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A THEORETICAL STUDY OF Xe '35 POISONING KINETICS IN

FLUID-FUELED, GAS-SPARGED NUCLEAR REACTORS

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A THEORETICAL STUDY OF Xe'35 POISONING KINETICS IN

FLUID-FUELED, GAS-SPARGED NUCLEAR REACTORS

M. T. Robinson

DATE ISSUED

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Operated by
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A Division of Union Carbide and Carbon Corporation
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CONTENTS
FEPOAUETION oottt ettt e et ettt e s et s ettt et r b e s shen st s ]
Derivation of the Differential EQuations ............oocoii i e 1
Relations Between the Various Phase-Transfer Rate Constants.............ccooveiiiiiiiiiiine, 4
Solution of the Differential EQUAtions ...........cooooiiiiiii e e, 6
Steady-State Operation of @ Reactor...........cocoiiiiiiiiiiiiciiiceecc e s 9
Kinetics of Xe 32 Poisoning inthe ARE ... e 12
Kinetics of Xe '3 Poisoning in the ART ..o 15
Kinetics of Xe'3% Poisoning During Shutdowns.........coco.ooviiiiiiece e, 18
Nomograms for Xenon-Poisoning Caleulations ...........cocooiiii 18

 

 
 

A THEORETICAL STUDY OF Xe '35 POISONING KINETICS IN FLUID-FUELED,

GAS-SPARGED NUCLEAR REACTORS
M. T. Robinson

1. INTRODUCTION

One of the substantial advantages claimed for liquid fuels in very-high-power nuclear reactors
is the easy removal of Xe'3® from the fuel, with the consequent gains in neutron economy.'
This claim is at least partly supported by operating experience with the ARE,?2 This report is

135 boisoning in a reactor in which this

concerned with a theoretical study of the kinetics of Xe
volatile poison is continuously removed by a stream of sparging gas. The theory is applied to
the experience with the ARE and is used to make predictions for the ART. Some comments on
full-scale aircraft power plants are also included.

The system is assumed to consist of two phases: the liquid fuel and the sparging gas. The
theory is concerned only with volume-averaged concentrations and neutron fluxes. Turbulent
motion of the two fluids is held to assure thorough mixing within each phase. The appropriate
differential equations which describe the behavior of the poisoning in such a system are derived
and solved. Steady-state behavior during high-power operation of the reactor is discussed.
Detailed kinetics of the poisoning during the approach to steady state are studied through a
series of calculations performed on the Oracle. A brief discussion of shutdown behavior follows,

135

A final section presents a rapid approximate method for calculating Xe poisoning in gas-

sparged fluid-fueled reactors.

2. DERIVATION OF THE DIFFERENTIAL EQUATIONS

135

The volume-averaged concentration of Xe in the fuel of a fluid-fueled nuclear reactor

changes because of a number of different processes, as shown schematically in Fig. 1. These

 

]W. R. Grimes et al., The Reactor Handbook, vol 2 (September 1953), p 973.

2M. T. Robinson, W. A, Brooksbank, and D. E. Guss, ANP Quar, Prog. Rep, Dec, 10, 1954, ORNL-
1816, p 124-125.

L ]
SSD-A-1167
ORNL—-LR~DWG 6430A
PROCUCTION FROM
135

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DECAY OF I
LOSS BY FLOW OF TRANSFER OF xe'33 ‘ LOSS BY
GAS FROM SYSTEM NRLLIY LIQUID TO GAS xe'33 N THERMAL-NEUTRON ABSORPTION
SPARGING - GAS y35 | LIQUID-FUEL
o LOSS BY PHASE TRANSFER OF Xe PHASE o
RADIOACT IVE DECAY GAS TO LIQUID * LOSS BY RADIOACTIVE DECAY

DIRECT PRODUCT!ION
IN FISSION

Fig. 1. Processes Governing Xe 133 Poisoning in Fluid-Fueled Reactors.

 

 
 

C e ‘:‘;
A

processes are as follows (see Table 1 for definitions of all symbols used):

1. direct production from fission,

 

 

Rate 1 = yxefifgfi ; (2.1)
2. production from decay of 1133,
-aat
Rate 2 = ylzfgb(] - e ) ; (2.2)
3. transfer from the gas phase to the liquid phase,
AegVe
Rate 3 = ——— (2.3)
VL
TABLE 1. DEFINITION OF SYMBOLS
English Greek
Lefl.ers Definifion Leflers Defin”ion
A Area of liquid-gas boundary surface % 100yxeaf/au
ag Activity of Xe'33 in the gas phase a, 100y|0’f/0'u
a, Activity of Xe '35 in the liquid phase a, )\Ir/,B)\J’r = RTS; see Eq. 3.4
cc Concentration of Xe '35 in gas phase a, Radioactive decay constant of 1133
c, Concentration of Xe '35 in liquid a, Radioactive decay constant of Xe!3%
phase
A V, Ve
&0 Concentration of 1'3% at ¢ = 0; see 135
Eq. 4.18 "1 Fission yield of |
T . 135
k* Mass-transfer film coefficient Yxe Fission yield of Xe
L o+ A+ A )\f Rate constant for transfer of xenon
! 4 / L from liquid to gas
k, a, + a2[3)\1r + Ag
)\g vG/VG
b Partial pressure of Xe'!3> in gas
phase A Txe®
0’ Rote of mass transfer A, Rate constant for transfer of xenon
from gas to liquid
R Universal gas constant
AgE 2 O'f Microscopic fission cross section
s Solubility coefficient of xenon in fuel of U233
T Absolute temperature Ef Macroscopic fission cross section
of fuel
t Time
. ) T, Microscopic neutron absorption cross
Ve Volumetric flow rate of sparging gas section of U235
Ve Volume of gas phase 2, Macroscopic neutron absorption cross
v, Volume of liquid phase section of fuel
. Xe 135 poisoning in fuel Oye Microscopic neutron absorption cross
. 135
section of Xe
y “Equivalent poisoning” in gas
phase; see Eq. 2.14 b Volume-averaged thermal-neutron flux

