- Email: [email protected]
An analytic solution to the time-dependent first-daughter fission-product plateout problem for multiregion isothermal slug flow
Ann. nucl. Energy, Vol. 12, No. 6, pp. 273-289, 1985 Printed in Great Britain
0306-4549/85 $3.00+0.00 Pergamon Press Ltd
A N A N A L Y T I C S O L U T I O N TO THE T I M E - D E P E N D E N T FIRST-DAUGHTER FISSION-PRODUCT PLATEOUT PROBLEM FOR MULTIREGION ISOTHERMAL SLUG FLOW J. W. DURKEE JR Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. C. E. LEE Department of Nuclear Engineering, Texas A&M University, College Station, TX 77840, U.S.A.
(Received 12 November 1984) Abstract--The time-dependent, axisymmetric, isothermal slug flow convective~tiffusion equation with radioactive decay is solved analyticallyto predict the behavior of a first-daughter fission-product undergoing gaseous transport through multiple materials in a cylindricalpipe. The integration coefficients are determined using the Davidon variable metric minimizationmethod. The behavior of fission-product material deposited on the conduit wall is described by a standard mass-transfer model. The time-dependent plateout rate behavior, determined previously for parent fission-product deposition, is again evident for daughter product plateout. Dominance of the daughter plateout by parent deposition characteristics is apparent. The determination of the daughter wall mass-transfer and diffusion coefficients using a least-squares analysis of measured data depends upon a reasonably low ratio of parent/daughter half-lives. This is illustrated with 137Cs/137Ba ( = 2 x 105)and 14°Ba/a4°La (= 7.6), where for 137Cs/137Ba the solution sensitivity to the 137Ba deposition parameters is small and for 14°Ba/14°La a reasonable solution is readily obtained. INTRODUCTION
The behavior ofsubmicron-sized particulate fission-product gaseous transport and plateout has been investigated since the early stages of high-temperature gas-cooled reactor development. Experimental studies, such as those reported by 6zi~ik and Neill (1966) and Burnette (1982), have often been conducted using diffusion tubes wherein the flow is rate controlled and filtered. Parameters which characterize the transport and deposition of fission products, such as diffusion coefficients and wall mass-transfer coefficients, are measured and used to extrapolate deposition behavior for differing flow conditions. The transport and deposition model to be used here for predicting fission-product plateout is developed using arguments which parallel those previously reported for the parent product by Venerus and 6zi~ik (1966) and Durkee and Lee (1984). The flow conditions frequently encountered in plateout experiments are nonturbulent (~e < 2000) so that the time rate of deposition is enhanced. It is postulated that, in light of previous experimental and theoretical studies, the slug flow (constant velocity) convective~:liffusion-decay equation suitably describes the transport of submicron-size fission-product particulate material through a cylindrical conduit containing one or more filtering material regions. The regions are distinguished in terms of their respective diffusion coefficients. It is assumed that the filtering material is not so dense that the volumetric flow rate is reduced or that a significant difference develops in the velocity of the carrier fluid and the fission-product material. The velocity profile Vo is treated as slug, or constant, in these regions. The stream concentration Cn,m(r , Z , t) for the parent, n = 0, and daughter, n = 1, in region m, extending over the interval &'era- 1 ~< Z ~< ~m (Ae0 = 0) for m = 1, 2 , . . . , M, is described by -
re2c..
1 ec.,Dr +
J
~2C n m]
r
,~.
nC..m+6L.2. ,C.-Lm,
(1)
where D,,m are the diffusion coefficients, 2. are the decay constants and a l""=
0, 1,
n = 0; n = 1.
(2)
The concentration W..m(Z,t) of fission-product material on the pipe wall depends upon diffusive transport and radioactive decay processes. Particles diffuse from the stream to the wall. Particles on the wall can decay or can ~,~ 12:6-^
273
274
J.W. DURKEEJR and C. E. LEE
desorb and diffuse back into the stream. The time rate of change of fission-product material on the wall is then described by 0W.,. Ot
D
OCn'm ] n,m - - -Or ],=e -2"W"'m+61'"2"-lW"-t'm"
(3)
I
The partial differential equations in equations (1) and (3) require M initial conditions, 2M radial boundary conditions, an inlet and an outlet boundary condition and 2 ( M - 1) interface conditions to determine a unique solution. For the solutions presented here, these conditions include : (1) The stipulation that the concentrations of radioactive material initially present in the stream and on the wall are zero so that the solution is applicable to the clean-pipe conditions present prior to reactor startup. (2) The assumption that an axisymmetric flow distribution develops within the pipe. (3) A relationship at the wall between the particulate concentration and the net radial current which is suitably described using the mass-transfer coefficients h .... as previously used by Venerus and 6zi~ik (1966), Kitahara et al. (1976) and Durkee and Lee (1984):
--Dn'm
Or"
(4)
= h.,.C.,.(R,Z,t).
(4) The assumption that fission-product material is well mixed in the reactor so that a spatially uniform distribution of fission-product material enters the pipe at a constant rate. (5) Treatment of the final region, region M, as semi-infinite in its extent since a space time concentration distribution at Z = LPM is not typically known. (6) The formulation of the convective transport interface conditions based on the assumption that a constant flow rate and profile is maintained as the fission products are convected from one material to the next. Diffusive mass-transport conservation is also required at each interface.
S O L U T I O N TO THE D A U G H T E R P R O D U C T M U L T I R E G I O N DEPOSITION M O D E L
The differential equations which describe the concentration of the parent and first-daughter fission-product material in the stream and on the wall in a region L,._ 1 ~ z ~ L m c a n be expressed in dimensionless form as oo.. o.
'
oo.. --
'
Oz
l OO.. +
L
+
o o. 7 ~[--KnOn
+
J
m-4-(~l n K n - l O n - 1,m
,
,
(5)
I O0..m ~e'.,. Op o=1 --K"tP"'m+61'"K"-lq~"-l'm'
(6)
and 0¢p.,,. Oz -
where the parent solution 0o,mis expressed using Jo, the Bessel function of the first kind of order zero, as (Durkee and Lee, 1984) N~
O0,m(p, Z, Z) = H(z -- z) ~ Jo(~i,mp) [c 1,,. exp (F 1,,.,z) + c2,,. exp (F2,, z)]
(7)
i=1
and ~£O,m "~ N / d'~O,m -I- 4( ~ £ o , m K 0 -t- ~2m) E l 2 ., i,m
2
(8)
The Heaviside function H(z--z)=
0, 1,
• < z; z >c z,
(9)
assures that the solution exists only for the axial locations z ~< z reached by the stream at time z. The eigenvalues
First-daughter fission-product behavior
275
~.., satisfy the mass-transfer condition at the wall "/[/'gg0,m ~i,mJl(O~i,m) = ~ Jo(o~i,m);
(10)
a finite number N= are determined in each material region so that computational consistency and feasibility can be realized. The integration coefficients c1,,= and c2,,= are determined using the axial boundary conditions and Davidon's method, as applied by Durkee and Lee (1984). The technique is outlined for the daughter solution. For the interval L,._ 1 ~< z ~< L,., the initial conditions are given by
O.,m(p,z,O) = 0
(i1)
q~.,=(z, 0) = 0,
(12)
60.,,.~p p=0 = 0
(13)
and
the multiregion radial boundary conditions by
and
dO.,,.Op p= 1
./Ira.,,.2 0.,,.(1, z, z)
(14)
and the inlet and outlet conditions by
O"a(P'O'z) - 01° =
1,
n = 0;
Cao/Coo,
n
1,
l i m On,M(p, Z, "C) = O. z~co
(15) (16)
The convective and diffusive conditions of continuity at each material interface are given by
O.,.,I.=L~ = O.,,.+llz=L~
(17)
and 1 O0.,m ~'.,,. &
z=L~-
1 9%,,. + 1
c~O"'m+Iz=l.~" Oz
(18)
The dimensionless variables are defined, for species n in region m, as O.,,.(p, z, 3) =
C.,,.(r, z , t)
the stream concentration,
(19)
the wall concentration,
(20)
the decay constant,
(21)
the material region interface,
(22)
the Nusselt number for mass transfer,
(23)
the P6clet number,
(24)
the radial variable,
(25)
Coo
~,,.(z, t)
q).,~(z, 3) = - -
Coo R
R2,,
Kn
190
£¢,. L m
--
R h,,,,,,R Dn,m Rv o
~e',m -- Dn,," r
R
276
J.W.
DURKEE JR a n d C. E. LEE
t/) o
- R
the time
(26)
the axial variable.
(27)
and z=
Z R
A solution to the daughter product convective~tiffusion equation with slug flow is obtained by constructing homogeneous and particular solutions to equation (5). Application of the method of separation of variables yields the homogeneous solution N#
01,m,h(P, Z, Z) = n('r-- z) ~ Jo([3j,,,,p) [c3~.~ exp (.c#lj. z) + c4j,~ exp (fq2j. z)],
(28)
j=l
where ~"1 ,ra -~-~ , 2 , m
"F-4 ( ~ 1 ,mg l -F flj2m)
c~l,2j,. =
2
(29)
The eigenvalues flj,,~ satisfy the condition J~/'¢~'1,m
~J'mJ l([JJ'm) --
2
JO(~j,ra),
(30)
and Na daughter eigenvalues are computed. The particular solution is obtained using the method of undetermined coefficients by postulating a solution with the same spatial structure as the inhomogeneous term in equation (5) and a set of constant coefficients to be determined using the orthogonality property of the radial eigenfunctions. Thus, for the inhomogeneous term N~
KoOo.m(p, z, z) = KoH(z - z) ~ Jo(o~i,mp) [c l,,. exp (F x,,~z) + c2,,~ exp (rz,,Z)],
(31)
i=1
a solution of the form Na
Ol.m.p(P, Z, Z) = n ( r -- z) ~ Jo(o~i.mP) [K 1,p,,~ exp (F 1,,z) + Kz,p,,~ exp (F2,, z)]
(32)
i-1
is postulated. The particular solution must however satisfy the radial boundary conditions for the daughter product, equations and (13) and (14). This is achieved by using the orthogonality property for Bessel functions to rewrite equation (32) in terms of the daughter product eigenvalues and eigenfunctions so that N~
i= 1
Ntt,Np
Jo(~i,,.p) = ~
i,j=l
~(~xi.,,.;flj.m)Jo(flj,mp),
(33)
where Na, N#
E
i,j=l
N~ Np
= E E,
(34)
i=lj=l
I 4~,j,m
~(0q.. ; flj,m) = ~i.j,. -- i5,,"
(35)
and 14..... = Jo ~e'~,,.pdo(flk,mp)do(~,.mp) dp = ~'~ m ~k.mJl(l~k.m)Jo(~j-- ~i.Mo(~k.~)Jl(~i.m!,
(36)
First-daughter fission-product behavior
277
and
P
IsJ'~ = ~o ~e'~.r~PJo(flk.,.P)Jo(flj,mp)dp
.i o(Pj.,,,) ~ I-o~
,-,~: =1.0,
//~ 1.,,,']2-] \
"
2
] J'
(37)
j~k.
