United States
Environmental Protection
Agency
Office of
Radiation Programs
Washington DC 20460
EPA 520/8-81-007
1981
Radiation
vvEPA
Short- and Long-Term Leach Rates of
Solidified Waste from a
Cylindrical Container
-------
EPA 520/8-81-007
Short- and Long-Term Leach Rates of Solidified
Waste from a Cylindrical Container
Kerry A. Landman
June 1981
Division of Statistics and Applied Mathematics
Office of Radiation Programs
U.S. Environmental Protection Agency
Washington, D.C. 20460
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ABSTRACT
The shor£- and long-term leach rates of radionuclides for the
three-dimensional diffusive flow and for purely axial diffusive flow
from a finite cylinder of solidified waste are determined here.
These analytical results are compared with the ones obtained
numerically by Hung (Hu80) for purely axial flow.
11
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CONTENTS
Page
1. Purely axial flow
«
*
1.1 Introduction 1
1.2 Solution to the model equation 1
1.3 Short-term leach rates 3
1.4 Long-term leach rates 6
1.5 Conclusions 6
Appendix A 7
2. Three-dimensional flow - 9
2.1 Introduction 9
2.2 Determination of the leach rate 9
2.3 Short-term leach rates 13
2.4 Long-term leach rates 15
Appendix B 17
References 23
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1. PURELY AXIAL FLOW
i
i.l Introduction
A common method for the disposal of radioactive waste is to
embed the radionuclides into a solid container. However, these
particles can still escape from the container and thus be a hazard
to the environment.
*
The short- and long-term leach rates of radionuclides for the
oiffusive flow from a finite cylinder of solidified waste are
discussed here. Their transport is governed by diffusion,
desorption, and radioactive decay, and is assumed to be in one
direction only. (For example, the system presented here models a
cylindrical container of length L, where all the faces, except for
one end, are insulated).
This same system was discussed by Hung (HuSO). The solution to
the basic differential equation with prescribed boundary conditions
was obtained by numerical analysis. However, the equation has a
fairly simple solution, which will be discussed below. Comparisions
of these results will be made with those of Hung, and also to the
solution for a semi-infinite cylinder with insulated sides.
1.2 Solution to the Model Equation
The basic equation for the transport of radionuclides through
the porous medium of the container (which has been immersed in an
aqueous solution), where diffusion, desorption, and radioactive
decay are present, is given by Hung (Hu80) as
- xdc , 0 <_ x < L , (1.1)
where c is the concentration of radionuclides (Ci/cm3) in the
aqueous solution which is saturating the waste solid, De is the
effective diffusivity through the solidified waste (cti^/year),
e is the porosity, R is the retardation factor defined in (HuSO),
and A(J is the decay constant for the radionucl ide.
The initial and boundary conditions on the container are
c(x,0) = CQ 0 < x < L, (1.2)
c(0,t) =0 t > 0, (1.3)
t>0. (1'4)
-------
Therefore, it is assumed that the equilibrium state of desorption
has been reached at time t=0, there is no accumulation of
radionuclides at the end x=0 and that the other end, x=L, is
insulated.
Tne equation (1.1) with conditions (1.2)-(1.4) can be solved by
the standard techniques of separation of variables and the super-
position of normal modes (Hi76). This gives
c(x,t)= p_ ^ 1 sin (2n+l),rx e , (1.5)
» n=0 2n+T 2L
where K » De/eR.
Now that the radionuclide concentration in the aqueous
solution, which saturates the solidified waste, is known, the rate
of leaching l(t) can be obtained from
l(tj=DeAr _ac_ (0,t) (Ci/year), (1.6)
where Ar is the cross-sectional area of the solid (cm2). It
is often useful to determine the rate of leaching as a fraction of
the total initial activity A0. Here
A0=eRC0ArL ; (1.7)
then from (1.5)-(1.7), the rate of radionuclide leaching as a
fraction of the initial activity per year is
e d
e (1.8)
n=0
Here a new parameter, the leaching number, Lc, has been
defined:
C e R L2
-------
In order,to conform to Hung's notation, we introduce a
dimensionless time variable t', a dimensionless leaching parameter
1* and a leaching number Ln:
f. =''°e t = Lt ,
A C
eRL2 1 = I , (1.10)
Ao De Ao Lc
and Ln = xd/Lc
Substitution of (1.10) into (1.8) gives
i t-
-Lnt
00 2
<- « i
2e e ' . (1.11)
In tne following two sections we consider approximations to 1' for
t1 small and for t1 sufficiently large (near one or greater).