 

 

 

 
4, loss by radioactive decay,
. Rate 4 = —oc, (2.4)
5. loss by absorption of thermal neutrons,
Rate 5 = Ty PC, (2.5)
6. loss by transfer to the gas phase,

Rate 6 = —)\ch . (2.6)

135

The over-all time dependence of the Xe concentration in the liquid phase is given by the

sum of these six rates:
VG

. -t
L= Vbt yE el - e ) +.7'L_A’CG - oy ogd + Adey . 27)

135

The processes which change the volume-averaged Xe '*” concentration in the gas phase are

as follows:

7. transfer from the liquid phase,

t\chVL
Rate 7 = —— (2.8)
VG
8. loss by radioactive decay,
Rate 8 = ~AyC o ; (2.9)
9. loss by transfer to the liquid phase,

Rate 9 = -A c. ; (2.10)

10. loss by flow of gas out of the reactor,

Yc‘G
Rate 10 = ~

 

v (2.11)

135

Several ways in which changes might occur in the concentration of Xe in the gas phase have

been specifically neglected; these are:

11. loss by absorption of thermal neutrons;

12. production from decay of 1'33 or from fission. This implies the neglect of transfer processes
(like 3, 7, 9, and 10) involving 1'3% or U235,

135

The over-all time dependence of the Xe concentration in the gas phase is given by the sum

of processes 7 through 10 to be

. Vi ( Vs
Cc = A c, = |ay + A+ — e - (2.12)
Ve / TV,

135

In this discussion of the behavior of a nuclear reactor, the behavior of the Xe poisoning

 
 

is of primary interest and is defined as

1000y, ¢, 2.1
X = ————Eu . .
The related quantity y is defined as
]OG'TXQCG .14
Yy = s . .

U

The virtue of this latter quantity stems from the identity

X CL

o (2.15)
y €c

 

which will be required in deriving a relationship between )\/ and A. By the use of Eqgs. 2.7,
2.12, 2.13, and 2.14 and some abbreviations from Table 1, the differential equations for the

poisoning are written as
. —ant
s o= oagh, +ad (1~ e 3) 4oy — (g + A+ A x (2.16)

y = BAx = (ag + a;BA, + Ay . (2.17)
The above equations apply during the nuclear power operation of a reactor. However, the
behavior of the poisoning during a shutdown must also be discussed. Inthis case it is necessary

to set A, = 0 and to replace the first two terms of Eq. 2.16 by the source term

agdge 3. (2.18)

The boundary conditions needed in solving Eqs. 2.16 and 2.17 are discussed in Sec. 4.

3. RELATIONS BETWEEN THE VARIOUS PHASE-TRANSFER RATE CONSTANTS

The problem of studying the kinetics of Xe!33

poisoning can be simplified by eliminating
one of the phase-transfer rate constants, defined in Egs. 2.6 and 2.10, The total rate of transfer
of xenon from the liquid phase to the gas phase is )\/VLCL. The total rate of transfer in the
reverse direction is AV c . Now, while it probably cannot be realized in practice, there exists
some pair of values (c’&, cz) corresponding to true thermodynamic equilibrium between the two

phases. The “‘law of mass action'’® requires that under these conditions the amount of material

entering a phase be the same as the amount leaving, that is, that

 

,\vac}: - )\'VGc*(‘;
or
VL CE,
A= A — — (3.1)
VG e

 

3C. M. Guldberg and P, Waage, Etudes sur les affinities chimiques, 1867,

 

 
The solubility coefficient of a gas in a liquid is the equilibrium concentration of solute in the

liquid phase when the partial pressure of the substance in the gas phase is 1 atm. That is,
c* = prS = cX RTS , (3.2)
where the ideal gas law has been used in the form

b = g RT

135

to relate the Xe pressure to its concentration in the gas phase. A combination of Egs. 2.1

and 3.2 gives the desired result:
vV

L
A= A, RTS — .
r f VG ’ (3 3)
whence
a, = RTS . (3.4)

Thus equilibrium solubility data may be used to eliminate the rate constant A
Also, a relation may be derived between the ‘‘true’’ rate constants, z\f and A, and the *‘ap-

parent’’ rate constant,? :\P. The latter is defined by

Net Xe '3% transfer rate = -z\ch . (3.5)

Equating this to the sum of rates defined in Egs. 2.3 and 2.6, it is found that

 

A, = A - ,\,fi ° (3.6)
Vi cp
or, introducing Eq. 3.3,
c
’\P = Af 1 ~ RTS? . (3.7)

If Egs. 3.4 and 2.15 are introduced, then

Qyy
/\p = A 1 - - . (3.8)

Thus experimentally derived values of )\p may be compared with values calculated from the

solutions to Eqs. 2.16 and 2.17.

 

The connection of the rate constant )\f to the usual mass-transfer film coefficient may be
shown by noting that the total net current of matter across the boundary between the liquid and

gas phases is

Q"= -NVie, + AVgeg = MV, e, ~ RTS¢g) (3.9)

r

According to the usual mass-transfer analysis,® the total current may be written as

Q" = —k"Ala;, - ag) . (3.10)

 

4. L. Meem, The Xenon Problem in the ART, ORNL CF-54-5-1 (May 3, 1954).
SG. G. Brown et al., Unit Operations, p 510 f, Wiley, New York, 1950.