Adopting this notation, so that only the daughter radial eigenfunction is explicitly expressed in the solution, the postulated particular solution becomes Na, N#
~i.j,mJo(flj,mp) EK!,p,,j,. exp ( r l l ,
O!,=,p(p,z, ~) = H ( ~ - z)
z ) q--
K2.p,,j,m exp (F2,, z)'].
(38)
i,j= 1
The constant coefficients Kl,pi,j,~ and K2,p,.j,~ are determined by inserting equation (38) into equation (5) and applying the orthogonality property for Bessel functions. Careful evaluation of the derivative of the Heaviside function (Sneddon, 1972) gives, for z > z+e, KoCli,m
KI'pI'J'~ -
1
rx,,.-
~,,~ rL~ + ~
~
1
~'el,m
2
= K0,,i, c!,.~
(39)
A -- go,.j.mC2,,m.
(40)
~j,m+ K!
and
l
g2"pi'J'm ~
Koc21.m 2 1
2
The general solution to equation (5) is therefore given by
Oi.m = Or.re.h+ 01.m.p
(41)
N#
= I-I(~- z) y" Jo(fl~.mO) {c3j,~ exp (Nl,,~z) + c4j,~ exp (~2j.~z) j=l Na
+ ~ ~i,j,m [Kl.p,,j.~ exp (FI,, z) + K2.p,,j,~ exp (F2,. z)] }.
(42)
The constants of integration Caj.~ and c4j,~ are determined using the axial boundary conditions. Application of the inlet condition gives
Jo(flj,!P) c3j.1 "]-C4j,I -~j=l
~i,j, 1 [-Kl,p,.j,, "t-g2.p,,~.l]
= 01o.
(43)
i=l
Continuity in the concentration of daughter product material convected across the interface locations Ls, m = 1, 2. . . . . M - - 1, is guaranteed by
Jo(fli,mP) [c3j,. exp (if!j, L,.) + c4j.. exp (ff2j, Lrn)] j=l
+ ~ ~ij.,[Kipj., i=1
, , ,~ exp (F1,,~Lm)+K2p, ,,j,~ exp (F2.Lm)]
}
= ~ Jo(flZm +1 P) ~[c37 . . . . exp (c~!7,.+, Lm) + c'*y. . . . exp (c~27. . . . Lm)] ]'=1
}
+ ~ ~i.Zm+ ! [K !.p~,~. . . . exp (F 1..... Lm) + K2,p,.~ . . . . exp (F 2. . . . . Lm) ] , i=1
(44)
278
J.W. DURKEE JR and C. E. LEE
while the expression
1 ,mj~-I N~ Jo(flLmP) { [ C 3 j , m f f l / , m
exp
(fflj,~L,.)+ c4j,~2~.~ exp (ff2j, Lm)]
)]}
No
+ ~ ~ij,,.[Kav. ,, , 1,j , m F1.l , m exp(F1.~,m L , . ) + K 2 v , t, j ,m F2 , , m exp(F2 t , m L,. i=1
1 ~2L., + 1
7~lJo(flZ.,+lP) [c37. . . . ff17. . . . exp (ff17. . . . Lm)+C4Z~+~(~2y .... exp (ff27. . . . L,.)]
+ ~ ~i,~,m+l [K1.p.,7 .... Fx,~+ ~exp(Fl..m.tLm)+K2.v.,7,~+.F2 ..... exp(F 2. . . . . L,.)]
(45)
i=1
insures conservation of mass for diffusive transport across each interface. The outlet boundary condition dictates that the concentration in region M vanish as z --, 0% which necessitates c3~,M = 0.
(46)
A set of linear algebraic equations to be solved for the c3j.~ and c4j.~ integration constants is obtained by premultiplying equations (43)-(45) by ~/1,mPJo(flk.m), integrating over p from 0 to 1 and utilizing the orthogonality property of Bessel functions. The potentially troublesome numerical difficulties expected to be encountered in solving the resulting stiffsystem are lessened by regrouping and redefining the matrix coefficients and constants of integration so that the arguments of all the remaining exponential terms are negative. Also, terms containing the integration constants for the parent solution, c~,,~ and c2,.~ are rewritten so that the values computed using Davidon's method are directly utilized. This alleviates potential problems with significance loss which could arise from intermediate computations. Letting c~'j,~ = exp (cgffl~,Lrn)C3j,~
(47)
exp (F1,.~Lm) Kl,p,,j,~ = Ko,,j,~cl ....
(48)
c'1,,~ = exp(Fl,.~L,,)cl,.~
(49)
c4~. . . . = exp (fizz. . . . L~) C4j . . . .
(50)
and
where
form = 1 , 2 , . . . , M - 2
and, for m = M - - l ,
and ^
exp (F 2..... L,,,) K 2,p,,7.... = Ko,,7.... (~2......
(51)
c2 . . . . . = exp (F z..... Lm) C2......
(52)
where
the system structure for computational enhancement becomes N~
. , exp(--ffllj,lL1)c*j,,+c%,, = 01 oI6~1/15jl-E
~i.Ll[exp
(--FI~,,L1)Ko~,j,,cx~,I ~ * +Ko,,j,,c2,.], ^
(53)
i=1 N#
Isj.m[c*j,.+exp(ffzj..Lm) C%.J-- ~ I7j,..y .... {exp [flit . . . . (Lm--L,,,+O] c3)'.+*, + e x p (c527,.+1Lm)C4.7,.~-1} j=l
Na
N#
= --Isj,m ~ ~,.j,m[Ko,.j,~C*,,m+exp(F2.,mLm) RO.,j.~Cz.,m-I+ ~.. I7j,~,7.... i=1
f= 1
Na
x ~ ¢i,7,.+ 1 {exp [F 1..... (L,,,--Lm+O] Ko,,7,.+- 1c1,,.*+, +6xp(F~ . . . . . Lm) K"o,d.... c2 . . . . . } i=1
(54)
First-daughter fission-product behavior
279
and 1
- -
Isj,. [ffa~, c ~ , . + fq2~,~exp (cff2~,Lrn) C4~,~] N#
1
~ l%,~,7.~+, {ff17. . . . exp [(¢17. . . . (L,,,--L,.+ 1)]c'~7.... + ff2y.... exp (ff27. . . . Lm)c'*7.... }
~'*'l.m+, 7=' 1
Na
1
15j
~g'l,m
,m
~ ~ jm[F, i= l
, ,
tm
/~0,~ c* , ,m
,,m
+F2
,m
exp(F2. Lm)/(0 j~c2 j + .
.
.
.
.
.
N# ~Ivj,~,7....
~i?il,m+ 1 f=
N~
x ~ ~i,Zm+I{F, . .+~exp[F . . . . 1. + (Lm-Lm+l)]Ko,5.~+:cl~*. . . . . + F 2 ..... exp(F2 ..... Lm)K'o,.7. . . . C2..... }' i=1
(55)
form = 1,2 . . . . . M - - 2 , and l~j,~ [c~'j,m+
exp
(q.~2.i,mLm)ca~.m]
--
~
Ivj,m,)- . . . .
547 . . . .
7=1
Na
N#
= --Isj.. ~ ~.j,m[go, j C * , + e x p ( r z , , L , . ) R o , j C 2 , , j + ~ i=1
Na
Ivj,., z.... ~ ~i.7,,.+IRo,,z .... F2...... j=l=
1
1
Is~'m[fqu'~c*~'~+fff2j'~exp(f~2j"L")c%'J 1
--
1
-
Np
"¢~e'X,m+l f=~l 17j'~'7 . . . .
~2y ....
?4j'....
N,
~el,,. +
(56)
i=l
15j,~ ~ ~i,j,,,,[F,,,~go,,j,.c*,,~+ F2,..exp(F2,,~L.,)Ko,,j,~c2,,J i=1
N#
N~
~ I%,~,7,-+, Z ~,,D.+,Fz . . . . . /2(o,,7.-+,F2. . . . .
~'*'1,,. + 1 7=1
(57)
i=,
for m = M - - 1. The integrals I6k,1 and Ivk.~,~-. . . . are
16k.1 =
=
0
~e'a,,PJo(flk,xP)dp
~" 1,1 J'(/~*") --
(58)
ilk,1
and
17.... ~.... = f ~ ~e',,mPJo(flk,mP)Jo(fly, m+ ,P) dp ~.
[flk,mJ,(fl*,,,,)Jo(flZm+ ,)--flY,,.+ lJo(flk,m)J,(flf.m+ ,)]
= O'el,m[
L-
R2
R2~
IJk,m--Pj,m+l
1"
(59)
I
The quasi-Newton Davidon (1959) iterative minimization technique is used to solve the system of equations (53)-(57). This is accomplished by constructing a function consisting of the sum of the squares of the residuals of the matrix-vector system. The least-squares function, f(X), is minimized by varying each constant of integration xl, x2 . . . . . Xm while holding all other parameters fixed using the Newton-like second-order form of functional minimization, X = X,. = X k --G-
lg,
(60)
where g and G - ' are the gradient and inverse Hessian matrix of f evaluated at X, respectively. The Davidon algorithm replaces the inverse Hessian matrix with a positive definite symmetric matrix which is constructed iteratively using functional and gradient information. Using this technique, the solution vector X can be obtained which, when substituted back into AX = B, satisfies the original equation to a prescribed accuracy. The structure of the residuals for the daughter solution is similar to that for the parent solution given previously
280
J.W. DURKEEJR and C. E. LEE
by Durkee and Lee (1984) with all inhomogeneous terms being nonzero for the daughter solution. The gradients are identical in structure, only the constant coefficients differ. The solution is computed by calculating the integration coefficients for the parent, constructing the system for the daughter using the necessary coefficients from the parent solution and applying Davidon's method for calculating the integration constants of the daughter solution.
Wall Concentration for Slug Flow in a Multiregion Pipe The expression for the concentration of the daughter product on the wall of the conduit is obtained by solving
~¢1,m
1 001~
.=x
(61)
where (Durkee and Lee, 1984) '~/'¢g0,m { 1 -- exp [K o(Zq~o,~-- 2~%,mK °
z)] }00,re(l, Z, Z).
(62)
Using the integrating factor u(z) = exp (Klz)
(63)
the solution is qN,,.(z, z) = ~
01,,. {1 - exp [K 1(z-- z)] }
1 -'}- 0 0 , m [ K
{1 --exp [ K , ( z - - 0 ] ,
1 {exp [Ko(z--0]--exp K1 --Ko
[Kx(z--z)]}~
]l
(64)
where JV~0'm 00,.,(1, Z, Z)
O0,m-- 2~O,m
(65)
gi/'~g I ,m 01 ,m -- 2 ~ 1 , m 01 ,m( 1, Z, "~).
