ORNL-TM-1626.txt

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LT

IRCULATION:

 
 

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s ot fl'!e Comml £510H, Re? 4Ry person Gc?mg o bfl%u%? of fhe Ccm i

| LEGAL MOTICE

:fepsri‘ wns prepared ag an ncc@un? of chemm@n‘? s;;eonsm*ed wark Nelfl’!ef ?he Umteé Srmes L

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‘Q i

ORNL-TM-1626

Contract No. W-T4(C5-eng-26

REACTOR DIVISION

PERIOD MEASUREMENTS ON THE MOLTEN SALT REACTOR
EXPERIMENT DURING FUEL CIRCULATION:
THECRY AND EXPERIMENT

 

E. B. Prince
NNCUNCEMENT

L e e B

LEGAL MOTICE
Thie repor{ wae prepared as an account of Government spongored werk. Neither the United
States, nor the Commisgion, nor any person acting on behalf of the Commisasion:
4, Mzkes sny warranty or represeitation, expressed or implied, with respect i the accu~
racy, completeness, or usefulness of the information centained in thie repert, or that the use
= of any information, appsratus, methoed, or process disclosed in thiz report may not infrings
privately owned righis; or
B, Aspumes any liabilitics with respect to the use of, or for damuges resulting {rom the
uze of any information, apparatus, method, or process disclesed in this report,
Az used in the sbove, ‘‘person zcting on behzif of the Cemmiseion® ircludes any em-
8 ployee or contractor of the Commission, or employee of such couiractor, te the cxtent that
such employee or contractor of the Commigsion, or employes of such contractor prepares,
disseminates, or provides access to, any information purauant to his employment or contract
with the Commigsion, or his employment with such contractor,

AR RO,
i
t—fl
b
o=
i
rai
o
v
e #’
l-:!

R S RN S AN

 

OCTOBER 1966

OAK RIDGE NATTONATL, LABORATCRY
Oak Ridge, Tennessee
operated by
UNION CARBIDE CORPCRATION
for the
U.5. ATOMIC ENERGY COMMISSION

 
 

 

 

 

 
|
e
-

 

CONTENTS

Page

ABSTRACT . v v v v v v e b 0 e e e e e e e e e e e e e e e e e 1
INTRODUCTION . o v v v v v v v e vt v e e e e e e e e e s e e e 1
THECRY OF ZERO POWER KINETICS DURING FUEL CIRCULATION . . . . . . 2
NUMERICAL AND EXPERIMENTAL RESULTS . . . + « « « « ¢ v v o o « o 27
Numerical GroundwOrK .« o v « « o v o o s v e e 6 e e e e e e 17
Experimental Results . . . . . « « « ¢ ¢ v v v o o s s . 25
DISCUSSION OF RESULTS . & © v v ¢ v v v v v e o o o o s v e o v s 29
Theoretical Model Verification . . . . . . « « + « & « « « & 29
Recommendations for the MSRE . . . . . ¢ + . « . « « « « . . 31
ACKNOWLEDGMENT e e e e e e e e e e e e e e e e e e e e e e 32

REFERENCES . .+ ¢« v« v v 0 v o e v v e o 6w o v 6 o o« v o 33

@
 

 

 

 
PERICD MEASUREMENTS ON THE MOLTEN SALT REACTOR
EXPERIMENT DURING FUEL CIRCULATIOCN:
THECRY AND EXPERIMENT

B. E. Prince

ABSTRACT

As an aid in interpreting the zero-power kinetics experi-
ments performed on the MSRE, a theory of perlod dependence on
the fuel circulaticon is developed from the general space de-
pendent reactor kinetics equations. A procedure for evaluat-
ing the resulting inhour-type eqguation by machine computation
is presented, together with numerical results relating the
reactivity tc the observed asymptotic period, hoth with the
fuel circulating and with it stationary. Based on this analy-
sis, the calculated reactivity difference between the time
independent flux conditions for the noncirculating and the
circulating fuel states is in close sgreement with the value
inferred from the MSRE rod calibration experiments. Rod-bump
period measurements made with the fuel circulating were con-
verted to differential rod worth by use of this model. These
results are compared with similar rod sensitivity measurements
made with the fuel stationary. The rod sensitivities measured
under these two conditions agree favorably, within the limits
of precision of the period measurements. Due to the problem
cf maintaining adeguate precision, however, the period-rod
sensitivity measurements provide & less conclusive test of the
theoretical model than the reactivity difference between the
time independent flux conditions. Suggestions are made for
improving the precision of the experiments to provide a more
rigid test of the theoretical model for the effects of cir-
culation on the delayed neutron kinetics.

1. INTRODUCTION

It is well recognized that the circulation of fuel in a liquid
fueled reactor introduces some unigue effects into its observable kinetic
behavior. Foremost in this category is the phenomenon of emission of
delayed neutrons in the psrt of the circulating system external to the
regctor core where they do not contribute to the chain reaction. The
literature in reactor kinetics contains several theoretical studies of
this effect,? 223 but there have been few copportunities for parallel

experimental studies. The Molten Salt Reactor Experiment, hereafter
Mo

 

referred to as the MSRE, has presented a urnique chance to develop and
test some aralytical models for the zero power kinetics of a circulating
fuel reactor. The experimental measurements discussed in this repcort
were made as part of an overall program to calibrate the control rods
in the MSEE.

In the first secticn fellowing, we develop the thecry of periocd
dependence on fuel circulation. The second section gives an account
of the results of applying this theory to the experimental measurements
mede during MSRE Run No. 3. Finally, in the third section, we discuss
the results of this work in terms of the general problem of kinetics
analysis for circulating fuel reactors Several questions are lelft open @

by the present study, and these are discussed in that section.