 
 

 

Both phases are assumed to be ideal. The xenon activity in the liquid may be replaced by the
concentration. Therefore the standard state in the gas phase must be considered as that pressure

of xenon in equilibrium with unit concentration in the liquid. Thus
a. = pS = RTSc. .
Then Eq. 3.10 becomes
Q" = ~k’Alc, - RTSc.) . (3.11)
Comparison of Eqs. 3.9 and 3.11 yields

k’A

 

(3.12)
L

In principle, the film coefficient 2" can be computed from the geometry of the system and

the physical properties and flow rate of the liquid fuel through a relation of the type

k’s

 

= f(SC, Re) , (3.]3)
L

135

where s is a characteristic dimension; D, is the diffusion coefficient of Xe in the liquid;

Re is the Reynolds number of the liquid; and Sc, the Schmidt number, is given by

vy

Sec =

- I
Dy
in which v, is the kinematic viscosity of the liquid. It does not appear practical to calculate

)\f in this way, because of the complicated geometry and flow regime obtaining in the ARE and

ART.

4. SOLUTION OF THE DIFFERENTIAL EQUATIONS

135

The time dependence of the poisoning of a nuclear reactor due to Xe may be expressed

by the differential equations

x = [ (1) + ahy — kyx (4.1)
and
y = Bx = kyy . (4.2)
The source term is
) = agh, + ap (1 = ¢ %) (4.3)

when the reactor is in operation and

fole) = aa&oe—aat (4.4)

|l35

otherwise. The quantity &0 is related to the amount of present at t = 0,

By solving Eq. 4.2 for x, differentiating with respect to t, and combining the results with

 

 
 

Eq. 4.1, the differential equation
y o+ (kg + k))y + (kiky = aBA)y = B ()
is obtained. The solution to this equation may be written as

--K.IZ

 

 

y = (I)n(t) + Ae + B_e .
where
1
2 2
q =5 by + &y + ik, — 57 + 40,803]
' J
Ky = (kg + Ry = Ve - k)2 B2
-~ gt
(g + a)BAA, B e

D) = i
kyky — a BAf (k) — ag)lk, ~ ay) - azfi)\?

 

I

B)lfaacfioe
(ky = ag)(ky - aj) ~ azfi)‘f

Combining these results with Eq. 4.2 yields

 

() =

1 -K]t 2

where

(CLO + al)kz)\L a])\L(

®](t) = -

kyky = aBAf k) = ag)(ky — a5) — a,BA]

 

...a.at

%&0 (ky ~ azle

@z(t) =
(k) — a3)lk, - a3) - 2'8)‘?

 

The most general boundary conditions are
As t — 0, X —> x, and Yy —>yg -
Inserting these conditions into Eas. 4.6 and 4.9, the integration constants become

ky = Ky B,
A, =——lyg =0, O] - —— {x, -~ 6(0)]

and

1
B, = ~——lyy = 9,00 + ———[x, - ©,0) .

(4.5)

(4.6)

(4.7a)

(4.7b)

(4.8a)

(4.9)

(4.10a)

(4.100)

(4.11)

(4.12a)

(4.125)
In this report three special cases will be considered:

Case |. Reactor operation starting with "‘clean’’ condition. ~ The poisoning is given by
Eq. 4.9, using the function 4.10a. The integration constants are found from Eqs. 4.12a and 5,
using x5 = yg = 0 and the quantities (D](O) and ®l(0)'

Case |l. Reactor operation at zero nuclear power, after a period of high-power operation. -~
The poisoning is given by Eq. 4.9, using the function 4,106, The initial conditions are found
from solutions to the problem of case I. In this instance A, = 0, and the quantity &0 is found

from the equation

ady = ari (1 - e , (4.13)

where A[ is the value of A, for the preceding period of high-power operation and ¢’ is the time

of operation.

Case lll. Sparging of reactor after a period of complete shutdown, during which no xenon is
removed. — The poisoning is given by Eq. 4.9, using the function @,(t). In this case A, =0,
Yo = 0, and &0 is found from

r’

, (4.14)

-a,t’ ~ant
a3&0=a/\(]-—-e ) 3
where ¢ ” is the time between the shutdown and the time ¢t = 0. The quantity x, is calculated
from
rr a A’ » rr »r
-a,tl 1L ~al —a.t -a,tl
x0=x0e4 +— (T -e 3)e 3 - %)

- o ' (4.15)

where xg is the poisoning at reactor shutdown, found from the solution to the problem of case .

In dealing with cases Il and Il above, it is of interest to know whether or not the quantities
x and y reach their extreme values {maxima) at the same time. When x reaches its maximum

value, from Eqgs. 4.1 and 4.2, then
ky* = BA [ = (kyky = azfi)\?)y* ' (4.16)

where the asterisks indicate the special time value. It can readily be shown that the coefficients
of /5 and of y* are both always positive quantities. Thus y* can vanish only if
BA;
y¥ = — 3 (4.17)
kik, — azfi)\f
Comparison of this equation with Eq. 4.6 shows that Eq. 4.17 cannot be satisfied in general, so
that the extreme behavior of x and ¥y cannot be examined by studying the differential equations

alone.