(66)
and
Predictions of fission-product accrual on a conduit wall may be compared with experimentally measured values once the pointwise expression, equation (64), is integrated to give the total concentration for a sectioned portion of pipe as Tll,m(V) = 2n
~01,,.(z,z)dz.
(67)
t i
For a section extending from Lt_ 1 to L~, integration of equation (64) gives Tl,t,,.(z ) = 2nH(z --z)
Jff~ l,,.
N~
II-E ~_2Ka~e.l,,..i= a
x { TI+Tz+
i u~=~jm[I~ojC = , , t m *i,m (T3-T,
+~i~lJo(o~i,,.)
) + K o st,
c*,,~ K ( T 3 - - T 4 )
1 A-c2"m[~ (TS- T6) KI--Ko
(Ts-T6)II
,m c2.~,m (Ts--
T6)]}
K ~ I_K (68)
First-daughter fission-product behavior
281
with T1 = c3j,~*
{exp [fflj,~(L~-Lm)] --exp [Nlj.~(LI- 1 - Lm)]} exp (--KlZ)
T2 = c,,j.~I~l--~-[exp(~z~,
{exp [KxLI+Wtj (Li--L,.)]--exp [K1Lt-l+f#lj
(L 1--L.)]}I,
(69)
exp (-- K 1z) K1 +(q2j.~ {exp [(K 1 +(qzj.~)Lt]--exp [(K1 +~2j,~)LI- 13}1,
(70)
II
L,)--exp(~zj, Lt_x)]
ll ~ 2 j , m
1
T• FI,,~ {exp IF l,,~(L -- L~)] -- exp [F,,.~(LI_I =
--
Lm)]},
(71)
T4 - exp (-- KlZ) {exp [K1LI + F~,,~(LI-Lm)]--exp [K,LI_, + F,,.~(LI_,- L,.)]}, K1 +FI,,~ 1
Ts =
[exp (Fzi, L/) -- exp (F2,.L 1_ , ) ' L
(72) (73)
l~2i,m
exp(-Klz)
T 6- K1 + ['2,,~ {exp [(Kx + F2,,m)LI] -- exp [(K1 + F2,,m)LI-1 ] }, Tv -
(74)
exp ( -- KoZ) {exp [KoLI + F I,.,(Lt -- L,,)] -- exp [KoLt-i + F I,,~(LI-a -- L~)] } Ko + Fx,.~
(75)
and Ts --
exp (-- Kor ) Ko + ['2i,m
{exp [(K o + F2,,m)Lt] - exp [(Ko + F2,,m)L,- 1]}
(76)
as the integrated plateout at time z, for the axial regions m = 1, 2 ..... M -- 1. For region m = M, the exponential arguments and constants of integration are different due to the regrouping for the variable metric algorithm so that the solution is of the form
T,.l,..(z) = 2,m(~-z)ll 2 ~
Jo(flj..,) Tg+ E ~ ,j...fgo,,.~z, m(Ylo--T103 ,
j=
i=1
+~i=12
Jo(cq,.,) F2,,~ ~-l(TIo--Tll)
K1
Ko (T12-Tll
(77)
with
F
T9 = F4~ m H ~
1
{exp [f¢2~ (LI-L.,_ 1 ) ] - e x p [f#2~,m(Ll_ 1 - L , . _ 1)]}
exp(--Kar) {exp [KILt+ff21,m(Ll--L,,,_ 1)] Kx "[- • 2 j , . 1
--exp [K1LI- 1+ ff2~,~(Lt- 1--L.,_ 1)]}11, 1
7"1° = F 2
(78)
{exp [F2,..(Lt--L,, ,_ 1)]--exp [F2~,~(L 1_ 1 --L,._ 0]},
(79)
i,m
exp (-- K l z) "/'11 -- Ka +F2,,. {exp [K1LI+ I"21,~(Lt--Lm_ 1)] --exp [K1Lt- 1+F21,~(Lt- aL,.- 1)]}
(80)
and
T~2ANE 12:6-B
exp ( -- Koz) {exp[KoLt+F2,,~(Li--Lm_O]--exp[KoLt_l+F2,,m(L Ko + F2~,~
x --L,._ x)]}.
(8l)
282
J.W. DURKEEJR and C. E. LEE
Activity units commensurate with measured values are readily obtained by premultiplying T~.t,~ by the decay constant of the daughter species and applying the appropriate units conversion factor. Deposition for the entire pipe is computed by summing the integrated contribution over all sections.
The Semi-infinite Pipe Daughter Product Deposition Model Solutions The single-region stream, wall and integrated deposition solutions are constructed by following a procedure similar to that for the multiregion solutions with the integration constants determined directly. The stream solution is given by
Ol(p,z,z ) = H(z--z) ~ Jo(flsP) 01,, j=l
exp(~2jz)+ ~ ~ijK,,,~[exp(F2(z)-exp(~2jz)] , i=1
(82)
where Kpl a -
go i
1 2 1 F2 i - ~1F2i-~ -
~lflJ
2
+ KI
'
4K o 1 K°' - JV~o F1 / 2~i \27
/
(83)
(84)
andF2,,ff2;¢i.jandl6s/Issarethesingle-region(M.= 1) values ofequations (8),(29),(35) and (58), respectively. The solution limits to that obtained by Venerus and Ozi~ik (1966) if axial diffusion is neglected. The wall concentration distribution is qg,(z, z) = K~iT,{1 - e x p [ K l(Z - Z)] } +0 °
I~- {1 -1 {exp [Ko(z--'r)] -- exp [K x(z-- z)] }1' (85) 1 - e x p [Kl(Z--z)] } - -K1--Ko
where Yg¢o
0o(1, z, z)
(86)
Yggl /7, = 2~f, 01(1,z,~).
(87)
Oo = ~ and
The time-dependent integrated plateout solution is
Tm(z ) = 2zrH(z--z)~ JV'ua ~ Jo(flj) ~'2+ ~'~ ~i,jKp,,j[~F9+(7"lo-- TI1)] ~_2K1~ex j="l i= 1
2 "o ~ I 1 (7.1o_7~1)
-]-~oi~=lc21gl
(88)
with 01o
~ - 1 [exp(fq2jLt)--exp(fqzjL, 1)]
exp ( -- K az) Klq-C~2j
{exp [(K 1 +ff2s)Lt] --exp [(K~ +ff2j)L~ 1]}1 , (89)
First-daughter fission-product behavior
7"9= - - I ~
283
p ( +- Ka32j l z ) {exp [(K1 + a32j)Lt]--exp [(K1 + c#2j)Lt-t]} 1 , (90) [exp (aJ2jLt)--exp ((q2jLt_ x ) ] - e xK1 1
7'to = ~2~ [exp (F2~Lt)--exp (F2,L t_ a)],
T12 -
(91)
exp (--Ktz) {exp [(K 1 + F 2)Lt] --exp [(K 1 + F 2)Ll_ 1] }, K1 +F2~
(92)
exp ( -- Koz) {exp [-(Ko + Fz)Lz] -- exp [(K o + F2)L l_ 1] } Ko+F2~
(93)
and 1
c2,-
(94)
{ 2~i "~2"
RESULTS
Representative computational results are presented for flow conditions similar to those reported by Burnette (1982) in the General Atomic Fort St. Vrain plateout probe experiment. A flow velocity of 10 cm s - 1, pipe radius of 0.5 cm and pipe length of 75.0 cm were selected to analyze the plateout of 137Cs (tho = 30.17 yr) and its daughter, 137Ba (th~ = 2.55 min). Single-region diffusion and wall mass-transfer coefficients D O = 0.004 cm 2 s - t, D 1 = 0.003 cm 2 s - 1 and h0 = 50.0 cm s - t, hi = 40.0 cm s - 1 were used. An equilibrium inlet concentration of 105 atom c m - 3 was assumed for 137Cs, while a zero concentration was assumed for 137Ba. The former assumption leads to a conservative overestimate of 137Cs plateout since the concentration from fission increases in time from zero. The latter assumption is plausible because the equilibrium ratio of the 137Ba to a37Cs due to decay is approx. 10- 7. The single-region solution is given for postulated open-pipe flow conditions. It is used to illustrate the proper limit for plateout calculated using a three-region solution with the perturbed diffusion coefficients listed in Table 1. The parent and daughter dimensionless pointwise wall concentrations, ~P0(P,z, z) and ~pt (P, z, z), are shown in Figs 1 and 2. The integrated tube total plateout profiles, T0(r) and Tl(Z), are displayed in Figs 3 and 4. The semi-log wall profiles are plotted as a function ofz for several values of the dimensionless time z. The log-log integrated profiles were calculated by the integration of the pointwise expressions over the length of the pipe, z e [0, 1501 with evaluation at times z = 1 0 3, 10'* . . . . , 1 0 1 2 . Selected daughter pointwise wall concentration and total tube plateout values are listed in Tables 2 and 3. The effect of positioning two filters in the pipe was simulated by reducing the parent and daughter diffusion coefficients by 100~ from region 1 to 2 and from region 2 to 3, as indicated in Table 1 (filtered). The wall masstransfer coefficients were held constant as the pipe wall properties were assumed to be invariant along the region of deposition. Selected values of the pointwise wall and tube total plateout for the daughter are listed in Table 4. Total steady-state deposited parent and daughter concentrations are reduced by 22.2 and 22.4~o to 4.22 x 101° and 6.87 × 103, respectively, for this parametric set, as compared to that for the open pipe. The tube total parent and daughter deposition profiles exhibit a linear behavior to z = 101°, at which time the profiles bend and approach constant totals of 5.44 x 101° and 8.51 × 103 for the parent and daughter, respectively. Table 1. Open- and filtered-pipe
137 Cs
and
137 Ba
diffusion coefficients
Diffusion coefficients (cm 2 s - 1) 137Cs
137Ba
Conditions
Do.l
Do.2
Do.3
D~.~
D 1.2
Dr,3
Open Filtered
0.004 0.004
0.004001 0.002
0.004002 0.001
0.003 0.003
0.003001 0.0015
0.003002 0.00075
284
J . W . DURKEE JR and C. E. LEE
o
t:
o
o
-~ '-~
-~
-~
~
'
o
o
-o
%
%
~o
~;
..o =
~,°
) y / y j j
~
,,,
o=
°
8
i8
o
g
-
~
o
i
° L
/
~
I 0
NOIIV~IN3
ONOD
I 0
~
.v
o
o
o
°
°
I
I
I
0
0
NOIIV~IN3DNOD
0
0
First-daughter fission-product behavior
285
eJ r
o
i\
o
T
o
\
o
\.
%
~e
\ e~e
\. I
\
.o
\ %
I
o
"\
%
\.
I
-- o
L
I
I
I
2 .I
\
o
o
..j
%
T
%
t
I
?
I
I
I
I
I
I
~o
I
\ \ \
~
•
•
\. \
~
%
\ I
L
I
I
I
I
i
%
I
t -
T
=
o
= o .~
l
\ "\ •
~
•
~ •
o
\
_~o
-\
O
~o %
"\ "\
~
% I
I
I
I
I
I
I
.~
I
I
I
I
I
I
I
L o
NOII~IN~DNO0 NOIIV~IN3
DNOD
%
I o
o
O
286
J. W .
DURKEE JR and
C.