2. THEORY OF ZERO POWER KINETICS DURING FUEL CIRCULATION

At negligible reactor power, the special effects produced by fuel
circulatior reduce essentially to (a) the transport of delayed neutron
precursors tc that part of the lcop external to the core where they
subsequently decay, and (b) the skhift of the spatial distribution of
delayed neutron precursors in the direction of circulation within the
reacter core, relative to the prompt neutron production. Although we
skall not attempt to review all eariier studies pertalning to these
effects, twe fairly recent studies made by Wolfe® and Haubenreich® are
pertinent to the present work., Wolfe employs a perturbation approach anc
obtzins an irhour-type eguaiion for ar infinite sliab reactcr, through which 6

“uel ecirculates in the direction of variation cf the neutron flux. Wolfe's

=

sppreach, while valusble, ig alwmost entirely formal, and requires some
modificaticr in corder to cbtain a procedure useful for the guantitative
analysis of experimental messurements. In the second of the above men-
Tioned works, Haubenreich considers an explicit analytical model rep-
reserting the MSRE in the circuiating, Jjust-critical conditicn. By means
of & modal analysis, he obtalns effective values for the delayed neutron

fractiong for this conditicn. He uses a bare cylinder approximation to

 

 

f ] L
For a comprehensive early study, see Ref. 1.
 

L

represent the MSRE core, with boundaries corresponding physically to the
channelled region of the actusl core.

We have combined and extended thege analyses to include the contri-
bution to the chain reaction of the delayed neutrons emitted while the
fuel 1s 1n the plenums Jjust sbove and below the graphite core, and o
include the case of the flux varying exponentially with a stable asymp-
totic period. It will be seen that an inhour-type equation results
which relates the period of a circulating fuel reactor to the static
reactivity of the same reactor configuration, but in which the fuel is

not circulating. The static reactivity, p_, is defined by the relation
w2

P, = — = 9 (1)

in which v 1s the physical, energy-averaged number of reutrons emitted
per fission, and V. is the fictitious wvalue for which the reactor with
the same geometric and material configuration would be critical with

the fuel stationary. One finds that as a result of this definition, the

 

reactivity for a critical, circulating fuel condition is greater than
zero (here, criticality derotes the condition of time independence of
the neutrcn and precursor concentrations). However, since the static
reactivity is the quantity normally obtained in reactor calculation
programs, it 1s most convenient to relate this quantity directly to the
asymptotic period.,

The theory following 1s developed by beginning along generasl lines
and delineating special assumptions as they are introduced. Several of
these simplifying approximaticns are suitable for MSRE anzlysis, both in
the neutronics model and the flow model for the circulating fuel.

The starting point of the analysis is the general time dependent
reactor equations, written fo include the transport of delayed neutron

precursors by fuel motion in the axial direction:

6
L@’é-(lmB)fP@f'!‘--!?‘,f C:v“lé?i (2)
T/ "p /071 Tdi i ot ’
Q/
!

i
ot ’

1 =1,2, « .« , 6 . (3)

 

9 _
By B8 - G - 55 (VCy) =

The symbeols ¢ and C represent the neutron flux and delayed neutron pre-
cursor densities. The operator L represents net neutron loss (leakage,
absorption, and energy transfer by scattering), and P represents the
production processes by fissicn. The explicit representation of these
operatcrs depends on the model used in analysis. ITf these processes
are given their most general representation in terms of the Boltzman
transport equation, the neutron flux, ¢, will be a function of poesition,
energy, direction, and time variables. In order to provide a framework
for discussion which can ke directly related to application, we shall
agsume that the angular variables have been integrated from the equation,
i.e., that a model such as multigroup diffusion thecry or its continuous
energy counterpart prcvides an adequate description of the neutron popu-
letion. In Eq. 3, azbove, P¢ is taken as zero in that part of the cir-
culating loop which is external to the chain reacting regions. The
remaining symbols in Egs. 2 and 3 are (1 - BT}, the fraction of all
neutrong from fission which are prompt, and Bi and Ki, the producticn
fraction and decay constant for the ith precursor group. The quantities
f  and f&i are energy spectrum operators which multiply the total volu-
metric production retes of prompt and delayed neutrons to obtain neutrons
of 2 specific energy. Finally, the symbols v and V represent the neutrcn
velocity and the fiuld velocity, respectively.

As applied to a fluid fueled reactor with a heterocgeneous structure
such as the MSRE {where the axizl fuel channels are located in a matrix
of solild graphite moderator}, the usual cellular homogenization must be

made on the neutron production and destruction rates. Thus, for example

(1 - BT) P% = local rate of production of prompt fission
neutrons, per unit cell volume

Xi C, = local rate of production of ith group of delayed
neutrons, per unit cell volume.

If we assume that the operators L and P are time independent, corre-

ponding to a fixed rcd position, we can investigate the conditions under
which the flux and precursor densities in the reactor and the external
. . . A wt , . -
circulating system vary in time as e . Assuming solutions of Egs. 2

and 3 exist of the form:

8(x, E, t) = #(x, B; v) & (1)
wt
Ci(Zi: t) = Ci(.}i; W) e 3 (5)
then,
6
2 4
Lfi+(1msT)prg+l>L+fdj_c1:v wid (6)
i=1
B. PZ - h.c mi(vc):wc i = 1,2 6 (7)
=4 i"i T 3z i i’ T e s ’ ‘

As previously stated, cur primary objective is to relate the observed

It
stable reactor period, w , to the static reactivity of the given reactor
configuration. The static reactivity defined by Eg. 1 is the algebra-

ically largest eigenvalue of the equation:

w,*(L-p ) TR A =0 (8)
where;
&
e — - % o /
R D L T (9)
i=1

Rather than attempting to solve the reactor eguations 6 and 7 directly,
we will make use of a procedure which is often useful for spatial reactor
kinetics problems. Several sources give details of similar analyses for
stationary fuel reactor systems (see, for example, Ref. 4). We multiply
the "local" equation for the neutron population by an appropriate weight-
ing function and integrate over the position and energy variables of the
neutron populaticn in order tc obtain a relation involving only “globsl"
or integral quantities. As seen below, by using the static adjoint flux,

+ . . . ‘ ]
g Qé, the sclution of the adjoint eguation corresponding to Eq. 8 as the
weighting function, the resulting relstion admits a direct physical inter-
pretation. The adjoint flux, also obtained in most reactor physics calcu-

lation programs in common usage, is the solution of

LT d (-0 )FR)T A =0, (10)

+ - F
where I, and (fP)

the ssme algebraically largest eigenvalue of Eq. 8. With appropriately

are operators adjoint to L and P of Eg. 8, and p, is

prescribed boundary conditions on the allowable functions on which these
operators are defined, the following relation® can be used to define the

*
sdjoint operator;

o0 = (8 A7) (11)

+
(7, A8) = (2
Here A represents abstractly either of the operators L and TP and

(Q:, A7) denotes the scalar product, i.e., the multiplication of A by
Qg and integration over the position and energy varisbles of the neutron
population.