 
 

5. STEADY-STATE OPERATION OF A REACTOR

For very long times of high-power operation, the poisoning reaches a steady-state value,
From Eqs. 4.12 and 4.7, the steady-state values of x and y are®
(an + a])kz)\L
- Tt 5.1)
2
k]kz - azfi)\f

and

 

(g + a]))\Lfi)\f B)\f
. =— X . (5.2)
kiky, ~ azfi)\f 2

These relations can be used in estimating the steady-state poisoning of a reactor under various

Yo =

conditions. The most convenient way to make these estimates is first to calculate

A (OL4 + A)
AL 59
a, + )\g + az,B)\f

This result is obtained by substituting Eqs. 5.1 and 5.2 into Eq. 3.8. If the mean lives

T: = )\% (5.4a)
v
and
]
7 = (5.45)
/
are now introduced, then
Ty =T +_a4 oy ) (5.54)

Since a, << )\g, this expression may be rewritten as

RTS v,
G
Then x__ is computed through the relationship
(ao + a]).\L
I —— (5.6)
an + A+ )\P

The data in Table 2 have been used to estimate the steady-state poisoning, x_, in the ART for
various assumed values of the phase-transfer mean life. The results are presented in Fig. 2.
It is of interest to examine briefly the expected behavior of ART-type reactors of higher

power, Although the poisoning of an unsparged reactor of this type is essentially independent

 

6By an argument similar to that at the end of Sec. 4, it may be shown that x and y do not reach steady-
state values simultaneously. Egquations 5.1 and 5.2 apply only after both quantities reach steady state.

 
 

10

TABLE 2. DATA FOR NUMERICAL CALCULATION

 

Numerical Data

ay = 0.254% R = 82.0567 cc-atm/mole/°K
a, = 4.74% T = 1033°K (1400°F)

a, = 0.0509 S =6 x 10~7 moles/cc-atm(®)
ay = 2.87 x 107% sec™! Oye = 1.7 x 108 barns(®)

a, = 2.09 x 1075 sec™!

Reactor Data

ARE (¢} ART(d)
vV, 5.35 f* 5.64 ft*
V 1 0.31 #*
ve 0.25 cc/sec 1000 STP liters/day
@ 8 x 10" neutrons/cm?/sec 1 x 10" neutrons/cm?/sec
B 5.35 18.2
A 1.36 x 1076 sec™! 1.7 x 104 sec™!

 

(a)R. F. Newton, personal communication.
“’)w. K. Ergen and H. W. Bertini, ANP Quar. Prog. Rep. March 10, 1955, ORNL-1864, p 16.
(C)J. L. Meem, personal communication and ARE Nuclear Log Book, ORNL Classified Notebook 4210,

(dy, T, Furgerson and J. L., Meem, personal communication.

of power, very large increases in poisoning are possible with increased power when efficient
sparging is employed. Only the most optimistic case will be considered, with 7= 0. Equation
5.6 may then be written as

(ao + a,))\L

- : 7
o "X, + (/RIS V,) &7

 

If there are no major differences in design of such a reactor and in particular if the fuel volume

and dilution factor are about the same as in the ART, the poisoning may be estimated on the

basis of ART data, by simply adjusting A, in proportion to the power change. The results are
presented in Fig. 3.7

 

TA scale for the amounts of helium required may be visualized by noting that an ordinary cylinder of
helium contains 220 scf of gas (6230 liters). !t may also be remarked that if’ff is obout 5 min, requiring
5000 STP liters of helium per day to maintain about 0.5% poisoning in a 300-Mw reactor, and if the aircraft
flies ot o speed of 1000 mph (about Mach 1.3 at sea level), the plane will get some 18 miles per gallon of

helium.

 

 
 

S!!-B-H 68

ORNL-LR-DWG-6434

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 1
<10 \\
g \
& \ = 20 min
|
=
S T —— =15 min
o - t
o
"o
> 05 \ --—-.__________ rf: 10 rl-nln
| \ &
*L \"‘"—-—.____ " T, 2 min
|
\________- rf= 0 nI“n
0 !
0 1000 2000 3000 4000 5000

SPARGING-GAS FLOW RATE (stp liters/day)

Fig. 2. Steady-State Xe!33 Poisoning in the ART as a Function of Sparging-Gas Flow Rate
for Various Assumed Values of the Phase-Transfer Mean Life Tpe

SSD-B-!!1O

ORNL-LR-DWG-7317

PHASE-TRANSFER MEAN LWE,I?'O
.Tafl
600 Mw

300 Mw
150 Mw
60 Mw

 

I

 

 

w
s

 

 

oS

 

 

 

 

Xe'>® POISONING (%)
o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

05N \\\\
\\\\\\»——-—
~
0

 

0 2000 4000 6000 8000 10,000
SPARGING GAS FLOW RATE (STP liters /day)

Fig. 3. Steady-State Xe '*° Poisoning in ART-Type Reactors
as a Function of Sparging-Gas Flow Rate for Yarious Assumed
Reactor Powers.

11
12

Appaangly, other things being equal, the sparging-gas flow rate must increase linearly with

power, to maintain constant poisoning.

It should be noted that while a decreased fuel volume

increases the term [vG/(RTS V)l in Eq. 5.7, this is roughly compensated by a corresponding

increase in AL’

which is proportional to the volume-averaged flux.

6. KINETICS OF Xe'35 POISONING IN THE ARE

An extensive series of calculations has been performed on the Oracle,®

the approach to steady state of the Xe'33

in Figs. 4 through 7.

Figure 4 illustrates the dependence of the Xe'®

to aid in studying

poisoning in the ARE. Typical results are presented

5 poisoning kinetics on the value of ?xf.