E.
LEE
Table 2. Selected single-region open-pipe pointwise and integrated ~37Ba plateout values Plateout (dimensionless) Time r :
105
108
10 ~
Location z 10
7.12x10
40 100 " 150
Total:
T
5
7 . 4 4 × 10 - 2
1.99×10 t
3.23 x 10 5 1.77 x 10 - 5 1.29x10 -5 3.04 × 10 - 2
3.37 x 10 - 2 1.85 x 10 2 1.35x10 2 3.18 x 101
9.03 x 10 ° 4.95 x 10 ° 3.62×10 ° 8.51 x 103
Table 3. Selected three-region open-pipe pointwise and integrated ~37Ba plateout values Plateout (dimensionless) Time ~ :
I115
10 8
101~
7.12xl0 5 3.23 x 10 - 5 1.75 x 10 - 5 1.27x10 5 3.03 x 10 - 2
7.44x10 -2 3.37 x 10 - 2 1.85 x 10 - 2 1.35x10 z 3.18 × 101
1.99x101 9.03 x 10 ° 4.95 × 10 ° 3.62×10 ° 8.51 × 103
Location z 10 40
100 150
Total:
T
Table 4. Selected filtered-pipe pointwise and integrated
~~7Ba
plateout values
Plateout (dimensionless) Time z:
10 5
108
101~
7 . I 2 x 10 - 5 3.23 x 10 - 5 1.11 x 10 _5 4.43x10 5 2.34 x 10 - 2
7.44 x 10 - 2 3.37 x 10 - 2 1.18 × 10 - 2 5.29x10 z 2.46 x 101
1.99 × 10 l 9.03 × 10 ° 3.17 × 10 ° 1.42×10 ° 6.61 × 10 ~
Location z 10 40
100 150
Total:
T
Steady-state conditions are approached when the deposition time nears the half-life of the parent species. Similar behavior is exhibited for shorter-lived parent species, with steady-state conditions being achieved at earlier times. Additional calculations indicate that the plateout behavior of 137Ba is dominated by 137Cs for a wide range of flow conditions. This results because of the large difference between the parent and daughter half-lives which leads to large differences in the magnitudes of the stream and wall concentrations. For instance, the parent and daughter stream concentrations at p = 0.5 and z = 25 for the open-pipe conditions were calculated to be 9.8 x 1 0 - 1 and 1.2 × 10 - 9 , respectively. The resulting dominance in the daughter wall behavior is evident in equations (64)-(66). Consequently, at least for the parametric range analyzed here, a filtration system designed to filter 137Cs will suitably filter la7Ba since so little 137Ba is created. This furthermore indicates, as has been demonstrated computationally, that the determination of the ~37Ba wall mass-transfer and diffusion coefficients using a leastsquares minimization to fit the analytic model to a set of measured data (Durkee and Lee, 1984) is not practical since the sensitivity of the 137Ba concentration to the plateout governing parameters is so small. The least-squares minimization technique was successfully applied to a set of fictitious data created for the decay chain 14°Ba (tho = 12.79 d) to 14°La (thi = 40.23 h). These isotopes possess half-lives whose magnitudes are much closer than those of 137Cs and 137Ba. Single-region diffusion and wall mass-transfer coefficients Do = 0.001 cm 2 s - 1, D x = 0.002 cm 2 s - 1 and ho = 1.0 cm s - t, h ~ = 10.0 cm s - 1 were used to generate a daughter plateout profile at z = 108.Aninletconcentrationofl05atomcm-3wasstipulatedfor 14°Ba.Azeroinletconcentrationwasassumed for ~4°La. It was expected that the change in the calculated diffusion and wall mass-transfer coefficients due to the
287
First-daughter fission-product behavior Table 5. Calculated best-fit parameters for the t4°La data analysis Wall mass-transfer coefficient (cm s- 1) Parent
1.2/10.0
Diffusion coefficient (cm2 s - 1)
Daughter
1.0/10.0
Inlet concentration (atom cm- 3)
Parent
Daughter
Parent
0.00127/0.00130
0.00195/0.002
8.7D4/8.8D4
time-dependent increase in t4°La arising from the decay of a4°Ba would be quite small ; only the calculated parent inlet concentration used for solution normalization should change appreciably. A set of 30 fictitious measured activities was then generated by randomly perturbing the analytic solution integrated plateout value in each section by a m a x i m u m of 20%. The least-squares fitting process consisted of fitting the data with analytic solutions generated using several sets of estimated diffusion and wall mass-transfer coefficients and inlet concentrations. The fitting procedure was not a strict minimization process due to the competing plateout effects between the diffusion and wall mass-transfer coefficients. A fit was deemed suitable when a parametric regime was located where a limited range of parent and daughter diffusion and wall mass-transfer coefficients led to a change in the least-squares fit. Table 5 contains the parameters which gave the'best fit to this data set. The solutions exhibited a greater sensitivity to changes in the diffusion coefficients than the wall mass-transfer coefficients. Figure 5 illustrates the data and the fit constructed using Do = 0.00128 cm 2 s - t , D1 = 0.002 cm 2 s - t , ho = 5.0 cm s -1 h a = 10.0 cm s -1, Coo = 8.8 x 104 atom cm -3 and Cao = 0.0 atom cm -3. For these parameters a plateout of 12.7 #Ci 14°La was predicted. Using these parameters with the equilibrium decay ratio of a4OLa!t 40Ba ' 0 ao = 0.13 I, as the dimensionless inlet concentration for ~4°La, the plateout was calculated to be 15.4/~Ci. The pointwise integrated activity profile, shown in Fig. 6, differs appreciably from that shown in Fig. 5 for a zero inlet concentration of *4°La only in its normalization. Figures 7-10 contain plots of the dimensionless parent and daughter stream and integrated tube total profiles
102
1°2I
101
101 101
i
LS :&
10 0
Z O
"r
E o ~5 ::k
I, •
F "m-m.m" mrm'm'mnmm~m-u . . .ram umm ammamlm,m.mm m mm
I - 10-1 Z hi O Z O ¢.3
10 °
~
Z 0 n- 10-1 Z
g o
10-2
10- 2 --
10-3 0.0
I 12.5
I 25.0 AXIAL
Fig. 5. Fitted
I 57,5
I 50.0
I 62.5
POSITION (cm)
14°La data for 01o = 0.0,
I 75.0
10-~ k 00
I
I
I
I
I
I
12.5
250
57.5
50.0
62.5
75.0
AXIAL
POSITION (cm)
Fig. 6. l*°La integrated profile using best-fit parameters with 01o = 0.131.
288
J . W . DURKEE JR and C. E. LEE 1.0 m z=l 0.9
1012 1011
08
101° 10 9
0.7
10 8
z
0 p <
0.6
~
0.5
o/o--o--o--e--
107
/
~) 10 6 Z © ~
rr
105
0.3--
Z~ 104 W 103 0 {.C' 102
0.2--
10o
04--
•
/ /"
101
10-1 0 1 -OC__ 0.0
10-2 10- 3 0.2
0.4
0.6
0.8
I
I
I
103 1 0 4 1 0 5 1 0 6
1.0
I
I
I
107 108109
I
I
I
101010111012
TIME
RADIAL POSITION
Fig. 7. Dimensionless 14°Ba stream concentration.
Fig. 8. Dimensionless 14°Ba integrated tube total plateout.
1.0 --
09
1012 10~1
0.8
1010 109
0.7
10 8 Z o
0.6
107 Z
0
r~ FZ 0.5 w 0 Z 0 0.4
o/o
106
O.3
--o--o
./
10 5 z 104 t.u o z 105 0 102 ,
--o--o
,/o
/
./
101 0.2
10° 10 1
01
z =150
6~
'
~
~
10-2
i-- 4 0 20 J 2 0 1 % ~ 4 0 0.0
10-3
I
0.0
0.2
0.4 0.6 RADIAL POSlTIDN
0.8
Fig. 9. Dimensionless 14°La stream concentration.
1.0
I I I l 103 104 1 0 5 1 0 6 1 0 7
I 1 I I I 108 109101010111012 TIME
F i g . 10. D i m e n s i o n l e s s
14°La
integrated tube total plateout.
First-daughter fission-product behavior
289
made using these fitting parameters and 01o = 0.131. The daughter stream profile exhibits the nonzero inlet concentration condition. The integrated profiles reached steady state by z = 10s, or roughly 58 days, a time much shorter than that for 137Cs and 137Ba due to the much shorter half-lives of 14°Ba and 14°La. The one- and three-region stream, wall and integrated solutions were generated using 19 parent and daughter eigenvalues with starting values of 1.0 used for the minimization. The one- and three-region tests required roughly 25 and 42 s ofCRAY CPU, respectively, to evaluate and plot the solutions at 10 points in time, 100 radial locations and 150 axial positions. The Davidon method effectively minimized the three-region, 95-equation residual functions for the parent and daughter species. For the filtered pipe study, calculated residual-functionminima were 3.9 x 10-7 and 1.3 x 10-12 for the parent and daughter, respectively. The least-squares data analysis was expeditiously conducted by generating the stream solution only at the wall for calculation of the integrated plateout solution. By analyzing the solution at only z = l0 s with nine parent and daughter engenvalues, the CPU requirement per solution was reduced to 1.2 s. Solution accuracy was verified using 19 eigenvalues and parameters within the bounds calculated to best fit the data. CONCLUSIONS
The time-dependent behavior of a first-daughter product in a flowing gas stream has been modeled using an analytic solution to a convective-diffusion-decaytransport and deposition model. The plateout behavior on the wall of a pipe containing different materials was modeled by the adjustment of the diffusion coefficients in each region. Using plateout parameters similar to those found in the literature for the submicron-sized particle deposition from an isothermal He stream, the behavior of the 137Cs to 137Ba and the 14°Ba to ~*°La decay chains were investigated. It was found that the plateout behavior of the daughter is increasingly influenced by the deposition characteristics of the parent as the parent to daughter half-life ratio increased. The parent and daughter integrated deposition profiles exhibited a time-dependent behavior. This behavior can be approximated using a model which assumes a constant rate of plateout, as in Burnette (1982), only for limited deposition times.
REFERENCES
Burnette R. D. (1982) Report GA-A16764. Davidon W. C. (1959) Report ANL-5990. Durkee J. W. Jr and Lee C. E. (1984) Ann. nucl. Energy 11(7), 347. Kitahara T., Yokoo H., Kaieda K., Toyoshima N. and Fukushima M. (1976) J. nucl. Sci. Technol. 13(3), 111. Ozi~ik M. N. and Neill F. H. (1966) Trans. Am. nucl. Soc. 9, 393. Sneddon I. N. (1972) The Use oflntegral Transforms, pp. 484-487. McGraw-Hill, New York. Venerus E. R. and (~zi~ikM. N. (1966) Nucl. Sci. Engng 26, 122.