1.3 Short-Time Leach Rates
An estimate of the dimensionless leach rate 1', given by (1.11)
is now required. This involves making an asymptotic approximation
to the summation S,
00 _ On+i ^ 2 £i
e ' (1.12)
n=0
for 0 < t' « 1. To do this, compare the summation with the
-------
area under the graph- of e for 0 <_ n £00 ; this is
illustrated in Figure 1. Since
/
2
oo - -nir- t
e dn . -- , (1.13)
. - u
we have, from consideration of Figure 1, the two unequal ities
s < 4- [ 1 + e ] < i [_L + i ] , d-14)
\£~F' " \£T'
and
S >-L_ [ 1 - 1] , d.15)
for all values of t' > 0. Therefore for 0 < t' « 1
S~ * . (1.16)
Combining the result with equation (1.11), and assuming that
Lnt' « 1, as well as t1 « 1,
(1.17)
Hence, we conclude that the short-time (dimensionless) leach
rate for the finite-dimensional geometry is the same as that for the
semi-infinite geometry (as shown in Appendix A). This confirms the
results of Hung (in his Figures 2 and 3).
-------
01234567
Figure 1. Graphs of e~(rbr/2^ t§ versus n for t1 > 0
-------
1.4 Long-Time Leach Rates
For sufficiently large values of (dimensionless) time, the
value of 1' can be approximated by the first term in the series:
. - [Ln +-L-?'
I1 -. 2*e 4 . (1.18)
For intermediate values of time, then the first few terms of the
series expansion (1.11) will be sufficient to give a good
approximation to 1 ' .
The values given by (1.18), for t1 near one, match well with
the values illustrated in Figure 3 by Hung, which were obtained by
numerically solving the set of difference equations associated with
equation (1.1).
1.5 Conclusions
The leach rates for the finite-dimensional model system can be
well approximated by the simple formulas:
1' - 1 for t' « 1 , (1.19)
and
2 e for t1 sufficiently (1.20)
large,
for the dimensionless variables 1' and t'. These results match well
with those of Hung, which were carried out more expensively by using
the computer to solve the basic equation.
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7
APPENDIX A
Tne analagous problem to the one discussed in Section 2, but
for 0 < x <» is
ac = a c - x CH ,
7 °
at ' ax
c(x,0) = CQ x > 0 , . (A.I)
c(0,t) =0 t > 0 ,
where * = De/eR. This has an exact solution
-A t
c(x,t) =CQ e erf( x , (A.2)
where erf(y) is the well-known error function
2
erf(y) = 2 e~ du . (A. 3)
f
/ ~ u
Then the leach rate from the face x=0 is just
-x t
r. _3C (0,t> = OeArCQ e . (A.4)
ax
-------
8
Using the notation of Section 1.2, this gives
I1 = 1 = e n . ' (A. 5)
For small t1 ^nd Lnt', this may be approximated as
I1 = . (A. 6)
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2. THREE-DIMENSIONAL FLOW
2.1 Introduction
Numerous studies have been made to determine the leach rates of
radionuclides from solid waste. However, the results obtained so
far are of limited use in practical applications because the basic
assumptions are often not physically acceptable.
For modelling purposes, it is usually assumed that the flow out
of the solidified waste is in one direction only. Both the
semi-infinite (Go74) and finite-dimensional (Hu80) models, with
uni-directional flow, have been studied. However, these geometries
are too simplistic to give reliable results.. A real canister is
usually cylindrical and it cannot be assumed that all of its
exterior surfaces, except for one end face, are insulated. In fact,
there will be a flux out of all the three faces of the cylinder,
giving rise to a three-dimensional flow.
This paper addresses the problem of short- and long-time leach
rates out of a cylinder, where the radionuclides can escape out of
all the sides. Some of the results will be compared with Hung's
(Hu80). He used a "sealing-effect" relationship, without adequate
justification, to approximate the three-dimensional flow from a
cylinder with the one-dimensional flow. However, we will show that
his results are not necessarily accurate.