Note that curves for all values of )tf > 6 x 1073 sec™! fall together on the scale chosen in the

 

8Coding and supervision of the calculations were done by C, L. Gerberich, ORNL Mathematics Panel,
Results were obtained by using Egs. 4.1, 4.2, and 3.8, together with numerical data from Table 2, except

as noted in the text.

SSD-B-I !'1

ORNL-LR-DWG-T7318

 

0.09 —
A= 210 % sec!
0.08 A
=2x10 !

~ 0.07

&

; 0.06 X, =6 x i0-4 Sec-i

=

z

3 0.05 A =6 x10°3 sec™

£ 004

"

—>°<’ 003 b e 0.25 cc/secs 1
0.02 cxe¢=|‘36 x 10" sec
0.01

0
0 10 20 30 40 50 60 70 80 90 100

" TIME {hr}
Fig. 4. Effect of )\/ on Xe '35 Purging in the
ARE.

550-B-1Z12
ORNL-LR-DWG-7398

=0

@

=(Q.25 cc/sec

=0.50 cc/sec
=1.00 cc/sec

Ar = 6x10°% sec’!

. -6 -
:rxe¢-l.36x10 sec

             

0 10 20 30 40 5 60 70 8 90 100
TIME (hr}

Fig. 5. Effect of Sparging-Gas Flow Rate
on Xe'33 Poisoning in the ARE.

SSD-B-
ORNL-LR-DWG-T 399
014
1

:2.80 x10° sec”

o
n

o O
o =
@ O

=1.36 x 105 sec”!

<
D

o o
Q
B

A =6 2104 sec!
=(.25 ¢c/sec

Xe'>> POISONING (%)

o
<
o

 

Q

70 80 90 100

o
35

20 30 40 5 60
TIME {hr)

Fig. 6. Effect of A, on Xe '35 Poisoning in
the ARE.

SSD-B- 4
ORNL-LR-DWG-T400

 

A (sec™) ¥, (cc/sec)
6x10* 1.00
6x10': 0.50
-3 ex1d 0.25
10 ex10? 0.00
_ 6x10> 0.25
T 1104 0.25
L
2
.Y
10
0O 10 20 30 40 50 60 70 80 90 100
TIME (hr)
Fig. 7. Variation of the Apparent Purge

Constant with Time in the ARE.

 
figure. This results from the small volumetric flow rate of off gas in the ARE. This flow rate
is rate-determining, making an accurate estimate of )\/ from experimental data difficuit.

Figure 5 illustrates the poisoning effects that occur as a result of variations in the sparging-
gas flow rate, v, ot a value )Lf = 6 x 1074 sec=!. A comparison of Figs. 3 and 4 shows that

at early times (up to 10 hr or so) the rate of Xe 133

removal is primarily governed by the rate of
phase transfer, while for longer times the gas flow rate becomes controlling. Thus, under ARE
conditions, fission-gas removal may be termed ‘‘off-gas controlled."’

Figure 6 illustrates the effects of A, on poisoning kinetics. As might be anticipated, the
results are roughly proportional to A, .

Figure 7 presents results on the time dependence of )\p, which is called here the ‘‘apparent
rate constant’’ for transfer of Xe'3® from fuel to off gas. The large decrease in )tp with time is
clearly evident. Note also that d)tp/dt is everywhere negative.

Theory and experiment may be compared as follows.? By employing the abbreviations

f6) = agh, + ap, (1 - e ) (6.1a)
and
gty = ay + A, + )\P(t) , (6.15)
the differential equation 2.16 may be written as
x(t) = f(t) - gle)x(e) . (6.2)

Expanding each of the functions in Eq. 6.2 about the origin,

e 2 .l.t3

 

 

 

xot xo

x(t) = x4t + e (6.32)
. fot?

fe) = fo + fot + + e, (6.3b)
. g ot

glt) = gg + gyt + + e, (6.3c)

 

where the subscript zero represents values at the origin (¢t = 0). If Egs. 6.3z, b, and c are

introduced into Eq. 6.2 and if the coefficient of each power of ¢ is equated to zero, then
x.‘o = /0 = O.DA.L ’ (6-40)

.’;0 = f;a ~ fo8g = (205 — aggoh, (6.4b)

 

This approach was suggested by D. K. Holmes, ORNL Solid State Division.

13

 
14

xg = fo = fo8g = foB2 + 28g) = ~[a;05 + a5, + (g2 + &N,  (6.4c)

Now, in principle, a set of experimental data may be fitted to a power series {Eq. 6.3a), and the
various coefficients of the series (Eq. 6.3c) can be determined from Eqs. 6.44, b, and c. Note
that from Eq. 3.8

AP(O) = )\/ ,

so that the value of the coefficient g, (i.e., the behavior of the poisoning near the origin) is of
primary concern.

The experimental data on poisoning in the ARE'? are given in Table 3, along with calculated

149 U235, The neutron capture cross section of Sm'4?

11

contributions due to Sm and to burnup of

was taken as 53,000 barns,'! and the burnup effect was calculated from

(Ak) 0.232AM
k burnup M

where % is the infinite multiplication constant and M is the mass of U?33 in the reactor. Other

149

data were taken from Table 2. Since the Sm and burnup contributions are well within the

experimental error in the total poisoning, the experimental results are taken to apply to Xe!33

poisoning alone.

The results from the ARE cannot be treated by the method described above for two major
reasons:

1. The ARE data are based on the assumption that the origin of the (x,?) coordinates was
at the start of the experiment. Since about 7 hr of high-power operation preceded the ‘‘xenon

135

experiment,’’ 1% both 1135 and Xe!35 were present in the core at the time “‘zero’’ in Table 3.