On forming the scalsr product of Eg. 6 with Q:} we obtain

+ -1 + , + )
(g, v = (1) + (1 - B, 2 e ¢

 

Similarly, forming the scalsr product of Eg. 10 with @ and making use of

Eq. 11 gives
’+ A =
(8., 18) « {1 - p M@, TPF) =0 . (13)

Combining Eq. 12 and Eg. 13 gives

 

 

¥For the purposes of reactor physics studies, we need to consider
oniy real valued functions.
 

+ -l A+ +
W v B = - (1 - 0 WF,, TRO) + (1 - B, £ P0) +
S
b { +
+ \ZJ f\,i(QfS? fdi Ci) .
i=1

By making use of Eg. 9, we may rewrite this equation as

 

 

. 6 6
+ -1 ' 4 : -
7 7 oy / = 3\
® i: i::l
P, =W + . (1h)
S + o= A =
(7, £ PF) (7, T PF)

The first term on the right hand side of Eg. 1L is simplified in appear-
ance 1f we make a conventional type of definition of +he proxpt neutron

generation time;

(7, v 9)

A = e
(7, T PF)

[ =3

: (15)

)
1

We may also note that the second term on the right hand side of Eq. 14
appears as the difference between the weighted total production of de-
layed neutrons and the weighted production of delayed neutrcone from

precursor decay within the resctor. In accordance with ifts definition

 

.
by Egs. 1 and 8, the static reactivity is completely determined by the

geometric and material configuration of the reactor, whether or not the
fuel is circulsting in the actual reactor. The relationship betwsen 0
and w expressed by Eqg. 1k, therefore, has an expliclt physical inter-
pretation. For example, if the fuel composition and contrel rod position
are such that the flux is time independent when the fuel is circulating,
w =0 1in Egs. 6 and 7. Eguation 14 then shows that the static reactivity
for the Jjust critical reactor is numericelly egusli to the net difference
in the producticn of delayed neutrons described above. In the nore gen-

k- eral case when the flux is varying in time according toc a stable periocd,
the first term on the right hand side of Eq. 14 will differ from zero,
and also the effective decrement in production of delayed neutrons will
differ numerically from the time independent case (ci and ¢ depend on
®w through Egs. 6 and 7. Eguation 14 is an inhour type relation which
can te used as & foundaticn for an appr?ximate determination of Py
given an cobserved asymptotic period, w . One may observe that it in-
cludes the usual inhour relation for the stationary fuel reactor as a
special case, simply by setting V = 0 in Eq. 7. Before discussing the
practical use of Eq. 1k, we can exhibit the previous concepts in an

explicit algebraic manner. From Eg. 7 we have,

ep = (w+ )7 (B0 - (Ve . (16)

Inserting this relationship into Ec. 14 gives

 

if we define

B, =8B, —————— (18)

 

Equaticon 17 may be written as

.........
 

 

6 = — 6
T (81 - 7]"_) —
g =Wl AA Lo w ot Ay " §j 1o (20)
121 * =]

This equatiorn has the appearance of the ordinary inhour relation for the

4

stationary fuel reactor,” except that the effective prcauction fractions,

§i are reduced by a quantity'7i which depenrnds on the circuistion rate and
also on w. In addition, the last fterm on the righlt hand side of Eq. 20
appears because the zeroc point of the static reactivity was chesen To
correspond to the cowmposition and geometry of the just-critical stationary
el reactor. The usual inhour relation for the stationary fuel reactor
is obtained by letting V —» O and 71 - 0 in Eg. 20. Althougkh Eq. 20 is
ingstructive in discussing the net effects of fuel circulaticn, Iits sim-
plicity is somewhat deceptive since ;£ depends in a complicated way on
w, through Eq. 7. Therefcore, it will be mcre converient tc discuss the
use of the inkcur relation starting with the form given in Eq. 1k.

At first appearance, in order to use Eq. 14 we are required to
solve Egs. 6 and 7 for g, Ci(é; w), and ¢(x, E; o), and also to solve
Eg. 10 for p_ and é;(g, E; Qs)" Both are eigenvalue prcblems in which
the algebraically largest real elgenvalues, ¢ ana Py are cf immediate
interest. Not only does Eq. 14 reduce tc an identity if this procedure
is ueed, but it is precisely the explicit solution of Egs. & and 7 that
we wish to avoid. The great usefulness of Eg. 1l is In the basis it pro-
vides for approximating the relation between P and the cbserved stable
period @ t. The basic simplification results from assuming that the
shape of the asymptotic flux distribution, ¢, is sufficiently well
approximated by the static fiux distribution, éss From & physical stand-
point, the validity of this appreximation is & consequence of the small-

ness of the delayed neutron fraction, £ If this approximation 1s made,

o
and ¢ is substituted for ¢ in Eq. 7, this equation car be integrated
arcund the circulating path of the fuel to cbtain the distributions of
gdelayed precursors, c. . Before completing the analysis, however, we
shall make a geccnd approximation to simplify the computation of the

integrals occurring in Eq. 14. It can be assumed that the correction
10

for the difference in energy spectra for emission of prompt and delayed
reutrons appearing in Eq. 14 can be calculated spproximately as a sepa-
rate step. This is done by reducing the age for the ith group of de-
layed neutrons from that of the prompt reutrons and modifying the static
delayed neutron fractions, Bi’ by the relative non-leaksge prcobability
factors approprizte to a bare reactor which approximates the actual core.
Such an appreximetion can alsc be justified from an objective standpoint,
since the correction for the differing "energy effectiveness” of delayed
neutronsg relative to prompt neutrons i1s small compared to the effect of
the spatial transport of precursors under considerstion. For the MSRE,
calculations of the former effect are given in Ref. 2. The net energy
correction changes the effective value of ST for B35 from 0.0064 to
0.00666 .

