diffusional release of a single component material

Ann. nucl. Ener.qy, Vol. 14, No. 6, pp. 283-294, 1987 Printed in Great Britain. All rights reserved

0306-4549/87 $3.00+0.00 Copyright ~ 1987 Pergamon Journals Ltd

D I F F U S I O N A L R E L E A S E OF A S I N G L E C O M P O N E N T MATERIAL FROM A FINITE CYLINDRICAL WASTE FORM G. F. THOMAS Ontario Hydro Research Division, 800 Kipling Avenue, Toronto, Ontario, Canada MSZ 5S4 (Received 28 July 1986; in revisedJbrm 20 Noeember 1986) Abstract--The diffusion of a single component material from a finite cylindrical waste form, initially containing a uniform concentration of the material, is investigated. Under the condition that the cylinder is maintained in a well-stirred bath, expressions for the fractional inventory leached and the leach rate are derived with allowance for the possible permanent immobilization of the diffusant through its decay to a stable product and/or its irreversible reaction with the waste form matrix. The usefulness of the reported results in nuclear waste disposal applications is emphasized. The results herein are related to those previously derived by Bell and Nestor. A numerical scheme involving the partial decoupling of nested infinite summations and the use of rapidly converging rational approximants is recommended for the efficient implementation of the expressions derived to obtain reliable estimates of the interstitial pore diffusion constant and the rate constant describing the diffusant-waste form interaction from laboratory data. The basis of a scheme to simulate the release of a single species from a corroding cylindrical waste form under high flow rate conditions is briefly described.

I. INTRODUCTION Associated with nuclear power is the generation o f potentially h a z a r d o u s radioactive wastes. Prior to their long-term interim storage or p e r m a n e n t disposal these wastes are to be encapsulated in tailored solid waste forms. This will ensure t h a t the wastes are immobilized for a long period of time d u r i n g which their activity will be significantly reduced. Eventually the waste repository will be breached with a c o n s e q u e n t g r o u n d w a t e r ingress. S o u n d engineering practice will ensure a predictable long-term p e r f o r m a n c e of a p r o p o s e d repository. However, in the e n v i r o n m e n t a l a n d safety assessment o f such a repository, defensible models t h a t have been validated against field d a t a are used to predict the long-term release a n d t r a n s p o r t of radionuclides from the vault t h r o u g h the geosphere a n d to the biosphere. Both the release of the encapsulated wastes from the waste form material a n d the integrity o f the waste form with respect to corrosion determine in part the so called near-field source term, i.e. the discharge flux of radioactive material at the v a u l t - g e o s p h e r e interface. The release or source term provides a b o u n d a r y condition necessary to solve the radionuclide m i g r a t i o n models. A n early I A E A (1968) sponsored symposium on the t r e a t m e n t o f low- a n d intermediate-level wastes resulted in the i n t r o d u c t i o n o f the first s t a n d a r d waste form leach test (Hespe, 1971) a n d the f o r m u l a t i o n of its m a t h e m a t i c a l basis by Bell (1971). Since then the general u n d e r s t a n d i n g of the factors affecting the leachability o f nuclear waste forms has increased significantly ( B a m b e r g e r et al., 1981) a n d a variety o f models (Anders et al., 1978; G o d b e e a n d Joy, 1974; G o d b e e et al., 1980; M achiels a n d Pescatore, 1981 ; Melling a n d Allnatt, 1980 ; M o o r e et al., 1977 ; Pescatore a n d Machiels, 1979, 1981, 1982a) has been developed to rationalize the leaching behaviors o f s u n d r y waste forms, especially glass. M o s t o f the models are based on mass t r a n s p o r t considerations a n d i n c o r p o r a t e the Fickian diffusion of a single encapsulated material t h r o u g h the bulk waste form as mediated by a c o n c e n t r a t i o n gradient. The more sophisticated models a t t e m p t to i n c o r p o r a t e o t h e r rate-controlling processes (such as the desorption of the diffusant at the waste f o r m - l e a c h a n t interface, redeposition of leached waste o n t o the solid surface, a n d the corrosion o f the waste form), consistent with mass balance a n d the electroneutrality o f the system. Increases in complexity ultimately lead to a n analytically intractable n o n - l i n e a r model. A c o m m o n feature o f the available models is that they are 1-D with the waste form treated as a semi-infinite medium. Pescatore a n d Machiels (1982b) have developed a conservative criterion relating leach duration, specimen radius, a n d effective bulk diffusion coefficient to the degree o f d e p a r t u r e from the semi-infinite m e d i u m behavior. In Section 2a, we investigate the leaching o f a single c o m p o n e n t material from a f i n i t e cylindrical waste 283

284

G.F. THOMAS

~

m

k

P

2£ I

f Fig. I. Cylindrical co-ordinates (p, z, ~b) of an arbitrary point r within a solid cylinder of radius a and height 21. form, initially containing a uniform concentration of the material. Expressions are derived for the fractional inventory leached and the leach rate for the worst case scenario wherein the noncorroding waste form, immersed in a well-stirred bath ofleachant, maintains a zero concentration of the waste at the surface boundary and the interstitial pore diffusion of the waste through the solid is the rate-controlling mechanism of release, although allowance is made for the possible permanent immobilization of the diffusant through its decay to a stable product and/or its irreversible reaction with the waste form matrix. In Section 2b, a numerical scheme involving the partial decoupling of nested infinite summations and the use of descending rational approximants is recommended for the efficient implementation of the expressions derived. Finally, in Section 3, the results reported here are related to those previously derived at Oak Ridge National Laboratory by Bell (1971) and a priori estimates of the truncation errors introduced on terminating the infinite sums in the various formulas are presented. 2. A N A L Y S I S

(a) Formal aspects (i) Introduction Diffusion of the material through the cylinder is governed by the equation (Carslaw and Jaeger, 1959; Crank, 1964 ; .lost, 1965) ¢? DV%(r, t) = otc(r, t), (1) where D is the diffusion constant (m ~' s ~) and c(r, t) is the spatial (r) and temporal (t) concentration (M) of the material in the cylinder, respectively. We assume that initially the cylinder contains a uniformly distributed inventory of the diffusing material, i.e. c(r, 0) = c,,.

(2)

Referring to Fig. 1, the cylindrical co-ordinates of an arbitrary point within a cylinder of length 2 l a n d diameter

Diffusional release from a finite cylindrical waste form

285

2a are r = p, ~b,z, where 0 ~< p ~< a, 0 ~< ~b < 2n, and Izl ~ l. If the cylinder is immersed in a well-stirred tank so that the external concentration of the material is zero then the appropriate boundary conditions are

c(a, z, t) = 0,

(3)

e(p, _+l, t) = 0,

(4)

for t > 0 and the ~b-dependence of c has been dropped since diffusion through the cylinder is isotropic. Transforming equation (1) to to-space gives

D

+

., + pep

c,,,(p,z) = toe.,(p,z)-eo,

(5)

where

c,o(p, z) =

dt exp ( - cot)c(p, z, t)

(6)

is the Laplace transform (Carslaw and Jaeger, 1949; McLachlan, 1939) of c(p, z, t). In what follows, equation (5) will be solved subject to the boundary conditions c,.(a, z) = c,o(p, 4-/) = 0 for to > 0 and the resultant solution e,,,(p, z) will be inverted to give c(p, z, t) through explicit evaluation of the Bromwich integral

I I ~+i~

c(p, z, t) = 2ni.)~ ,