 

ARE Nuclear Log Book, ORNL Classified Notebook 4210,

”S. Glasstone and M, C. Edlund, Elements of Nuclear Reactor Theory, p 338, VYan Nostrand, New
York, 1952,

TABLE 3. EXPERIMENTAL DATA ON ARE POISONING

 

Calculated Poisoning (%)

 

 

Time Total Poisoning
(hr) (%) Burnup 5, 149 xo 135
0 0 0 0 0
1.3 0,003 £ 0,001 0.0001 0.0000 0.004
12,7 0.006 * 0,002 0.0006 0.0003 0.110
13.7 0.009 * 0.002 0.0006 0.0003 0.119
16.0 0.012 £ 0,002 0.0007 0.0004 0.144
20,2 0.015 * 0,003 0.0009 0.0006 0.182

 

 

 
2, Applicatiamatmthe method outlined above requires knowledge of /\L. This quantity
governs the scale of the x-coordinate. For the present calculations, a value of 1.36 x 10~¢
sec™! was assumed, based on 1.7 x 10% barns for the Xe!3% cross section and 8 x 101!
neutrons/cm?/sec for the ARE thermal flux.

135 cross section in the ARE is nearer 1.4 x 10°

It has recently been shown that the Xe
barns.'2 The flux value employed was based on the values 575 barns for the U233 fission cross
section; 173 Mev per fission absorbed in the reactor; fuel density 3.24 g/cc; composition 13.59
wt % uranium, 93.4% enriched; and 2 Mw reactor power. The resulting value for the flux is not
more precise than 120%. It does not appear possible to expect agreement better than about a
factor of 2 between theory and experiment.

On this basis, results from the ARE have merely been compared with calculated curves

similar to those of Fig. 4. It is concluded that )\f must be larger than about 5 x 10~4 sec~! and

is probably around 1 x 1073 sec™!.

7. KINETICS OF Xe'35 POISONING IN THE ART

In this section the results of Oracle computations of the time dependence of the Xe'3®
poisoning in the ART are presented and discussed. The data employed are those of Table 2,

except as noted. Typical results are shown in Figs. 8 through 12.

 

2w, K. Ergen and H. W. Bertini, ANP Quar. Prog. Rep. March 10, 1955, ORNL-1864, p 16.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Y A
* o S$SD-C-1223
ORNL-LR-DWG-7677
| I l l
1.2 Ar= 6x10% sec! (7,228 min) _
10 // A,=8x10'4 sec’ {z=21 min) _
- | [ L L
[l ]
] Ar=1 x10° se¢! {r,=17 min)
~ 0.8 A
s / //
o0
206 /// A= 2x10°3 sec? (1,283 min) ]
Z /// Fx2x sec ' (1,=8.3 min
® / —— i i i f
O .
O ng /// // )‘,=4:(10'3 sec ! (r,=42 Imin)—fl
n 4 t t T
"o » /,/ A= 8x10° sec! (r,=2.4 min)—
0.2 //,/"'—'—f | ]
/A/ )~,,=6x10'2 sec’ (r/=0.3min) _|
0 L]
oy, =1.70 x10% sec’ xg=4.53n:10'3 sec’’ ’
(,=1030 STP liiters /day)
| b
0 20 40 60 80 100

TIME (hr)

Fig. 8. Effect of )t/ on Xe 135 Poisoning Kinetics in the ART.

15

 

 
16

Yy
SSD-B-1222

 

/o [ T

[

ORNL-LR-DWG-7676
] i !

 

W
o

 

 

na W
) o
\

|
\
\
|
\

 

 

/éz =0 { STEADY*STATE POISONING =3.98 %)
|

- T4 o
UXe¢ =1.70 x10 " sec

|
T

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

° A=A x 10 ® sec”
(_2?20 (lrf-17m1n)l ‘
=z = | '
S ] | |
2 '@ = 510 litersgrp / day
X 1.0 / | I
‘ :
05 \l{q = 103i0 liters g1 / day
. \
o 10 20 30 40 50 60 70 80 90
TIME (hr)
Fig. 9. Effect of Sparging-Gas Flow Rate on ART Poisoning Kinetics.
o s
SSD-B-1224
ORNL-LR-DWG-7678
1.2 T 7 T | I ! [
v, = C (STEADY-STATE POISONING =3.97 % )
1.0 ;
208 y, = 260 STP liters / day
(29 /’ —
2 06—~
v v, =510 STP liters /day
2 -/ L
8 04 | v, = 1030 STP lifers /day
2 /S ————
/// aXeqb=1.TOx1O sec
0.2 -
/ Ar =4 x10 " sec
( Tf =4.2 min )
O | ] !

 

 

 

 

 

O {0 20 30 40 50 60

TIME (hr)

70 80 90 100

Fig. 10. Effect of Sparging-Gas Flow Rate on ART Poisoning Kinetics.