Orne further remark should be made concerning Eq. 14. A given
static resgctivity, ps’ corresponding to some time independent resactor
configuration is also related through Eg. 14 to gll physicaliy allowable
transient mcdes present ir the neutron flux after the final resctor con-
figuration has been established. In this study, we have chosen to
emphasize only the relstion between the circulation and the asymptotic
mode. An spproximete analysis indicates that the remaining elgenvalues
end eigenfunctions of Egs. € and 7 differ in certain fundamental respects
from those of stationery fuel rezactors. For the purpose of this report,
we ghsll not attempt to demonstrate this. Except for a brief return to
this topic in a later seciicon, we will restrict attention entirely to
the analysis of the stable asymptotic period measurements.

The ccmpletion of the required anslysis consists of the integration
of Bg. 7 arcund the circulating psth of the fuel. Obviously, even if @
is replaced by QS calculated from Eg. 8, further simplifications in the
flow model are necegsary before the problem becomes amenable to practical
computation. The approximationsg we have used in representing the circu-
lating loop of the MSERE are shown schematically in Fig. 1. The model
consists of a three region approximation to the actual core, representing
the lower cr entrance plenum, the graphite modersted region, and the
upper, or exit plenum, respectively. The core is represented as a right

cylinder with volumes in the upper and lower plenums equal to those of

 
V.

 

11

 

ORNL-DWG 66-5561

e s —— T o |

  
 

~—EXTERNAL PIPING
{INCLUDING FUEL

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 I PUMP AND HEAT
IN REACTOR QUTLET EXCHANGER)
z=H
UPPER
TOP GF
3 INLET FLOW PLENUM )
10 - PR ROD 7 DISTRIBUTOR E 2% He
° T ANC DCWNCOMER
2 09 7
_ | /
£ F
| 08 1 —-CONTROL ROD
e L ——CONTROL. ROD PP THIMBLES
€ o7 THIVMBLE
x (TYPICAL}
£ 06 7 }
3 CHANNELED
W 05 [ REGION
w0
&
& o4l
5
203}
o DRIVEN ROD
z 02 LOWER LIMIT
e SCRAMMED ROD
g ¢l LOWER LIMIT
x \J
o BOTTOM QF MOST GRAPHITE 220
LOWER
PLENUM
(@) MSRE CORE GEOMETRY {SCHEMATIC) (6} MODEL FOR NEUTRON:CS CALCULATIONS

Fig. 1. Gecmetry of MSRE Core and Three Region Core Model Used in

Physics Calculations.
1

Tthe actual MSRE ccre. This three-region model is a simplified version
of that used for all previous MSRE core physice computations.?

Fluld dynamics studies with the MSRE core mockup have indicated
that the fuel velocity withian the graphite moderated regicn is very
negrly constant over & large part of the core. Higher velocities occur
in a small region about the core axis and near the outer radius. In the
upper plenum, the flow is nearly laminar, whereas in the bottom head the
flow distribution is complex due to the reversal in the flow direction
between the peripheral downcomer and the graphite moderated region.

In the present study, we have assumed the flow velccity in each
region except the lower plenum to be constant (plug flow)} with the

magnitude cf the linear velccity determined by

Ve =5 (21)

wWieTe

Lk is the axial length of the kth region, and tk is the residence
time of the fluid in the kth region. It will be seen lster that the
precursor censitles in the regions of zero neutron flux (external loop)
depenc only on the fluld residence time in these regions.

Although the lower plenum is in a region of relatively low neutron
lmportance so that several approximations are allowable, rather than
assign a linear velocity tc this region it was considered more realistic
s

from 2 physical standpoint Lo treat the region as = "well stirred tank”

with an average neutren flux and importance (adjoint flux) assigned to

Within the graphite moderated region, the primary difference in the
spatial distributions of prompt and delayed neutrons can be expected to
be 1n the direction of fuel salt flow. As g first approximation, we have
agsumed the velocity profile to be flat in the radisl direction across
the entire core, and have neglected the radial averaging of the neutron
production rates implied in the scalar product integrals in Eg. 14. This
is equivalent to sssuming radial and axial separability of the neutron
producticn rates and adjoint fiuxes. Thus, if we preaverage the fluxes

over the radial coordinate, and comsider only the axial (Z) dependence

 

 
of PQ% in the integration of Eq. 7, the problem reduces entirely to a
one-dimensional calculation (Line mo@.el).‘)é

With these simplifications, the required integration of Eg. 7 can
ncew be completed. As written in the preceeding formulas, the precursor
densities and neutron reaction rates sre the homogenized values, 1.e.,
normalized to a unit cell volume of the reactor. To integrate Eq. 7.
account must be taken of the variation in the fiuid volume fracticn over
the path of flow throcugh the reactor. IFf o is the volume fraction of
fuel in the kth region and superscript (o) is used to indicate the pre-

cursor densities and prompt neutron production rate in the fuel;

c.(z)

1

1l

o el(z) (22)

fl

o
PP (z) = a P& (2) . (23)
The values of o used for numerical calculations with the model of the
MSRE core shown in Fig. 1b were 0.225 for the graphite moderated region,
and 1.0 and C.91 for the top and bottom plenums, respectively. The

explicit forms of Eq. 7 for the various regions become:

a) Craphite modersted region

o o 9
_ = s < < !
B, P2 (2) — (O + ) c; =V, 53 O<z<H, , (2h)

{ b) Top plenum

o o G

BiPfis(z)-—-(ki-Fw) cs =V, B, <z<H , (25)
c) External piping

.

—(xi+w)ci=vexa-£— H<z =L | (26)

 

*

The validity of this approximaticn is not immediately obvious;
however, it appears to be adequate in the light of results presented
later.

 
d) Bottom plenum

(P°F_

i s (27)

t

\ 0 :
- (k. + T = e : - .
>E (Ai W) ¢, = 3T 0=t < t,
In the sbove equations, Hc 1s the height of the graphite moderated region,
Hwfic is the thickness of the upper plenum, and L-H is the effective length
of the region representing the external piping and heat exchanger (see
Fig. ib). As descrited above, the lower plenum region is treated by means

cf & well mixed tank approximation, in contrast to the pilug flow model for

L]
i

the remaining regions. In Eg. 27, tg is the average residence time and

 

(PQfié)E is the average fission production rate for each element of fluid
in the lower plenum.