ANE
12:6-C
0306-4549/85 $3.00+0.00 Pergamon Press Ltd
A N A N A L Y T I C S O L U T I O N TO THE T I M E - D E P E N D E N T FIRST-DAUGHTER FISSION-PRODUCT PLATEOUT PROBLEM FOR MULTIREGION ISOTHERMAL SLUG FLOW J. W. DURKEE JR Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. C. E. LEE Department of Nuclear Engineering, Texas A&M University, College Station, TX 77840, U.S.A.
(Received 12 November 1984) Abstract--The time-dependent, axisymmetric, isothermal slug flow convective~tiffusion equation with radioactive decay is solved analyticallyto predict the behavior of a first-daughter fission-product undergoing gaseous transport through multiple materials in a cylindricalpipe. The integration coefficients are determined using the Davidon variable metric minimizationmethod. The behavior of fission-product material deposited on the conduit wall is described by a standard mass-transfer model. The time-dependent plateout rate behavior, determined previously for parent fission-product deposition, is again evident for daughter product plateout. Dominance of the daughter plateout by parent deposition characteristics is apparent. The determination of the daughter wall mass-transfer and diffusion coefficients using a least-squares analysis of measured data depends upon a reasonably low ratio of parent/daughter half-lives. This is illustrated with 137Cs/137Ba ( = 2 x 105)and 14°Ba/a4°La (= 7.6), where for 137Cs/137Ba the solution sensitivity to the 137Ba deposition parameters is small and for 14°Ba/14°La a reasonable solution is readily obtained. INTRODUCTION
The behavior ofsubmicron-sized particulate fission-product gaseous transport and plateout has been investigated since the early stages of high-temperature gas-cooled reactor development. Experimental studies, such as those reported by 6zi~ik and Neill (1966) and Burnette (1982), have often been conducted using diffusion tubes wherein the flow is rate controlled and filtered. Parameters which characterize the transport and deposition of fission products, such as diffusion coefficients and wall mass-transfer coefficients, are measured and used to extrapolate deposition behavior for differing flow conditions. The transport and deposition model to be used here for predicting fission-product plateout is developed using arguments which parallel those previously reported for the parent product by Venerus and 6zi~ik (1966) and Durkee and Lee (1984). The flow conditions frequently encountered in plateout experiments are nonturbulent (~e < 2000) so that the time rate of deposition is enhanced. It is postulated that, in light of previous experimental and theoretical studies, the slug flow (constant velocity) convective~:liffusion-decay equation suitably describes the transport of submicron-size fission-product particulate material through a cylindrical conduit containing one or more filtering material regions. The regions are distinguished in terms of their respective diffusion coefficients. It is assumed that the filtering material is not so dense that the volumetric flow rate is reduced or that a significant difference develops in the velocity of the carrier fluid and the fission-product material. The velocity profile Vo is treated as slug, or constant, in these regions. The stream concentration Cn,m(r , Z , t) for the parent, n = 0, and daughter, n = 1, in region m, extending over the interval &'era- 1 ~< Z ~< ~m (Ae0 = 0) for m = 1, 2 , . . . , M, is described by -
re2c..
1 ec.,Dr +
J
~2C n m]
r
,~.
nC..m+6L.2. ,C.-Lm,
(1)
where D,,m are the diffusion coefficients, 2. are the decay constants and a l""=
0, 1,
n = 0; n = 1.
(2)
The concentration W..m(Z,t) of fission-product material on the pipe wall depends upon diffusive transport and radioactive decay processes. Particles diffuse from the stream to the wall. Particles on the wall can decay or can ~,~ 12:6-^
273
274
J.W. DURKEEJR and C. E. LEE
desorb and diffuse back into the stream. The time rate of change of fission-product material on the wall is then described by 0W.,. Ot
D
OCn'm ] n,m - - -Or ],=e -2"W"'m+61'"2"-lW"-t'm"
(3)
I
The partial differential equations in equations (1) and (3) require M initial conditions, 2M radial boundary conditions, an inlet and an outlet boundary condition and 2 ( M - 1) interface conditions to determine a unique solution. For the solutions presented here, these conditions include : (1) The stipulation that the concentrations of radioactive material initially present in the stream and on the wall are zero so that the solution is applicable to the clean-pipe conditions present prior to reactor startup. (2) The assumption that an axisymmetric flow distribution develops within the pipe. (3) A relationship at the wall between the particulate concentration and the net radial current which is suitably described using the mass-transfer coefficients h .... as previously used by Venerus and 6zi~ik (1966), Kitahara et al. (1976) and Durkee and Lee (1984):
--Dn'm
Or"
(4)
= h.,.C.,.(R,Z,t).
(4) The assumption that fission-product material is well mixed in the reactor so that a spatially uniform distribution of fission-product material enters the pipe at a constant rate. (5) Treatment of the final region, region M, as semi-infinite in its extent since a space time concentration distribution at Z = LPM is not typically known. (6) The formulation of the convective transport interface conditions based on the assumption that a constant flow rate and profile is maintained as the fission products are convected from one material to the next. Diffusive mass-transport conservation is also required at each interface.
S O L U T I O N TO THE D A U G H T E R P R O D U C T M U L T I R E G I O N DEPOSITION M O D E L
The differential equations which describe the concentration of the parent and first-daughter fission-product material in the stream and on the wall in a region L,._ 1 ~ z ~ L m c a n be expressed in dimensionless form as oo.. o.
'
oo.. --
'
Oz
l OO.. +
L
+
o o. 7 ~[--KnOn
+
J
m-4-(~l n K n - l O n - 1,m
,
,
(5)
I O0..m ~e'.,. Op o=1 --K"tP"'m+61'"K"-lq~"-l'm'
(6)
and 0¢p.,,. Oz -
where the parent solution 0o,mis expressed using Jo, the Bessel function of the first kind of order zero, as (Durkee and Lee, 1984) N~
O0,m(p, Z, Z) = H(z -- z) ~ Jo(~i,mp) [c 1,,. exp (F 1,,.,z) + c2,,. exp (F2,, z)]
(7)
i=1
and ~£O,m "~ N / d'~O,m -I- 4( ~ £ o , m K 0 -t- ~2m) E l 2 ., i,m
2
(8)
The Heaviside function H(z--z)=
0, 1,
• < z; z >c z,
(9)
assures that the solution exists only for the axial locations z ~< z reached by the stream at time z. The eigenvalues
First-daughter fission-product behavior
275
~.., satisfy the mass-transfer condition at the wall "/[/'gg0,m ~i,mJl(O~i,m) = ~ Jo(o~i,m);
(10)
a finite number N= are determined in each material region so that computational consistency and feasibility can be realized. The integration coefficients c1,,= and c2,,= are determined using the axial boundary conditions and Davidon's method, as applied by Durkee and Lee (1984). The technique is outlined for the daughter solution. For the interval L,._ 1 ~< z ~< L,., the initial conditions are given by
O.,m(p,z,O) = 0
(i1)
q~.,=(z, 0) = 0,
(12)
60.,,.~p p=0 = 0
(13)
and
the multiregion radial boundary conditions by
and
dO.,,.Op p= 1
./Ira.,,.2 0.,,.(1, z, z)
(14)
and the inlet and outlet conditions by
O"a(P'O'z) - 01° =
1,
n = 0;
Cao/Coo,
n
1,
l i m On,M(p, Z, "C) = O. z~co
(15) (16)
The convective and diffusive conditions of continuity at each material interface are given by
O.,.,I.=L~ = O.,,.+llz=L~
(17)
and 1 O0.,m ~'.,,. &
z=L~-
1 9%,,. + 1
c~O"'m+Iz=l.~" Oz
(18)
The dimensionless variables are defined, for species n in region m, as O.,,.(p, z, 3) =
C.,,.(r, z , t)
the stream concentration,
(19)
the wall concentration,
(20)
the decay constant,
(21)
the material region interface,
(22)
the Nusselt number for mass transfer,
(23)
the P6clet number,
(24)
the radial variable,
(25)
Coo
~,,.(z, t)
q).,~(z, 3) = - -
Coo R
R2,,
Kn
190
£¢,. L m
--
R h,,,,,,R Dn,m Rv o
~e',m -- Dn,," r
R
276
J.W.
DURKEE JR a n d C. E. LEE
t/) o
- R
the time
(26)
the axial variable.
(27)
and z=
Z R
A solution to the daughter product convective~tiffusion equation with slug flow is obtained by constructing homogeneous and particular solutions to equation (5). Application of the method of separation of variables yields the homogeneous solution N#
01,m,h(P, Z, Z) = n('r-- z) ~ Jo([3j,,,,p) [c3~.~ exp (.c#lj. z) + c4j,~ exp (fq2j. z)],
(28)
j=l
where ~"1 ,ra -~-~ , 2 , m
"F-4 ( ~ 1 ,mg l -F flj2m)
c~l,2j,. =
2
(29)
The eigenvalues flj,,~ satisfy the condition J~/'¢~'1,m
~J'mJ l([JJ'm) --
2
JO(~j,ra),
(30)
and Na daughter eigenvalues are computed. The particular solution is obtained using the method of undetermined coefficients by postulating a solution with the same spatial structure as the inhomogeneous term in equation (5) and a set of constant coefficients to be determined using the orthogonality property of the radial eigenfunctions. Thus, for the inhomogeneous term N~
KoOo.m(p, z, z) = KoH(z - z) ~ Jo(o~i,mp) [c l,,. exp (F x,,~z) + c2,,~ exp (rz,,Z)],
(31)
i=1
a solution of the form Na
Ol.m.p(P, Z, Z) = n ( r -- z) ~ Jo(o~i.mP) [K 1,p,,~ exp (F 1,,z) + Kz,p,,~ exp (F2,, z)]
(32)
i-1
is postulated. The particular solution must however satisfy the radial boundary conditions for the daughter product, equations and (13) and (14). This is achieved by using the orthogonality property for Bessel functions to rewrite equation (32) in terms of the daughter product eigenvalues and eigenfunctions so that N~
i= 1
Ntt,Np
Jo(~i,,.p) = ~
i,j=l
~(~xi.,,.;flj.m)Jo(flj,mp),
(33)
where Na, N#
E
i,j=l
N~ Np
= E E,
(34)
i=lj=l
I 4~,j,m
~(0q.. ; flj,m) = ~i.j,. -- i5,,"
(35)
and 14..... = Jo ~e'~,,.pdo(flk,mp)do(~,.mp) dp = ~'~ m ~k.mJl(l~k.m)Jo(~j-- ~i.Mo(~k.~)Jl(~i.m!,
(36)
First-daughter fission-product behavior
277
and
P
IsJ'~ = ~o ~e'~.r~PJo(flk.,.P)Jo(flj,mp)dp
.i o(Pj.,,,) ~ I-o~
,-,~: =1.0,
//~ 1.,,,']2-] \
"
2
] J'
(37)
j~k.