 

 
 

o=

-B-1225
ORNL - LR-DWG-

SSD-A-1226
_q ORNL- L R-DWG-7680
Ty ®=1.70x10 " sec
¥, =1030 STP liters/day

     

102

 

EACH CURVE IS LABELED WITH THE
APPROPRIATE VALUE OF X,

 

<1030 STP liters/ day 1 GxTO-Z sec!
= 2 _ .-
¥* 510 STP liters/day I ZxICEZ sec_1
I 110" sec
3 %+ 260 STP liters /day K 6xG° sec’
¥ 4x10° sec’
T o 2x0° sed’
3
o $=170x10 e T o “(L sec_1
- a o < YO 8xi0° sec
- - = o - -
§ Xy = 4x10 Tsec E X 6!104 seci
:q {rp= 4.2 min} R
15t
163
0 10 20 30 40 50 €0 0 10 20 30
TIME (hr} TIME (hr)
Fig. 11. Time Dependence of the Apparent Fig. 12, Time Dependence of the Apparent
Rate Constant I\.P in the ART. Rate Constant Ap in the ART,

Figure 8 illustrates the dependence of poisoning kinetics on the value of )lf, for a value of
)\g = 4.53 x 10~3 sec™! (vg = 1030 STP liters/day).!3 The effects of sparging-gas flow rate
are presented in Figs. 9 and 10 for two different values of )\./. Because of the much higher
sparging-gas flow rates, the ART will not be as insensitive to the rate of phase transfer as was
the ARE. Examination of Figs. 9 and 10 shows that the reactor will be more sensitive to off-
gas flow rate if )\f is comparatively small than it will if )tf is comparatively large. Poisoning
kinetics in the ART can be termed neither ‘‘off-gas controlled’’ nor ‘‘phase-transfer controlled,”’

both processes being appreciably rate determining.

The time behavior of the apparent rate constant, )\p, is somewhat different from that in the
ARE, because of the much greater sparging-gas flow rate in the ART, Examination of Figs. 11
and 12 shows that at high gas flow rates )\p reaches its steady-state value very rapidly ~ only
about 3 hr being required, compared with about 40 hr in the ARE. Physically, this means that
the gas phase in the ART reaches a steady state with the fuel phase very rapidly.

Recause of the rapid approach to steady state of )\P, it is possible to use the approximate

method of Sec. 9 for rapid calculations of ART poisoning kinetics.

 

]3|n converting the values of A_ to the values of v quoted, it has been assumed that the gas pressure

&g
in the ART swirl chamber was about 2 psig. Then Ve (STP liters/day) = 2.27 X ]05 Ag (sec_]).

17

 
 

18

e 8. KINETICS OF Xe'3® POISONING DURINE SHUTDOWNS'

In this section a brief analysis will be made of the expected behavior of the Xe!33

poisoning
of the ART during shutdowns. For this purpose the equations derived in Sec. 4 for cases |l and
{1l will be employed.

First to be considered is a shutdown of nuclear power during which fuel flow and sparging
are continued. The reactor is assumed to have been at steady state prior to shutdown. The
data given in Table 2 for the ART are chosen, with 7 taken as 5 min. The result is not shown
since values for all the terms other than the one for 135 decay are always negligible. Under
the assumed conditions, the poisoning will not rise by as much as 1 or 2% of the steady-state
valve. It is thus concluded that decreases in reactor power will cause no troublesome transient

increase in the Xe!33

poisoning in reactors of the ART type.

A more serious problem is concerned with the growth of xenon during a total shutdown. The
behavior of the ART is examined in this regard by assuming that after the reactor reaches steady-
state operation it is shut down totally and the xenon is allowed to grow in until it reaches its
maximum concentration. At this point, sparging is started and continued, at zero nuclear power.
It is necessary to determine how rapidly the poisoning can be reduced to the high-power steady-
state level. The behavior in this respect governs in large part the amount of ‘“xenon override’’
which must be built into the reactor. The data used are from Table 2, with '7;,=5 min. The
maximum poisoning was calculated from Eq. 4.15,

After the reactor is shut down, the Xe'3% poisoning rises to a maximum of about 12%. If no
sparging were used, it would then decrease slowly, reaching the full-power steady-state value
in about 70 hr. During almost all this time, operation of the reactor would be impossible with
the control rod presently proposed for the ART. However, if sparging is started at the time of

135 concentration (11.2 hr after shutdown), rapid reduction in poisoning occurs.

maximum Xe
Figure 13 shows that Xe'33 is reduced to the full-power steady-state value in about 36 min.
Since this time is less than that necessary to start up the ART after a total shutdown, '® there
seems to be no reason to provide large amounts of ‘xenon override’ in the control rod. This

statemen? remains true even if T is significantly larger than the value used here.

9. NOMOGRAMS FOR XENON-POISONING CAL CUL ATIONS

Two simple nomograms have been constructed to speed rough calculations of Xe!33 poisoning.

Nomogram 1 (Fig. 14) describes the steady-state poisoning

(ao + cr.]))\L
X & me—— (9.1)

)\L+a4+)tp

 

14 o . . . . . . . . .
Although it is somewhat illogical to use the term *‘poisoning’’ in discussing conditions during a
reactor shutdown, it is convenient to do so. Difficulties are thus avoided in comparing shutdown con-
ditions with those during nuclear power operation,

1 . .
5W. B. Cottrell, personal communication.

 

 
 

SSD-C-1209
ORNL-LR-DWG-7316

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T
'4"tN(]:RELSEI 0|-'I POIISOJ\IIN(l MIAXILAU,\!,, P!OIS(!)NII\!IG @) —
L DURING TOTAL SHUTDOWN
(ZERO POWER; NO SPARGING)A 1
{2 —= i
L~ e
— //
¥ 10 ,
;’ /]
2 i
Z 8
@
£ /
o ° 7
M y
¥
4 / CHANGE OF POISONING DURING
SPARGING AT ZERO POWER,
SEE EXPANDED PLOT IN (£)~.]
A Ly
STEADY-STATE POISONING | |
/T4~ AT FULL POWER
L= It b b L]
0 2 4 6 8 10 12 14 16 18
TIME (hr)
12 (&) 1+
10 \\
8 \
g \
()
2
=
3 5 REMOVAL OF Xe'>® DURING ZERO-POWER
S SPARGING AFTER A TOTAL SHUTDOWN.
T, = 5 min
8
T
x \
4
\\
> \\
STEADY -STATE POlS(m
AT FULL POWER ———_
O J l I \\---:‘::-:hfl"
0 5 10 15 20 25 20 15

Fig. 13. Behavior of X
During Shutdowns.