The boundary conditions for each region reqguire that the precursor
concentrations in the salt, Cg} are continuous along the path of flow.

Ls applied tc Eq. 27, these conditions are;

cf(z = Q) = cg(z =Ly , (28)
ci(t =t,) =cl(z=0) . (29)

These conditions, Jogether with the continuity conditions given sbove,
corpletely determine the solution of Ras. 24 through 27. The results

cf iategrating the =bove differentiel eguatiocns are:

 

 

Z Z
_(Rg TW) s . (A, w)( 7 )
S o, - e P O , c dz' -
e (2) = ¢(0)e £y B (20)e 7 - (30)
0 c
H <2z < H;
C
2=, z-7"
n o+ W) - -
o o '\}l + W)( 1;‘,7'1-\‘1 ) ?Z o ()\‘l + w)(vu )dzF
e (z) = c/(H Je 1 B (2 )e + » (31)
1 1 C !JH S /

 
 

 

 

 

H= 2 <1
z-H
o) - ome T Ve
c;(z) = ci(H)e , (32)
=L =0
0 0 " +W)tfi o, Fi - e_(Kjfiw{)f}i’-a
Ci(o) = Ci(L)@ + fii(P Qé)g “”““x;“$f;““““ 5 (33)
0. O {h, W)t
(“3} _ Bl(P Qé)fi . O(L) _ Bi(P ¢é>£ 1—-e U ¢ (3h)
Ciig ™ NoFw Ici X (h, + W)T, ° 3

The last of the equations given above results from averaging the precursor
concentration obtained from solving Eg. 27 over the residence time tg'

By use of the continuity conditions for the entrance and exit concentrsa-
tions in each region, the above equations may be used to solve for c?(o),

The result is-

 

 

 

H «z'
C .
. f 5, . E(Ki+w)( v, * tu+tex+tfi) g
e, o) = j B.F Qg(z')e T
1 o .
H-27
E . -(y ) 7 byt .
* J BiP fié(z‘)e v
H u
c
§
- 1 - e“(hi+w)tfl "(K1+W)tcirc
+L-° . [ o
+ Bi(P Qé)fl | Xi o 1L —e > (35)

where the following definitions of the residence times have been used:
 

Hc
‘:C:{F’} (36)
C
H - HC
o= s (37)
Q
L —H
tex‘“ ~ (38)
ex
% =4+t o+ .
cire c * tu tex * tfl (39)

The computational procedure developed in this section may be sum-
marized as follows:

1. Calculate spproximations to the static flux and adjoint flux
axlal distributions by standard techniques of core physics analysis.

2. From a specified assymptotic inverse period, w, and the static
flux distribution, € ., corresponding to the same reactor state, calculate
ci(@;w) using Fg. 35. Note that the absolute normalization of the flux
e arbitrary since Eg. l4 is independent of the flux normelization.

3. Calculate the axial distritution of precursor densitles in the
salt, cg(z;w)? by means cof formulas 30 through 34. FEither a numerical
integration procedure or analytical approximations for £ (z) can be used
in evaluating the integrzls. The former method was used in the work
described in the fcollowing section.

L. Calculate Py by performing the integrations in Eg. 14,

t is obvicus that the only means of practical calculation with this
scheme is the digital computer. A calculation program based on this scheme
was written for the IBM-7090. Specific details concerning the numerical
resuite, and the application of this analysis to the MSRE experiments are

2s

-t

given in Section
 

17

3. NUMERICAL AND EXPERIMENTAL, RESULTS

Numerical Groundwork

 

Our intention in this section is to apply Eq. 14 to the analysis
of a series of rod bump-stable period measurements made with the MSRE
during fuel circulation. These measurements were taken at various con-
trol rod insertions during the course of enrichment of the fuel salt in
Run Ko. 3. It has been seen from the analysis presented in Section 2
that the quantities required to relate the measured asymptotic periocd
to reactivity are the static distributions of the precursor concentration
and the fission production rate together with the asgscciated adjoint flux
distribution. We have already introduced several approximations in order
to simplify calculations with Eg. 14. We now make explicit use of the
group diffusion medel for the neutron fluxes. The flux distributions we
shall use were obtained from standard core physics analysis with one-
dimensional multigroup and two-dimensional, few group diffusion models
for the neutronic behavior. In performing a "first round" analysis of
the experimental measurements, the attempt was made to maintain as much
computational simplicity as possible without discarding the essential
features of the reactor and fuel circulation models.

It has been shown that the static distribution PO¢S and @Q occurring
in Eq. 1k are those appropriate to the asymptotic state, 1.e., the
reactor-rod configuration during the period measurement. However, the
three control rods in the MSRE are in a highly localized cluster about
the center of the core (Fig. 2), and the insertion of a single rod
produces a perturbation in the thermal flux distribution which is fairly
well localized in the radial direction. Some calculational results in
support of this conclusion are shown in Fig. 3. These are plots of the
radial variation of the thermal flux, taken at an azimuthal angle half
way between control rods 2 and 3. The calculsations were made with the
two dimensional diffusion program EXTERMINATOR,® using an R-6 model of
the MSRE core gecmetry. The perturbed fluxes with rod No. 2 inserted

and with all three rods inserted are compared with the fluxes when all
18

OREL TWG 64-8214

 

 

 

 

TYPICAL FUEL PASSAGE

MOTE: STRINGERS NOS. 7.60 AND &t {(FIVE! ARE
REMOVABLE.

 

 

 
    
  
  
 
   

74—~ CONTROL ROD

! 77

9,
\GUIDE BAR
NN

REACTOR
vJgfg’c‘:Em'ERLmIE

 

 

i
™

2

| THREE GRAPHITE AND
INOR-8 REMOVABLE
SAMPLE BASKETS

 

 

 

 

 

s

AN
/|
A

|

5
NN

   

 

 

 

REACTOR CENTERLINE

Fig. 2. Iattice Arrangement of MSRZ Control Rods.
 

19

CRNL-DWG 66-8563

 

 

 

 

 

 

 

  
  

_S><-ROD NO. 2 INSERTED
» - i SAMPLE HOLDER . _|

- |
CONTROL RCD AND

CONFIGURATICN FOR
FLUX CALCULATIONS

 

LTHREE RODS INSERTED

 

 

 

@

 

 

 

THERMAL. FLUX AT 10 MW (normatized to axial position of maximum flux)

 

 

 

 

 

 

 

 

 

N

 

Fig. 3.
lated Curves).