The other assumptions often made are that the chemical and
physical characteristics of the solidified waste stay the same over
long time periods. This may also be simplistic.
2.2 Determination of the Leach Rate
The basic equation for the transport of radionuclides through
the porous medium of the container (which has been immersed in an
aqueous solution), where diffusion, desorption, and radioactive
decay are present, is given by Hung (Hu80). For a cylindrical
geometry, with axial symmetry, this is most conveniently given in
cylindrical co-ordinates:
3C. Un A O _ 0V. j. O t , _ (? \\
__ __
-------
10
where r and z are the radial and axial co-ordvicites, respectively.
Here c is the concentration of radionuclides in the aqueous solution
which saturates the solid waste, De is the effective diffusivity,
c is the porosity, R is the retardation factor defined in (Hu80),
and xa is the decay constant for the radionuclide.
The initial and boundary conditions for the container are:
r
c = CQ 't = 0, 0 £ r < a, 0 < z < L, (2.2)
c » 0 z = 0, z = L, 0 < r < a, t > 0, (2.3)
c = 0 r = a, 0 < z < L, t > 0. (2.4).
Therefore, it is assumed that the equilibrium state of
desorption has been reached at time t = 0 and that there is no
accumulation of radionucl ides on the exterior surfaces of the
container.
The equation (2.1) with boundary conditions (2.3) - (2.4) can
be solved by the standard techniques of separation of variables and
the superposition of normal modes as
00 00
c(r.z.t) . e ^ - BmnJ0 (umr)sin n*z e -~r , (2.5)
m=l n= 1 L
where K= De/eR. The yma are the set of positive zeros of J0(x)
and the Bmn are constants which are determined by the initial
condition (2.2) as
.
mn ^
r a r l
I i"J,,(u r) dr I sin nnz dz
b 4 T"
-------
11
LHere the orthogonality relations for the J0(ymr) and
sin nitz functions are used (Hi76)]. Evaluation of the integrals gives
L
(2.7)
Bmn
0 n even
Substitution of (2.7) into (2.5) then gives the solution as
00
c(r,z,t) =^o_e u /_^i VV ' sin n,z e u "' (2.8)
wa m,n=l np J-, (y a) L
n odd
Now that the radionuclide concentration is known, the rate of
leaching l(t) can be obtained. Since the flux of radionuelides,, ^3 ,
is given by "*
0 = - Dg grad c » -Oe( ac r + ^c z) , (2.9)
^ 3 P U 7
0 I O fc
the concentration of radionuclides which passes out of the cylinder per
unit time is
l(t) = -// J . n dS , (2.10)
where n is the unit normal of the surface in the outward direction
ana the integral is taken over the whole surface. Therefore, the
integral in (2.10) has three parts - one corresponding to the end
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12
face z = 0, a similar one for z = L, and one for the surface r
writing this out more fully we must evaluate
0 D
r dr de
z=0
.2* /.a
0 3£
3Z
r dr de
z=L
(2.11)
.2w
0 ac
e 3?
a dz de
r=a
From equation (2.8), it is easily verified that
00
l(t) =32
L
i r/*\ . -I .
e T m
m,n=l
n odd
It is often useful to determine the rate of leaching as a fraction
of tne total initial activity A0. Here
A0 = cRCQira2 L ,
(2.13)
so that
00
ill)
A_
,2 A 2
m,n=l /nit \
n odd ( l'
. (2.14)
In the following two sections we consider approximations to
l(t)/A0 for small and long periods of time and compare our results
to Hurig's for a particular example he discusses.
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13
2.3 Short-Time Leach Rates
An estimate of the rate of radionuclide leaching as a fraction
of the initial activity, given by equation (2.14) is now required.
This involves making an asymptotic approximation to the double
summation
00
S . . * e-T (2.15)
n.n-1 (n.) ' .
n oda v LJ
for t sufficiently small. With some manipulation this may be rewritten as
2
oo nir, rt oo 2 ..
n=l x
n odd
00
00
, .
111=1 n odd
It is convenient to define two dimensionless time scales
T = ict and T = Kt . (2.17)
7 7
In Appendix B, the asymptotic forms of each of the series in
equation (2.16) is determined for small values of T and T. The results
there give
L2 , T, T«l
^^M
8 (2.18)
S
- 1 .