TIME (min)

e 135 Poisoning in the ART

19

 
#‘3'

UNCLASSIFIED

 

 

 

 

 

 

 

 

 

 

 

 

 

SSD-A-1157
ORNL -LR-DWG-6011
O'Xe4> -Dk/k ¢
(sec’) (%) (n-cm 2 sec™)
O-Xe —
(cm?) (barns) —
o7 10° - 2 (499 Y
;'."10 limit ) —10'° (sec™)
— Q.
— 4.9 —10°
\\ = 4 /® —10
1018 106\ - e —
Q ; 1014 =
\ :—_ //162 ; -2
& l
0'° 10° N 4 —
=10~ 10 3 =
— 10 -
_ \166 =
= =— |
-20 4 —
10 10 — -
— 108 \ 12
— 10
A B 01 6'2 D

 

20

Fig. 14. Nomogram 1: Steady-State Xe 37 Poisoning.

 

 
 

where all symbols are defined in Table 1 and values are given in Table 2. To derive the nomo-

gram, let
@y + a,
U= log |21 _ | (9.2)
xw
V = log (,\p + a.4) , (9.25)
W = log AL . (9.2¢)
Then, introducing Egs. 9.2 into Eq. 9.1, the equation for bars B, C,., D of the nomogram is
Uu=v-w, (9.3)
Furthermore, by letting
Z = logoy, , (9.42)
S = logd , (9.4b)

the equation of bars A, B, C, may be written
W=24+5. (9.5)

The five bars are laid off with linear scales in the variables S, Z, U, V, and W. The distances

between bars is

AB = BC = CD . (9.6)

To use nomogram 1, lay a straightedge from the value of Oy, ©n bar A to the value of ¢ on
bar C,, locating their product (A, ) on bar B. Lay the straightedge from this point on bar B to
the value of )Lp on bar D, locating x, on bar C,. The procedure is illustrated by an example
shown on the nomogram by faint dashed lines.

Nomogram 2 (Fig. 15} describes the approach of the poisoning to its steady-state value.

The equation employed is

b -
E= o memat LD (TR ey (9.7)
xoo a - a3
where
a =0, + A, + )Lp ,
%
b = — = 0.949 .,
D + %
To derive the nomogram, let
U= e~% , (98a)
vo=e 3 (9.8b)
ab
Wes— (9.8¢)
a — 9%

21

 
UNCLASSIFIED
SSD-C-1158
ORNL-LR-DWG-6012

 

\

A

/

 

 

at t ! a
(hr) O'Xed) +)\p

(sec)

O
>
—_
O
O
Q
g

&4
o
|
N
o

I[I|||I|III| IIII|

o
on

O
IllJIIlLlilJIJI'IIl_lIllllJ .
|'|'|'||||||||||J/'“']""| '|[|'|’|'|'|
N
O
LD Til
| |

o

o
\
A
o
O\
QO
]

\

l
~_

  
   

CLyv L

Q\

]
\

\

ll\lllll_LIm

\

i

 

 

 

 

”llllllll_lj“’l T I'I'I'I'I'I'l
o

|
gun

 

Jlllll/lflill
- "lllnll/lullllluu I
o
N
T
O — o w
*P”L%mlll]l L]

o

 

Fig. 15. Nomogram 2: Time Dependence of Xe 37 Poisoning.

 

 
 

E=1-U-WU-V). (9.9)

Then Eq. 9.7 becomes

Further, let
Z=WU-V)=1-U~ ¢, (9.10)

Bars C,, E,, D, are linear scales in the variables U, V, and (U - V), respectively, the sub-
traction being performed on this subnomogram. Since the numerical value of (U ~ V) is not

required, no scale is inscribed on bar D,. Bars A, F, D, constitute a nomogram for the operation

]
Z o —= (U ~V). (9.11)

W
Bar A is a linear scale in Z. Bar F is a scale in (1/W), constructed to obey Eq. 9.11, with the
Z and (U - V) scales both linear. Bars A, C,, and B constitute a nomogram for the operation
Eel-v-1z. 9.12)
Bar B is linear in the variable £. The bars C,, D,, E, are used as a subsidiary nomogram to

perform the operation

log {at) = loga + logt . (9.13)

These bars are linear in the variables log ¢, log at, and log a, respectively. The distances

between the five vertical bars are

AB = BC = CD = DE . (9.14)
Bar F is laid off between the origins of bar A (Z = 0) and bar D, (v - V) = 0].

To use nomogram 2, proceed as follows: From bar B of nomogram 1, read the value of

L
edge from the desired time on bar C, to the value on bar E,, locating the value (at) on bar D..

A, =0y, ® and add to it the value A_. Enter this result on both bars E, and F. Lay a straight-
° P 2

Transfer this value to bar C,. Now lay the straightedge from bar C, to the desired time on bar
E,, locating a point on bar D,. Lay the straightedge from this point on bar D, to the value
marked on bar F, locating a point on bar A, Finally, lay the straightedge from bar A to the point
marked on bar C,, locating the desired value £ on bar B. The procedure is illustrated by an
example shown on the nomogram by faint dashed lines.

The accuracy of these nomograms is limited by the precision of the input data, the process
of drawing, and the means of reproduction. It is believed that the versions given in this report

are accurate to around +5%.

23