20

30 40 50
RADIUS (cm)

&0 70 80

Radial Distribution of Thermel Flux in MSRE Core (Calcu-
SN
ot

 

rods are fully withdrewn.* In g1l cases, the fluxes are normalized to
a core power level of 10 Mw. It may be seen that the shape of the flux
is relatively undisturbed over a large fraction of the radial cross
secticn of the core.

One may also note that the total neutron production rates in the numer-
ator and dencminator of Eq. 1li are multiplied by the energy spectrums of
neutrcn production. Hence, when performing the integration over energy
in evalusting the scalar products in Eg. 14, only the high erergy portion
(vhere T(E} # 0) contributes tc the result. We therefore need only
approximate the fast group adjoint flux distribution.

Based on the preceeding observetions, as a final set of simplifying .
approximations we have used only the unperturbed distributions of fisslon
production and fast adjeint flux, corresponding to the case of the rods
fully withdrswn from the core. This simpliification is consistent with
the neglect cf the radial variations of flux and precursor ccuncentrations
over the core which was discussed in Sectlion 2.

The calculated unperturbed axisl distributions of fission production
rates and fast adjoint flux are given in Fig. 4. These were used for
the calculstion of the axial distributicns of the six delayed neutron
precursor groups, in accordance with the scheme developed in Section 2.
The Individual precursocor ccncentratious, cg(z)g were multiplied by Ki
and summed to obtszin the total locsl rete of emission of delsyed neutrons
along the flow path through the resctor core. These resulting totsl dis-

tributions are shown in Fig. 5

©
#

for the particular cases of w = 0
(circulating critical) and w = O.1 SGCHA(EO sec stable period), and also
for the case when the fuel is not circulating. All three distributions
are normalized to the same total production rate, Pofién The bottom
pienum is omitted from Fig. 5 since the well stirred tank model was
used to represent the delayed neutron production in this region.

Baced on these sarme unperturbed fluxes, the relationship between the
stablie inverse period, wfilg and the increment in static reactivity

corresponding to the rod bump is shown in Fig. 6. For comparison

 

¥The "dip" in the thermal flux when the three rods are fully
ithdrawn is caused by the presence of the INOR-8 rod thinmbles.

 
 

ORNL-DWG 66-8582

 

 

b T

 

3 |

12

 

 

_EOTTOM
"~ PLENLM

| PLENUM

 

 

 

CHANNELED REGION - TOPmB{

 

 

 

 

0%

08

 

 

 

 

 

 

06 }

AMPLITUDE (normalized)

04

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 4.

20 40 &0 80 100 120 140 160 180
AXIAL DISTANCE FROM BOTTOM OF GRAPHITE {cm)

Axial Distributions of Fission Density in Fuel Salt and

Fast Groups Adjoint Flux (Calculated Curves).

200
M
Do

ORNL-DWG 65-i0654R

 

 

 

5.5 ; T : T T T T
e ——— - CHANNELED — |_——’+ TOP —f==-
' REGION ! ‘ PLENUM

50 ------ —_ e T—-- -
e, |
4.5 __I —e __ L 4}_ _l— ,,,,,, j

 

 

\ i
/ STATIONARY l ! |
so| Lo N
? | | I

30— e A |
CIRCULATING, w =0 " | T

2.8 koo e e T T Nt _%_
/ / . ) i

—

 

40 b e A N e
|

 

 

 

 

 

 

 

 

 

| % NG N
, : _/ CIRCULATING, w = 01 sec™! \ \ l
.5 — f e S i e by }

 
  

 

 

 

TOTAL. DENSITY OF DELAYED NEUTRON EMISSION IN SALT {arbitrary units)

 

 

 

0 2C 40 €C 80 100 420 140 10 180 200
DISTANCE FROM BOTTOM OF GRAPHITE (cm)

FPig. 5. Axial Distribution of Net Scurce of Delayed Neutrons in
MSRE (Calculated Curves).

 
ORNL-DWG 65-10

 

 

100 10f

107"
INVERSE PERIOD 7 (sec™)

 

10°¢

 

 

nverse Period of MSRE (Calculated

table T

Reactivity vs S

Fig. 6.

Curves ).

 
 

purposes, & calculated curve is also given for the standard inhour rels-
tior*?8when the fuel is stationary in the core. All of these calculations
are based on the static velues of Bi and. Ai and the fuel residence times

given in Table 1.

Table 1. Static Delayed Neutron Precursor
Characteristics and Tuel Residence
Times used in Calculations

 

Delsyed Group¥ 1 2 3 L 5 6 )
10t B, (n/10% n} 2.23 1L.57 13.07 26 .28 7.66 2.80

Ag (Secml) 0.0124  0.0305 0.111hk ©.3013 1.140 3.010

Fuel Residence Times {sec)®

 

Core (graphite mcderated region) G.h
Upper plenum 3.9
External system 8.1
Lower plenum 3.8
Total 25.2
#*

B, corrected for the difference between the statlc energy

spectrums of prompt and delsyed neubron emission.®

As the curves in Fig. & illustrate, because the margin between the prompt
and delayved neutron sources has been increased by the spatial transport
of the precursors, the same static reactivity increment produces a faster
rate of rise of both neutron and precursor densities. The calculated
curves extend somewhat beyond the region of practical interest in experi-
mental measurements of the stable period. They are presented mainly for
the purpoeses of illustraticon. In analyzing the experimentsl data, it
proved to be most expedient tTo carry out machine calculations of the

reactivity increments corresponding to each observed stable period.

 
 

M2
A

Experimental Results

 

In the chronology of the MSRE zero power experiments, the measure-
ments of interest in this study began at the point of reaching the mini-
mum #3250 concentration required for criticality with circulation stopped
and all control rods fully withdrawn. A summary of the zero power experi-
ments 1s given elsewhere,iO and we consider only those details pertinent
to the teopic of this report.