4V7T
- _1 r
leViT
a2.
4
£
Vf
+ 1
2v
-------
14
Therefore, for small values of Kt/1.2, Kt/a2 and x^t, the
rate of leaching as a fraction of the initial activity is
This asymptotic form shows that the rate of leaching for the
three-dimensional flow from a cylinder is made up of two
contributions one that would occur from the purely radial flow and
tne other from the purely axial flow.
Indeed, since the radius a of a canister used for disposal
purposes is usually smaller than the length L, choosing a time t
such tnat ict/a2 « l shows us that the dominant contribution to
the leach rate is from the radial flow.
This shows that the one-dimensional axial flow from a cylinder
will not accurately determine the short time leach rates from a
cylinder, since the radial flow will usually contribute a larger
amount .
If T and T are small, but the decay constant is large enough so
that xdt is no longer small, then the fractional leach rate is
"Xd
lill - 2 e r 1 + 1 1 (2.20)
AQ >»t L a L J
and the above remarks remain valid. In fact, the example presented
by Hung (Hu80) for the predicted leach rate at time t = 100 years is
really a problem of determining the short-term leach rate for the
three-dimensional flow described here. The parameters are as
follows: the number < , which measures the physical and chemical
properties of the radionuclides in the solidified waste medium, can
-------
15
oe determined from the leaching constant for the 5-cm long test
piece, so that
K = 3.74 x 10-6 x (5)2 = 9.35 x 10~5 cm2/day. (2.21)
Also
xd = 6.33 x 10~b/day
a ' s 30 cm
L . = 90 cm (2.22)
t =3.65 x 104 day.
Therefore
T = *t . 3.79 x 10"3 > Kt (2.23)
""
hence both the dimensionless time scales r and T are small. However,
xdt . 2.31, (2.24)
so that the short term leach rate as given by equation (2.20) applies. This
gives
l(t) = 2.51 x 10"7/day; (2.25)
o
so that, for this particular choice of the parameters, the predicted
fractional leach rate has the same order of magnitude as that
obtained by Hung (his equation (32)), with the use of his scaling
effect argument. However, this will not always be true, the
relative size of the parameters is important.
2.4 Long-Term Leach Rates
For sufficiently large values of the dimensionless time
variable
-------
16
wnere £j = 2.405 is the first positive zero of J0(x). For
intermediate values of (dimensionless) time, the first few terms of the
series expansion (2.14) will be sufficient to give a good approximation
to 1/A0. /
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17
APPENDIX B
i. Consider the purely axial flow from a cylinder, radius a and
length L. Suppose u(z,t) satisfies
u = au 0<*< L -,
' . (B.I)
u(*.0) = 'UQ 0 < * < L ,
u(0,t) = u(L,t)= 0 t > 0 .
The solution to this problem, when obtained by separation of variables
and superposition of the normal modes, is
oo ,nn»2 0 ,
j ~"~L 1^
rlo.^ K
** , (B.4)
u = 0 , * = 0,L
-------
18
The solution to (B.4) is
U r, sinhqz + sinhq(L-fr)
u ~ * ~ sinhqL
Then,
1 uj ~ - 1 u.
*o
- * i coshqL-1
K q * sinhqL
L
q
-I
QC
(B.6)
JL ri + 2 , , .
K q L n=l J
Taking the inverse transform of these standard Laplace transform
functions (Ab72) gives
n2!2
oo - n L
i| =_i£l - ° [1 * 2
3* -feO a* I *=L A i r n=l
22
A i r
VlT
-------
19
Similarly, from (B.2) we can evaluate the average concentration in
tne cylinder as a function of time:
-ij U(*.t)d» = f!^_ 2J i
_ e u . (B.10)
n=l
n odd
But from (B.5) and then (B.6),
-if ~(-t ) d* - Uo fl fcosncll1\
v = -j- / * '"' ~ " Lq * sinhqL
0 P
oo
. (B.ll)
Taking the inverse transform gives
00
v = u - 4Ufl,<.t,1/2MT-1/2 + 2 Z(-Dn ierfc n -, . (B.12)
Again comparison of (B.10) with-the above equation gives, for small values
of r ,
oo o ?
a-nSr 2 . (B.13)
E
1 e"
ri
n odd
2. Consider the purely radial flow from a cylinder of radius a and length
L. Suppose u(r,t) satisfies
~ K L L- (r 1 0 < r < a ,
at r ar » ar'
(8.14)
u(r,0) = UQ 0 <_ r < a- ,
u(a,t) =0 t > 0 .