Adopting this notation, so that only the daughter radial eigenfunction is explicitly expressed in the solution, the postulated particular solution becomes Na, N#
~i.j,mJo(flj,mp) EK!,p,,j,. exp ( r l l ,
O!,=,p(p,z, ~) = H ( ~ - z)
z ) q--
K2.p,,j,m exp (F2,, z)'].
(38)
i,j= 1
The constant coefficients Kl,pi,j,~ and K2,p,.j,~ are determined by inserting equation (38) into equation (5) and applying the orthogonality property for Bessel functions. Careful evaluation of the derivative of the Heaviside function (Sneddon, 1972) gives, for z > z+e, KoCli,m
KI'pI'J'~ -
1
rx,,.-
~,,~ rL~ + ~
~
1
~'el,m
2
= K0,,i, c!,.~
(39)
A -- go,.j.mC2,,m.
(40)
~j,m+ K!
and
l
g2"pi'J'm ~
Koc21.m 2 1
2
The general solution to equation (5) is therefore given by
Oi.m = Or.re.h+ 01.m.p
(41)
N#
= I-I(~- z) y" Jo(fl~.mO) {c3j,~ exp (Nl,,~z) + c4j,~ exp (~2j.~z) j=l Na
+ ~ ~i,j,m [Kl.p,,j.~ exp (FI,, z) + K2.p,,j,~ exp (F2,. z)] }.
(42)
The constants of integration Caj.~ and c4j,~ are determined using the axial boundary conditions. Application of the inlet condition gives
Jo(flj,!P) c3j.1 "]-C4j,I -~j=l
~i,j, 1 [-Kl,p,.j,, "t-g2.p,,~.l]
= 01o.
(43)
i=l
Continuity in the concentration of daughter product material convected across the interface locations Ls, m = 1, 2. . . . . M - - 1, is guaranteed by
Jo(fli,mP) [c3j,. exp (if!j, L,.) + c4j.. exp (ff2j, Lrn)] j=l
+ ~ ~ij.,[Kipj., i=1
, , ,~ exp (F1,,~Lm)+K2p, ,,j,~ exp (F2.Lm)]
}
= ~ Jo(flZm +1 P) ~[c37 . . . . exp (c~!7,.+, Lm) + c'*y. . . . exp (c~27. . . . Lm)] ]'=1
}
+ ~ ~i.Zm+ ! [K !.p~,~. . . . exp (F 1..... Lm) + K2,p,.~ . . . . exp (F 2. . . . . Lm) ] , i=1
(44)
278
J.W. DURKEE JR and C. E. LEE
while the expression
1 ,mj~-I N~ Jo(flLmP) { [ C 3 j , m f f l / , m
exp
(fflj,~L,.)+ c4j,~2~.~ exp (ff2j, Lm)]
)]}
No
+ ~ ~ij,,.[Kav. ,, , 1,j , m F1.l , m exp(F1.~,m L , . ) + K 2 v , t, j ,m F2 , , m exp(F2 t , m L,. i=1
1 ~2L., + 1
7~lJo(flZ.,+lP) [c37. . . . ff17. . . . exp (ff17. . . . Lm)+C4Z~+~(~2y .... exp (ff27. . . . L,.)]
+ ~ ~i,~,m+l [K1.p.,7 .... Fx,~+ ~exp(Fl..m.tLm)+K2.v.,7,~+.F2 ..... exp(F 2. . . . . L,.)]
(45)
i=1
insures conservation of mass for diffusive transport across each interface. The outlet boundary condition dictates that the concentration in region M vanish as z --, 0% which necessitates c3~,M = 0.
(46)
A set of linear algebraic equations to be solved for the c3j.~ and c4j.~ integration constants is obtained by premultiplying equations (43)-(45) by ~/1,mPJo(flk.m), integrating over p from 0 to 1 and utilizing the orthogonality property of Bessel functions. The potentially troublesome numerical difficulties expected to be encountered in solving the resulting stiffsystem are lessened by regrouping and redefining the matrix coefficients and constants of integration so that the arguments of all the remaining exponential terms are negative. Also, terms containing the integration constants for the parent solution, c~,,~ and c2,.~ are rewritten so that the values computed using Davidon's method are directly utilized. This alleviates potential problems with significance loss which could arise from intermediate computations. Letting c~'j,~ = exp (cgffl~,Lrn)C3j,~
(47)
exp (F1,.~Lm) Kl,p,,j,~ = Ko,,j,~cl ....
(48)
c'1,,~ = exp(Fl,.~L,,)cl,.~
(49)
c4~. . . . = exp (fizz. . . . L~) C4j . . . .
(50)
and
where
form = 1 , 2 , . . . , M - 2
and, for m = M - - l ,
and ^
exp (F 2..... L,,,) K 2,p,,7.... = Ko,,7.... (~2......
(51)
c2 . . . . . = exp (F z..... Lm) C2......
(52)
where
the system structure for computational enhancement becomes N~
. , exp(--ffllj,lL1)c*j,,+c%,, = 01 oI6~1/15jl-E
~i.Ll[exp
(--FI~,,L1)Ko~,j,,cx~,I ~ * +Ko,,j,,c2,.], ^
(53)
i=1 N#
Isj.m[c*j,.+exp(ffzj..Lm) C%.J-- ~ I7j,..y .... {exp [flit . . . . (Lm--L,,,+O] c3)'.+*, + e x p (c527,.+1Lm)C4.7,.~-1} j=l
Na
N#
= --Isj,m ~ ~,.j,m[Ko,.j,~C*,,m+exp(F2.,mLm) RO.,j.~Cz.,m-I+ ~.. I7j,~,7.... i=1
f= 1
Na
x ~ ¢i,7,.+ 1 {exp [F 1..... (L,,,--Lm+O] Ko,,7,.+- 1c1,,.*+, +6xp(F~ . . . . . Lm) K"o,d.... c2 . . . . . } i=1
(54)
First-daughter fission-product behavior
279
and 1
- -
Isj,. [ffa~, c ~ , . + fq2~,~exp (cff2~,Lrn) C4~,~] N#
1
~ l%,~,7.~+, {ff17. . . . exp [(¢17. . . . (L,,,--L,.+ 1)]c'~7.... + ff2y.... exp (ff27. . . . Lm)c'*7.... }
~'*'l.m+, 7=' 1
Na
1
15j
~g'l,m
,m
~ ~ jm[F, i= l
, ,
tm
/~0,~ c* , ,m
,,m
+F2
,m
exp(F2. Lm)/(0 j~c2 j + .
.
.
.
.
.
N# ~Ivj,~,7....
~i?il,m+ 1 f=
N~
x ~ ~i,Zm+I{F, . .+~exp[F . . . . 1. + (Lm-Lm+l)]Ko,5.~+:cl~*. . . . . + F 2 ..... exp(F2 ..... Lm)K'o,.7. . . . C2..... }' i=1
(55)
form = 1,2 . . . . . M - - 2 , and l~j,~ [c~'j,m+
exp
(q.~2.i,mLm)ca~.m]
--
~
Ivj,m,)- . . . .
547 . . . .
7=1
Na
N#
= --Isj.. ~ ~.j,m[go, j C * , + e x p ( r z , , L , . ) R o , j C 2 , , j + ~ i=1
Na
Ivj,., z.... ~ ~i.7,,.+IRo,,z .... F2...... j=l=
1
1
Is~'m[fqu'~c*~'~+fff2j'~exp(f~2j"L")c%'J 1
--
1
-
Np
"¢~e'X,m+l f=~l 17j'~'7 . . . .
~2y ....
?4j'....
N,
~el,,. +
(56)
i=l
15j,~ ~ ~i,j,,,,[F,,,~go,,j,.c*,,~+ F2,..exp(F2,,~L.,)Ko,,j,~c2,,J i=1
N#
N~
~ I%,~,7,-+, Z ~,,D.+,Fz . . . . . /2(o,,7.-+,F2. . . . .
~'*'1,,. + 1 7=1
(57)
i=,
for m = M - - 1. The integrals I6k,1 and Ivk.~,~-. . . . are
16k.1 =
=
0
~e'a,,PJo(flk,xP)dp
~" 1,1 J'(/~*") --
(58)
ilk,1
and
17.... ~.... = f ~ ~e',,mPJo(flk,mP)Jo(fly, m+ ,P) dp ~.
[flk,mJ,(fl*,,,,)Jo(flZm+ ,)--flY,,.+ lJo(flk,m)J,(flf.m+ ,)]
= O'el,m[
L-
R2
R2~
IJk,m--Pj,m+l
1"
(59)
I
The quasi-Newton Davidon (1959) iterative minimization technique is used to solve the system of equations (53)-(57). This is accomplished by constructing a function consisting of the sum of the squares of the residuals of the matrix-vector system. The least-squares function, f(X), is minimized by varying each constant of integration xl, x2 . . . . . Xm while holding all other parameters fixed using the Newton-like second-order form of functional minimization, X = X,. = X k --G-
lg,
(60)
where g and G - ' are the gradient and inverse Hessian matrix of f evaluated at X, respectively. The Davidon algorithm replaces the inverse Hessian matrix with a positive definite symmetric matrix which is constructed iteratively using functional and gradient information. Using this technique, the solution vector X can be obtained which, when substituted back into AX = B, satisfies the original equation to a prescribed accuracy. The structure of the residuals for the daughter solution is similar to that for the parent solution given previously
280
J.W. DURKEEJR and C. E. LEE
by Durkee and Lee (1984) with all inhomogeneous terms being nonzero for the daughter solution. The gradients are identical in structure, only the constant coefficients differ. The solution is computed by calculating the integration coefficients for the parent, constructing the system for the daughter using the necessary coefficients from the parent solution and applying Davidon's method for calculating the integration constants of the daughter solution.
Wall Concentration for Slug Flow in a Multiregion Pipe The expression for the concentration of the daughter product on the wall of the conduit is obtained by solving
~¢1,m
1 001~
.=x
(61)
where (Durkee and Lee, 1984) '~/'¢g0,m { 1 -- exp [K o(Zq~o,~-- 2~%,mK °
z)] }00,re(l, Z, Z).
(62)
Using the integrating factor u(z) = exp (Klz)
(63)
the solution is qN,,.(z, z) = ~
01,,. {1 - exp [K 1(z-- z)] }
1 -'}- 0 0 , m [ K
{1 --exp [ K , ( z - - 0 ] ,
1 {exp [Ko(z--0]--exp K1 --Ko
[Kx(z--z)]}~
]l
(64)
where JV~0'm 00,.,(1, Z, Z)
O0,m-- 2~O,m
(65)
gi/'~g I ,m 01 ,m -- 2 ~ 1 , m 01 ,m( 1, Z, "~).
(66)
and
Predictions of fission-product accrual on a conduit wall may be compared with experimentally measured values once the pointwise expression, equation (64), is integrated to give the total concentration for a sectioned portion of pipe as Tll,m(V) = 2n
~01,,.(z,z)dz.
