kinetic
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H. Soodak, "Pile Kinetics'", Chapter 8 in The Science and Engineering
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A. F. Henry, "The Application of Reactor Kinetics to the Analysis of
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W. K. Ergen, "Kinetics of the Circulating-Fuel Nuclear Reactor'",
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J. A. Fleck, Jr., "Kinetics of Circulating Reactors at Low Power",
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B. Wolfe, "Reactivity Effects Produced by Fluid Motion in a Reactor
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M. J. Kolar and F. D. Miraldi, "The Temperature-Dependent Kinetics of
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C. W. Nestor, Jr., "ZORCH-An IBM-7090 Program for the Analysis of
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B. E. Prince, "Period Measurements on the Molten-Salt Reactor Experi-
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(October 1966).
33

16, L. A. Zadeh and C. A. Desoer, Linear System Theory: The State Space
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17. B. E. Prince and J. R. Engel, "Temperature and Reactivity Coefficient
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19. MSR Program Semiannu. Progr. Rep., Feb. 29, 1968, ORNL-4254,
pp. 48-49.
- - -

Lo~~~ T wWwMND P

e
WN ko

14-18.

58.
59.

60,
61l.
62.
63.

64,
65"67 .
68-69.
70-86.

87.
88-89,

35

ORNL-TM~4179

 

Internal Distribution
J. L., Anderson 27. A, J, Miller
S. J. Ball 28, R. L. Moore
S. E. Beall 29, C. W. Nestor
E. S. Bettis 30. L. C, Oakes
E. G. Bohlmann 31. A. M. Perry
R. B. Briggs 32-36., B. E. Prince
0. W, Burke 37. J. C., Robinson
W. B. Cottrell 38-39. M. W. Rosenthal
F. L. Culler 40. Dunlap Scott
S. J. Ditto 41, Myrtleen Sheldon
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A, P, Fraas 43. M. J. Skinner
D. N, Fry 44, 0. L. Smith
P, N. Haubenreich 45, I. Spiewak
H., W, Hoffman 46, D. A, Sundberg
P. R. Kasten 47. D. B, Trauger
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External Distribution

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