Once this first basepcoint 2357 concentration had been reached, the
concentration was further increased by the addition of capsules of en-
riched fuel salt containing approximately 85g amounts of 23%U. The
amount of control rod insertion required to compensate for these additions
was meagsured, and period measurements about these new critical rod posi-
tions were made. Care was taken to determine the minimum extra addition
of #3%U required to reach criticality with the fuel circulating and all
rods fully withdrawn. Once this second basepoint was reached, the
critical rod position and period messgurements were made first with the
pump stopped then with the fuel circulating. The measurements were
terminated when enough ®2%U had been added to calibrate one rod over its
entire length of travel.

Most of the period measurements were made in pairs. The rod whose
sensitivity was to be measured was first adjusted to make the reactor
critical at about 10 watts. Then it was pulled a prescribed distance
and held there until the power had increased by sbout two decades. The
rod was then inserted to bring the power back to 10 watts and the measure-
ment was repeated at a somewhat shorter stable period. Two fission cham-
bers driving log-count-rate meters and & two-pen recorder were used to
measure the period. The stable period was determined by averaging the
slopes of the two curves (which usually agreed within about 2%). Periods
observed were generally in the range of 30 to 150 sec.

Prior to pulling the rod for each period measurement, the attempt
was made to hold the power level at 10 watts for at least 3 minutes, in
an effort te assure initial equilibrium of the delayed neutron precursors.
Generally, however, it was difficult to prevent a slight initial drift

in the power level as observed on a linear recorder, snd corrections were
therefore introduced for this initial period. The difference between
the reactivity during the transient and the initial reactivity, as com-
puted from the in-hour relstion, was divided by the rod mevement and
this sensitivity was ascribed to the rod at the mean position.

The results of the rod sensitivity measurements made with the fuel
stationary are shown in Fig. 7. Since the rod resctivity worth Is
affected by the 235U concentration, theoretical corrections have been
applied to the raw data to account for the fact that the 23517 mass was
continually being increasgsed during the course of these experiments. The
rod sensitivities shown in Fig. 7 have been normalized to & single 235U
concentration, that at the beginning of the calibration measurements.

Tne calibration curve in Fig. T constitutes a convenient reference
from which the static reactivity egquivalent of the 23%U additions can be
determined, simply by relating the integral under the curve between the
fully withdrawn position and the critical rod position to the 23U nmass.
By applying this egquivalence to the minimum ®25U increment between criti-
cality with the fuel stationary and with it circulating, one obtains an
experimental determination of the resctivity loss increment due to cir-
culation. In addition, the curve in Fig. 7 is a convenient compariscn
curve for the rod sensitiviiy data taken while the fuel was circulating.

In the first of the two above cases, the reactivity increment
cbtained from experiment was 0.212 * 0.004% 6k/k. This was found to
compere with 0.222% cslculated directly from Eq. 1k, with w = 0. In the
second case, the rod sensitivities deternmined from period measurements
during circulation and Eg. 14 are shown in Fig. 8. Note that the solid
curves in Figures 7 and 8 are identical. The experimental points were
found to be distributed fairly closely about the reference curve, but the
scatter of points relative to this curve is somewnat larger than observed
with the data points of Fig. 7. Suggestions concerning the possible
sources of this scatter, and methods for improving the precision of these

measurements are included in the following section.

 

 
@

 

 

0.C7 l»/

27

 

 

 

0.06 |

005 - ———- Tyflf———l S———
0

 

e
1B e B io
g e ®le

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o
£
™
S
=
“)
3
I
—' 004 g
Eo |
1
= 003 .% .
= / |
E # J
z / ‘ |
u 0.02/ booom
L
L b
I ! @
% ootp o
]
0 5 g 12 16 20 24 28 32 36 40 44 48 52

ROD POSITION (in.)

Fig. 7. Differential Worth of MSRE Control Rod No. 1, Measured with

Fuel Stationary.

(Normalized to initial critical 227U loading).
Do
o

 

 

 

   
 

 

 

 

0.07 (— : - e — : P S
__ 006 Lo o +#_
%\ 008 i———— AR L L NG e ————y
o
T 004 —
o
O
=
4 CO3 e 4
2 FUEL STAT:ONARY
=
Lt
& ooplf LTl NG
L
L
o
201k
"o 4 8 2 6 20 24 28 2 36 40 44 48 52
R3OD POSTTION (im)
Fig. 8. Differential Worth of MSRE Contrecl Rod Ne. 1, Measured with
Fuel Circulating. (Nermalized to initial critical ?3°U loading).

 
N
N

L, DISCUSSION OF RESULTS

Thecretical Meodel Verification

 

In keeping with the order of presentation of the preceeding part of
this memorandum, we shall first discuss those aspects of the results
which relste directly to the theoretical model. Following this, some
suggestions concerning the MSRE experiments will be made.

As a test of the theoretical model for determining the effects of
fuel circulation on the delayed neutron precursors, quite satisfactory
results were obtained for the reactivity increment between the just-
critical states with the pump stopped aend with the fuel in steady circu-
lation. Here, the principal difference in our model and an earlier
calculation by Haubenreich® is the specific inclusion of the top and
bottom reactor plenums as additional sources of delayed neutron emission
which contribute to the chain reaction. As was shown in Fig. 5, the
combined effect of fuel flow and radioactive decay cof the precursors is
to "skew" the distribution of delayed emission toward the upper core and
the top plenum. It is important to notice that this relative increase
in emission of delayed neutrons in the top plenum is further accentuated
by the much larger fuel volume fracticn in this region (the effective
neutron preduction is the product of the source in the fuel salt times
the fuel volume fraction). Because Haubenreich's model accounted for
the ultimate contribution to fission of only those delayed neutrons
emitted in the channelled region of the core, he obtained a somewhat
larger effective loss in delayed neutrons due to circulaticn.

Although these results seem to constitute an adeguate test of the
model for the steady state condition, we cannot extend this claim to the
stable asymptotic periods measured over the entire range of rod travel,
due primarily to the lack of precisiocn in the period measurements. How-
ever, until more precise measurements can be made, this calculational
model gppears to be an adequate description of the delayed neutron pre-
cursor transport due to circulation. |

Mcre generally, one may pose the guestion of the need for sensitive

S

e tests and models of the delsyed neutron kinetics during fuel circulation
at zero power. For the rod calibration work, it is always possible to
stop the fuel pump, perform the tests with the fuel stationary in the
core, and use simpler, well developed methods for the analysis of stable
periods. Alsc, the questions of ultimgte safety and stability of the
reactor generally involve large reactivity sdditions which take place on
a time scale short compared tc the core transit time, and greatly in-
fluence the kinetics of the prompt neutron generation. These questions
also involve nonlinesr interacticn effects with temperature at high
power.