(67)
t i
For a section extending from Lt_ 1 to L~, integration of equation (64) gives Tl,t,,.(z ) = 2nH(z --z)
Jff~ l,,.
N~
II-E ~_2Ka~e.l,,..i= a
x { TI+Tz+
i u~=~jm[I~ojC = , , t m *i,m (T3-T,
+~i~lJo(o~i,,.)
) + K o st,
c*,,~ K ( T 3 - - T 4 )
1 A-c2"m[~ (TS- T6) KI--Ko
(Ts-T6)II
,m c2.~,m (Ts--
T6)]}
K ~ I_K (68)
First-daughter fission-product behavior
281
with T1 = c3j,~*
{exp [fflj,~(L~-Lm)] --exp [Nlj.~(LI- 1 - Lm)]} exp (--KlZ)
T2 = c,,j.~I~l--~-[exp(~z~,
{exp [KxLI+Wtj (Li--L,.)]--exp [K1Lt-l+f#lj
(L 1--L.)]}I,
(69)
exp (-- K 1z) K1 +(q2j.~ {exp [(K 1 +(qzj.~)Lt]--exp [(K1 +~2j,~)LI- 13}1,
(70)
II
L,)--exp(~zj, Lt_x)]
ll ~ 2 j , m
1
T• FI,,~ {exp IF l,,~(L -- L~)] -- exp [F,,.~(LI_I =
--
Lm)]},
(71)
T4 - exp (-- KlZ) {exp [K1LI + F~,,~(LI-Lm)]--exp [K,LI_, + F,,.~(LI_,- L,.)]}, K1 +FI,,~ 1
Ts =
[exp (Fzi, L/) -- exp (F2,.L 1_ , ) ' L
(72) (73)
l~2i,m
exp(-Klz)
T 6- K1 + ['2,,~ {exp [(Kx + F2,,m)LI] -- exp [(K1 + F2,,m)LI-1 ] }, Tv -
(74)
exp ( -- KoZ) {exp [KoLI + F I,.,(Lt -- L,,)] -- exp [KoLt-i + F I,,~(LI-a -- L~)] } Ko + Fx,.~
(75)
and Ts --
exp (-- Kor ) Ko + ['2i,m
{exp [(K o + F2,,m)Lt] - exp [(Ko + F2,,m)L,- 1]}
(76)
as the integrated plateout at time z, for the axial regions m = 1, 2 ..... M -- 1. For region m = M, the exponential arguments and constants of integration are different due to the regrouping for the variable metric algorithm so that the solution is of the form
T,.l,..(z) = 2,m(~-z)ll 2 ~
Jo(flj..,) Tg+ E ~ ,j...fgo,,.~z, m(Ylo--T103 ,
j=
i=1
+~i=12
Jo(cq,.,) F2,,~ ~-l(TIo--Tll)
K1
Ko (T12-Tll
(77)
with
F
T9 = F4~ m H ~
1
{exp [f¢2~ (LI-L.,_ 1 ) ] - e x p [f#2~,m(Ll_ 1 - L , . _ 1)]}
exp(--Kar) {exp [KILt+ff21,m(Ll--L,,,_ 1)] Kx "[- • 2 j , . 1
--exp [K1LI- 1+ ff2~,~(Lt- 1--L.,_ 1)]}11, 1
7"1° = F 2
(78)
{exp [F2,..(Lt--L,, ,_ 1)]--exp [F2~,~(L 1_ 1 --L,._ 0]},
(79)
i,m
exp (-- K l z) "/'11 -- Ka +F2,,. {exp [K1LI+ I"21,~(Lt--Lm_ 1)] --exp [K1Lt- 1+F21,~(Lt- aL,.- 1)]}
(80)
and
T~2ANE 12:6-B
exp ( -- Koz) {exp[KoLt+F2,,~(Li--Lm_O]--exp[KoLt_l+F2,,m(L Ko + F2~,~
x --L,._ x)]}.
(8l)
282
J.W. DURKEEJR and C. E. LEE
Activity units commensurate with measured values are readily obtained by premultiplying T~.t,~ by the decay constant of the daughter species and applying the appropriate units conversion factor. Deposition for the entire pipe is computed by summing the integrated contribution over all sections.
The Semi-infinite Pipe Daughter Product Deposition Model Solutions The single-region stream, wall and integrated deposition solutions are constructed by following a procedure similar to that for the multiregion solutions with the integration constants determined directly. The stream solution is given by
Ol(p,z,z ) = H(z--z) ~ Jo(flsP) 01,, j=l
exp(~2jz)+ ~ ~ijK,,,~[exp(F2(z)-exp(~2jz)] , i=1
(82)
where Kpl a -
go i
1 2 1 F2 i - ~1F2i-~ -
~lflJ
2
+ KI
'
4K o 1 K°' - JV~o F1 / 2~i \27
/
(83)
(84)
andF2,,ff2;¢i.jandl6s/Issarethesingle-region(M.= 1) values ofequations (8),(29),(35) and (58), respectively. The solution limits to that obtained by Venerus and Ozi~ik (1966) if axial diffusion is neglected. The wall concentration distribution is qg,(z, z) = K~iT,{1 - e x p [ K l(Z - Z)] } +0 °
I~- {1 -1 {exp [Ko(z--'r)] -- exp [K x(z-- z)] }1' (85) 1 - e x p [Kl(Z--z)] } - -K1--Ko
where Yg¢o
0o(1, z, z)
(86)
Yggl /7, = 2~f, 01(1,z,~).
(87)
Oo = ~ and
The time-dependent integrated plateout solution is
Tm(z ) = 2zrH(z--z)~ JV'ua ~ Jo(flj) ~'2+ ~'~ ~i,jKp,,j[~F9+(7"lo-- TI1)] ~_2K1~ex j="l i= 1
2 "o ~ I 1 (7.1o_7~1)
-]-~oi~=lc21gl
(88)
with 01o
~ - 1 [exp(fq2jLt)--exp(fqzjL, 1)]
exp ( -- K az) Klq-C~2j
{exp [(K 1 +ff2s)Lt] --exp [(K~ +ff2j)L~ 1]}1 , (89)
First-daughter fission-product behavior
7"9= - - I ~
283
p ( +- Ka32j l z ) {exp [(K1 + a32j)Lt]--exp [(K1 + c#2j)Lt-t]} 1 , (90) [exp (aJ2jLt)--exp ((q2jLt_ x ) ] - e xK1 1
7'to = ~2~ [exp (F2~Lt)--exp (F2,L t_ a)],
T12 -
(91)
exp (--Ktz) {exp [(K 1 + F 2)Lt] --exp [(K 1 + F 2)Ll_ 1] }, K1 +F2~
(92)
exp ( -- Koz) {exp [-(Ko + Fz)Lz] -- exp [(K o + F2)L l_ 1] } Ko+F2~
(93)
and 1
c2,-
(94)
{ 2~i "~2"
RESULTS
Representative computational results are presented for flow conditions similar to those reported by Burnette (1982) in the General Atomic Fort St. Vrain plateout probe experiment. A flow velocity of 10 cm s - 1, pipe radius of 0.5 cm and pipe length of 75.0 cm were selected to analyze the plateout of 137Cs (tho = 30.17 yr) and its daughter, 137Ba (th~ = 2.55 min). Single-region diffusion and wall mass-transfer coefficients D O = 0.004 cm 2 s - t, D 1 = 0.003 cm 2 s - 1 and h0 = 50.0 cm s - t, hi = 40.0 cm s - 1 were used. An equilibrium inlet concentration of 105 atom c m - 3 was assumed for 137Cs, while a zero concentration was assumed for 137Ba. The former assumption leads to a conservative overestimate of 137Cs plateout since the concentration from fission increases in time from zero. The latter assumption is plausible because the equilibrium ratio of the 137Ba to a37Cs due to decay is approx. 10- 7. The single-region solution is given for postulated open-pipe flow conditions. It is used to illustrate the proper limit for plateout calculated using a three-region solution with the perturbed diffusion coefficients listed in Table 1. The parent and daughter dimensionless pointwise wall concentrations, ~P0(P,z, z) and ~pt (P, z, z), are shown in Figs 1 and 2. The integrated tube total plateout profiles, T0(r) and Tl(Z), are displayed in Figs 3 and 4. The semi-log wall profiles are plotted as a function ofz for several values of the dimensionless time z. The log-log integrated profiles were calculated by the integration of the pointwise expressions over the length of the pipe, z e [0, 1501 with evaluation at times z = 1 0 3, 10'* . . . . , 1 0 1 2 . Selected daughter pointwise wall concentration and total tube plateout values are listed in Tables 2 and 3. The effect of positioning two filters in the pipe was simulated by reducing the parent and daughter diffusion coefficients by 100~ from region 1 to 2 and from region 2 to 3, as indicated in Table 1 (filtered). The wall masstransfer coefficients were held constant as the pipe wall properties were assumed to be invariant along the region of deposition. Selected values of the pointwise wall and tube total plateout for the daughter are listed in Table 4. Total steady-state deposited parent and daughter concentrations are reduced by 22.2 and 22.4~o to 4.22 x 101° and 6.87 × 103, respectively, for this parametric set, as compared to that for the open pipe. The tube total parent and daughter deposition profiles exhibit a linear behavior to z = 101°, at which time the profiles bend and approach constant totals of 5.44 x 101° and 8.51 × 103 for the parent and daughter, respectively. Table 1. Open- and filtered-pipe
137 Cs
and
137 Ba
diffusion coefficients
Diffusion coefficients (cm 2 s - 1) 137Cs
137Ba
Conditions
Do.l
Do.2
Do.3
D~.~
D 1.2
Dr,3
Open Filtered
0.004 0.004
0.004001 0.002
0.004002 0.001
0.003 0.003
0.003001 0.0015
0.003002 0.00075
284
J . W . DURKEE JR and C. E. LEE
o
t:
o
o
-~ '-~
-~
-~
~
'
o
o
-o
%
%
~o
~;
..o =
~,°
) y / y j j
~
,,,
o=
°
8
i8
o
g
-
~
o
i
° L
/
~
I 0
NOIIV~IN3
ONOD
I 0
~
.v
o
o
o
°
°
I
I
I
0
0
NOIIV~IN3DNOD
0
0
First-daughter fission-product behavior
285
eJ r
o
i\
o
T
o
\
o
\.
%
~e
\ e~e
\. I
\
.o
\ %
I
o
"\
%
\.
I
-- o
L
I
I
I
2 .I
\
o
o
..j
%
T
%
t
I
?
I
I
I
I
I
I
~o
I
\ \ \
~
•
•
\. \
~
%
\ I
L
I
I
I
I
i
%
I
t -
T
=
o
= o .~
l
\ "\ •
~
•
~ •
o
\
_~o
-\
O
~o %
"\ "\
~
% I
I
I
I
I
I
I
.~
I
I
I
I
I
I
I
L o
NOII~IN~DNO0 NOIIV~IN3
DNOD
%
I o
o
O
286
J. W .
DURKEE JR and
C.