-------
20
The solution to this problem, using the separation of variable technique,
is
00
2 Kt
u(r,t) = 2UQ ]T
m=l
(B.15)
wnere J0(O=Q» O 0. Then,
00
7
(B.16)
m=l
However, the series in (B.15) and (8.16) converge very slowly for
small values of Kt/a2. The Laplace transform method is better for
this case; then (B.14) becomes:
2-
d u
- 2-
1 du - q u
r dr
U
us 0, r=a
The solution to (B.17) is
(B.17)
(B.18)
Therefore,
a |i.
r=a p y^
Using the expansion of I (x) for large values of x,
(B.19)
Iu(x) . ex fl - 4v2-l
» r^ir L 1 I U
(4v2-l)(4v2-32)
^7
2: (ax) *
(B.20)
-------
21
equation (B.19) becomes
r=a
= q
p
[i
L1
o
~ ql -
+ 1 +
Sqa
I28q2a2 +
9 +
...j
...-,
Taking the inverse transform (Ab72), gives
ar
r=a
Hence,
a
(B.21)
(8.22)
Compare this with (8.16); we see that for small values of ict/a2
following asymptotic approximation is valid:
oo
m
(B.23)
T, the
(B.24)
Similarly, from (B.15) we can evaluate the average concentration in the
cylinder as a function of time:
ru(r,t)dr = 4 UQ ^ _1_
m=l *r
- o
e m a2
(B.25)
But from (B.18) we also have
a
v = 2
7
rdr =
ai I0(
-------
22
Using equations (B.19) and (6.20), this has the expansion
2U°
apq
-
2qa 8q2fl2
Taking the .inverse transform, the average concentration is
v = UQ-U
"' ad vS" \a' a-
Compare this with (B.25); we see that for small values of
m=l
m
(B.27)
(B.28)
(B.29)
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23
REFERENCES
Ab72 Abramowitz, M. and Stegun, I.A., Handbook of Mathematical
Functions wjth Formulas, Graphs, and Mathematical Tables,
National .bureau of Standards, Applied Mathematics Series 55, 1972.
Go74 Goodbee, H.W. and Joy D.C., Assessment of the loss of radio-
active isotopes from waste solids to the environment. Part I:
Background and theory. Oak Ridge National Laboratory, ORNL-TM-
4333, 1974.
Hi76 Hildebrand, F.B., Advanced Calculus for Applications,
Prentice Hall, Inc., 1976.
Hu80 Hung, C.Y., Prediction of long-term Teachability of a
solidified radioactive waste from a short-term Teachability
test by a similitude law for leaching systems. U. S. Environ-
mental Protection Agency, Office ofRadiation Programs, Washington,
D.C. 20460, 1980.
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO. ' 2.
EPA 520/8-81-007
4. TITLE AND SUBTITLE
Short- and Long-Term Leach Rates of Solidified
Waste from a Cylindrical Container
7. AUTHOR(S)
Kerry Landman
9. PERFORMING ORGANIZATION NAME.ANO ADDRESS
Office of Radiation Programs
Environmental Protection Agency
Washington, D.C. 20460
12. SPONSORING AGENCY NAME AND ADDRESS
Same
3. RECIPIENT'S ACCESSION NO.
5. REPORT DATE
June 1981
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO
10, PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The short- and long-term leach rates of radionuclides for the
three-dimensional diffusive flow and for purely axial diffusive flow
from a finite cylinder of solidified waste are determined here.
These analytical results are compared with the ones obtained
numerically by Hung for purely axial flow.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
radionuc i ides
axial flow
three-dimensional flow
leach rates
18. DISTRIBUTION STATEMENT
b.lOENTIFIERS/OPEN ENDED TERMS
19. SECURITY CLASS I This Report/
Unclassified
20. SECURITY CLASS iTMs page/
Unclassified
c. COSATI Field/Group
21. NO. OF PAGES
26
22. PRICE
SPA form 22241
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