In addition to the theoretical interest, there are several contri-
butions & good understanding of the effect of circulation on the delayed
neutron kinetice can make from the viewpoint of resctor operation. For
example, the observed differences between the fuel stationary and circu-
lating critical conditions can comstitute a sensitive method of determin-
ing fuel circulaticn rate, amount of circulating gas bubbles, or any
gpecial characteristic due to the circulation. Necessary control system
sensitivities and response times for normal operation can be specified
with less uncertainity in the desigr of other molten salt reactors.

Although muck of this study coverlaps that of earlier work, it is
useful to re-emphasize here that the kinetics of circulating fuel systems
belong to a mathematical category walch is fundamentally different from
that of stationary fuelied reacters. This is particularly important in
the description of reasctor transients cccurring on a time scale com-
parable with the fuel transit times. It is appropriate to refer to the
circulating fuel systems as "time-lag"’ systems, governed by partial
differential equations for the delayed emitter ccncentrations, rather
than by ordinary differential equations. For many purposes, spproximate
reductions of the eguations to a form similer to the kinetics equations
of the staticnary fueled systems are adequate (e.g., the replacement of
the individual delayed fractions, Si, by effective reduced values, Biff;
in the conventional reactor kineties equations)a This reduction, however,
does not seem tTo be fully Jjustified for the gnalysis of an arbitrary
transient. In this study, the only conditions we have fully examined are
the Jjust critical state and the state of flux changing with a stable

asymptotic period. For a complete thecretical understanding of the

 

 
 

 

%U-
L

coupling of the circulation with the delayed neutron kinetics, we need
slso to analyze the transient delayed neutron modes, mentioned briefly
in Section 2. This transient coupling determines the effective time to
establish the asymptotic period, and should slsc provide a basis for
approximations made in the analysis of arbitrary transients. Scome pre-
timinary work has been done on this problem, and we hope to make this

the subject of s future memorandum.

Recommendations for the MSRE

 

One reason for the lack of precision of the rod bump-period measure-
ment of control rod sensitivity 1s that the latter is the ratio of two
amall quantities. For best results, this requires the experimenter to
maintain high precision in both the measurement of the increment in rod
position and the slope of the log n vs time curve. Early in the course
of thesge experiments it was decided that only the regular reactor instru-
mentation would be used for the rod sensitivity measurements. Determi-
nation of the period in each measurement involved laying a straight-edge
along the pen line record on the log n chart and reading the time interval
graphically along the horizontal scale which corresponded to a change of
severagl decades in the neutron level. OSince these charts, together with
the pen speeds, are subject to variations, this is a probable source of
error in the rod sensitivity measurements. A second possible socurce of
error, probably less important, lles in the measurement of the increments
in the rod pcsitions. It has been seen that the scatter in the experi-
mental points was larger for the sensitivity mesasurements made during
circulation. This is expected, since similar increments in rod with-
drawal result in a shorter stable period under these conditions.

Although a complete error analysis has not been carried out, these
considerations would seem to be obvious starting polnts in any future
attempt to obtaln more precise measurements of the rod sensitivities.

At the time these experiments were performed, the MSRE on-line computer
was unaveilable for the automatic recording of experimental data. It
would be useful, during future MSRE operation, to repeat some of the

period measurements while at zero power, using the computer in the data
logging mode to sutomatically record the log n and time intervaels. The
shim rods could be adjusted to vary the initiasl critical position of the
regulating rod for the period messurements. The immediate need for more
precise experiments is not great, since the precision of the rod cali-
bration cobtained from the measurements with the fuel stationary is

Judged adequate for further operation.

ACKNOWLEDGMENT

The cooperation of J. R. Engel and various members of the MSRE
operating staff in performing the experiments and aiding in analysis
is gratefully acknowledged. Also, the author is indebted to J. L. Lucius
cf the Central Data Processing Facility for programming the machine

computatvions to implement the analysis.

 

 
33
REFERENCES
1. J. A, Fleck, Jr., Thecory of Low Pcwer Kinetics of Circulating Fuel
Reactors With Several Groups of Delayed Neutrons, USAEC Report
BNL-334 (7-57), Brookhaven National Labecratory, 1955.

2. B. Wolfe, Reactivity Effects Produced by Fluid Mction in a Reacter
Core, Nucl. Sci. Eng., 13(2): 80-90 (June 1962).

 

P. N. Haubenreich, Prediction of Effective Yields of Delayed
Neutrons in MSRE, USAEC Report ORNL~TM-380, Oak Ridge Naticral
Laboratory, Octover 13, 1962,

A

i. T. Gozani, The Concept of Reactivity and Its Application to
Kinetics Measuremerts, Nukleonik, 5(2): 55 (1963).

5. B. Friedman, Principles and Techniques of Applied Mathematics,
Chapter 1, Jchn Wiley & Sons, Inc., New York {1956).

6. T. B. Fowler et al., EXTERMINATOR — A Multigroup Code for Solving
Neutren Diffusion Problems in One and Twe Dimensions, ORNL-TM-342,
February 1965.

7. P. N. Hauvenreich et al., MSRE Design and Operaticns Report,
Part ITT: DNuclear Analysis, USAEC Report ORNL-TM-73C, Cak Ridge
National Laboratory, February 3, 1964.

8. S. Glasstone and M. C. Edlund, The Elements of Nuclear Reactor
Theory, Chapter 10, Van Nostrand, New York {(1952).

9. R. C. Robertson, MSRE Design and Operations Report, Part I:
Degcription of Reactor Design, ORNL-TM ra% Janudrg 1965, p. 102.

1C. P. N. Hauvenreich et al., MSRE Zero Power Physics Experiments,
Ozk Ridge Nat iocnal Laboratory Report in preparation.

...........
Paiepazasaraaial
 
 
 

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