E.
LEE
Table 2. Selected single-region open-pipe pointwise and integrated ~37Ba plateout values Plateout (dimensionless) Time r :
105
108
10 ~
Location z 10
7.12x10
40 100 " 150
Total:
T
5
7 . 4 4 × 10 - 2
1.99×10 t
3.23 x 10 5 1.77 x 10 - 5 1.29x10 -5 3.04 × 10 - 2
3.37 x 10 - 2 1.85 x 10 2 1.35x10 2 3.18 x 101
9.03 x 10 ° 4.95 x 10 ° 3.62×10 ° 8.51 x 103
Table 3. Selected three-region open-pipe pointwise and integrated ~37Ba plateout values Plateout (dimensionless) Time ~ :
I115
10 8
101~
7.12xl0 5 3.23 x 10 - 5 1.75 x 10 - 5 1.27x10 5 3.03 x 10 - 2
7.44x10 -2 3.37 x 10 - 2 1.85 x 10 - 2 1.35x10 z 3.18 × 101
1.99x101 9.03 x 10 ° 4.95 × 10 ° 3.62×10 ° 8.51 × 103
Location z 10 40
100 150
Total:
T
Table 4. Selected filtered-pipe pointwise and integrated
~~7Ba
plateout values
Plateout (dimensionless) Time z:
10 5
108
101~
7 . I 2 x 10 - 5 3.23 x 10 - 5 1.11 x 10 _5 4.43x10 5 2.34 x 10 - 2
7.44 x 10 - 2 3.37 x 10 - 2 1.18 × 10 - 2 5.29x10 z 2.46 x 101
1.99 × 10 l 9.03 × 10 ° 3.17 × 10 ° 1.42×10 ° 6.61 × 10 ~
Location z 10 40
100 150
Total:
T
Steady-state conditions are approached when the deposition time nears the half-life of the parent species. Similar behavior is exhibited for shorter-lived parent species, with steady-state conditions being achieved at earlier times. Additional calculations indicate that the plateout behavior of 137Ba is dominated by 137Cs for a wide range of flow conditions. This results because of the large difference between the parent and daughter half-lives which leads to large differences in the magnitudes of the stream and wall concentrations. For instance, the parent and daughter stream concentrations at p = 0.5 and z = 25 for the open-pipe conditions were calculated to be 9.8 x 1 0 - 1 and 1.2 × 10 - 9 , respectively. The resulting dominance in the daughter wall behavior is evident in equations (64)-(66). Consequently, at least for the parametric range analyzed here, a filtration system designed to filter 137Cs will suitably filter la7Ba since so little 137Ba is created. This furthermore indicates, as has been demonstrated computationally, that the determination of the ~37Ba wall mass-transfer and diffusion coefficients using a leastsquares minimization to fit the analytic model to a set of measured data (Durkee and Lee, 1984) is not practical since the sensitivity of the 137Ba concentration to the plateout governing parameters is so small. The least-squares minimization technique was successfully applied to a set of fictitious data created for the decay chain 14°Ba (tho = 12.79 d) to 14°La (thi = 40.23 h). These isotopes possess half-lives whose magnitudes are much closer than those of 137Cs and 137Ba. Single-region diffusion and wall mass-transfer coefficients Do = 0.001 cm 2 s - 1, D x = 0.002 cm 2 s - 1 and ho = 1.0 cm s - t, h ~ = 10.0 cm s - 1 were used to generate a daughter plateout profile at z = 108.Aninletconcentrationofl05atomcm-3wasstipulatedfor 14°Ba.Azeroinletconcentrationwasassumed for ~4°La. It was expected that the change in the calculated diffusion and wall mass-transfer coefficients due to the
287
First-daughter fission-product behavior Table 5. Calculated best-fit parameters for the t4°La data analysis Wall mass-transfer coefficient (cm s- 1) Parent
1.2/10.0
Diffusion coefficient (cm2 s - 1)
Daughter
1.0/10.0
Inlet concentration (atom cm- 3)
Parent
Daughter
Parent
0.00127/0.00130
0.00195/0.002
8.7D4/8.8D4
time-dependent increase in t4°La arising from the decay of a4°Ba would be quite small ; only the calculated parent inlet concentration used for solution normalization should change appreciably. A set of 30 fictitious measured activities was then generated by randomly perturbing the analytic solution integrated plateout value in each section by a m a x i m u m of 20%. The least-squares fitting process consisted of fitting the data with analytic solutions generated using several sets of estimated diffusion and wall mass-transfer coefficients and inlet concentrations. The fitting procedure was not a strict minimization process due to the competing plateout effects between the diffusion and wall mass-transfer coefficients. A fit was deemed suitable when a parametric regime was located where a limited range of parent and daughter diffusion and wall mass-transfer coefficients led to a change in the least-squares fit. Table 5 contains the parameters which gave the'best fit to this data set. The solutions exhibited a greater sensitivity to changes in the diffusion coefficients than the wall mass-transfer coefficients. Figure 5 illustrates the data and the fit constructed using Do = 0.00128 cm 2 s - t , D1 = 0.002 cm 2 s - t , ho = 5.0 cm s -1 h a = 10.0 cm s -1, Coo = 8.8 x 104 atom cm -3 and Cao = 0.0 atom cm -3. For these parameters a plateout of 12.7 #Ci 14°La was predicted. Using these parameters with the equilibrium decay ratio of a4OLa!t 40Ba ' 0 ao = 0.13 I, as the dimensionless inlet concentration for ~4°La, the plateout was calculated to be 15.4/~Ci. The pointwise integrated activity profile, shown in Fig. 6, differs appreciably from that shown in Fig. 5 for a zero inlet concentration of *4°La only in its normalization. Figures 7-10 contain plots of the dimensionless parent and daughter stream and integrated tube total profiles
102
1°2I
101
101 101
i
LS :&
10 0
Z O
"r
E o ~5 ::k
I, •
F "m-m.m" mrm'm'mnmm~m-u . . .ram umm ammamlm,m.mm m mm
I - 10-1 Z hi O Z O ¢.3
10 °
~
Z 0 n- 10-1 Z
g o
10-2
10- 2 --
10-3 0.0
I 12.5
I 25.0 AXIAL
Fig. 5. Fitted
I 57,5
I 50.0
I 62.5
POSITION (cm)
14°La data for 01o = 0.0,
I 75.0
10-~ k 00
I
I
I
I
I
I
12.5
250
57.5
50.0
62.5
75.0
AXIAL
POSITION (cm)
Fig. 6. l*°La integrated profile using best-fit parameters with 01o = 0.131.
288
J . W . DURKEE JR and C. E. LEE 1.0 m z=l 0.9
1012 1011
08
101° 10 9
0.7
10 8
z
0 p <
0.6
~
0.5
o/o--o--o--e--
107
/
~) 10 6 Z © ~
rr
105
0.3--
Z~ 104 W 103 0 {.C' 102
0.2--
10o
04--
•
/ /"
101
10-1 0 1 -OC__ 0.0
10-2 10- 3 0.2
0.4
0.6
0.8
I
I
I
103 1 0 4 1 0 5 1 0 6
1.0
I
I
I
107 108109
I
I
I
101010111012
TIME
RADIAL POSITION
Fig. 7. Dimensionless 14°Ba stream concentration.
Fig. 8. Dimensionless 14°Ba integrated tube total plateout.
1.0 --
09
1012 10~1
0.8
1010 109
0.7
10 8 Z o
0.6
107 Z
0
r~ FZ 0.5 w 0 Z 0 0.4
o/o
106
O.3
--o--o
./
10 5 z 104 t.u o z 105 0 102 ,
--o--o
,/o
/
./
101 0.2
10° 10 1
01
z =150
6~
'
~
~
10-2
i-- 4 0 20 J 2 0 1 % ~ 4 0 0.0
10-3
I
0.0
0.2
0.4 0.6 RADIAL POSlTIDN
0.8
Fig. 9. Dimensionless 14°La stream concentration.
1.0
I I I l 103 104 1 0 5 1 0 6 1 0 7
I 1 I I I 108 109101010111012 TIME
F i g . 10. D i m e n s i o n l e s s
14°La
integrated tube total plateout.
First-daughter fission-product behavior
289
made using these fitting parameters and 01o = 0.131. The daughter stream profile exhibits the nonzero inlet concentration condition. The integrated profiles reached steady state by z = 10s, or roughly 58 days, a time much shorter than that for 137Cs and 137Ba due to the much shorter half-lives of 14°Ba and 14°La. The one- and three-region stream, wall and integrated solutions were generated using 19 parent and daughter eigenvalues with starting values of 1.0 used for the minimization. The one- and three-region tests required roughly 25 and 42 s ofCRAY CPU, respectively, to evaluate and plot the solutions at 10 points in time, 100 radial locations and 150 axial positions. The Davidon method effectively minimized the three-region, 95-equation residual functions for the parent and daughter species. For the filtered pipe study, calculated residual-functionminima were 3.9 x 10-7 and 1.3 x 10-12 for the parent and daughter, respectively. The least-squares data analysis was expeditiously conducted by generating the stream solution only at the wall for calculation of the integrated plateout solution. By analyzing the solution at only z = l0 s with nine parent and daughter engenvalues, the CPU requirement per solution was reduced to 1.2 s. Solution accuracy was verified using 19 eigenvalues and parameters within the bounds calculated to best fit the data. CONCLUSIONS
The time-dependent behavior of a first-daughter product in a flowing gas stream has been modeled using an analytic solution to a convective-diffusion-decaytransport and deposition model. The plateout behavior on the wall of a pipe containing different materials was modeled by the adjustment of the diffusion coefficients in each region. Using plateout parameters similar to those found in the literature for the submicron-sized particle deposition from an isothermal He stream, the behavior of the 137Cs to 137Ba and the 14°Ba to ~*°La decay chains were investigated. It was found that the plateout behavior of the daughter is increasingly influenced by the deposition characteristics of the parent as the parent to daughter half-life ratio increased. The parent and daughter integrated deposition profiles exhibited a time-dependent behavior. This behavior can be approximated using a model which assumes a constant rate of plateout, as in Burnette (1982), only for limited deposition times.
REFERENCES
Burnette R. D. (1982) Report GA-A16764. Davidon W. C. (1959) Report ANL-5990. Durkee J. W. Jr and Lee C. E. (1984) Ann. nucl. Energy 11(7), 347. Kitahara T., Yokoo H., Kaieda K., Toyoshima N. and Fukushima M. (1976) J. nucl. Sci. Technol. 13(3), 111. Ozi~ik M. N. and Neill F. H. (1966) Trans. Am. nucl. Soc. 9, 393. Sneddon I. N. (1972) The Use oflntegral Transforms, pp. 484-487. McGraw-Hill, New York. Venerus E. R. and (~zi~ikM. N. (1966) Nucl. Sci. Engng 26, 122.
ANE
12:6-C