J. B. Condon
392 (36) (37) (38) (39) (40)
R. P. Buck, J. Nectroanal. Chem., 23,219 (1969). F. B. Llewellyn, Proc. /R€, 21, 1532 (1933). R. de Levie, J. Electroanal. Chem., 25, 257 (1970). T. Teorell, J. Gen. Physiol., 42, 831 (1959).
S. R. Caplan and D. C. Mlkulecky In "Ion Exchange," Voi. 1, J. A. Marinsky, Ed., Marcel Dekker, New York, 1966, p 46.
(41) Reference 35, p 81. (42) J. Crank, "The Mathematicsof Diffusion," Oxford University Press, New York, N.Y., 1957, Chapter 10. (43) 8. Carnahan, H. A. Luther, and S. 0. Wllkes, "Applied Numerical Methods," Wiley, New York, N.Y., 1969. Chapter 7. (44) R. P. Buck and J. R. Sandifer, J. fhys. Chem., 77, 2122 (1973).
Calculated vs. Experimental Hydrogen Reaction Rates with Uranium J. B. Condon' Union Carbide Corporation, Nuclear Division, Oak Ridge, Tennessee 37830 (ReceivedAugust 22, 1974) Publicationcosts assisted by Union Carbide Corporation, Nuclear Division
Reaction rates for the hydrogen attack on uranium samples were both measured and calculated. The calculations were made using the model involving the diffusion equation and the equation expressing the kinetics of hydriding. All parameters but one were specified by previous experiments. The model was found to be in excellent agreement with reaction rates determined experimentally provided caution was exercised to randomize the grain structure of the starting sample.
Introduction The purpose of this paper is to reconcile the previously observedza first-order reaction rate with the normally observed linear reaction rates between hydrogen and uranium samples of reasonable size. To do this, calculations are made utilizing the proposed reactant-diffusion model as presented in the previous paper. These calculations are required for most uranium samples, other than fine powders, to determine reaction rates. The ranges of sizes for which these calculations are needed is dependent upon temperature and hydrogen pressure. This is rigorously defined in a subsequent report,zbwhich demarks the range of temperatures, pressures, and sample thicknesses in which calculations become necessary. With the assumption of high tenacity for uranium (at least on the time scale of the reaction), the parabolic, first-order and mixed-parabolic and first-order reaction rates were derived. Tenacity of uranium under stress, however, is not normally observed. The rates of overall reactions are slow enough and the sample size large enough to allow either spalling from a stressed surface or at least loss of integrity. The consequence of this more realistic assumption is a reaction rate which is linear with time and surface area. Most reaction rates reported in the literature are of the linear type. The exceptions include parabolic rates of Albrecht and Mallett3 and Alire, et ~ 1 . first-order ; ~ rates by Condon and Larson;2a and S-shaped dependence by Amilhat-Caralp, et aL5 The former two types were dealt with fair success in the mentioned report;zbwhereas, the latter is impossible to compare since sample shapes were not specified. Svec and Duke6 did not report a uranium or surface area dependence but did report an activation energy of 8.2 kcal. Reports of linear dependence include data by Wicke and Otto,7 Albrecht and Mallett,3 and Galkin, et a1.* Reported herein are the results of some additional experThe Journal of Physical Chemistry, Voi. 79,No. 4, 1975
iments along with the method and results of the linear reaction-rate calculations. Comparisons of the experimental results with the calculated results should reveal the range of validity of the proposed model.
Method of Calculation Basic Model Equations. The reactant-phase-diffusion model involves a simple diffusion equation with a sink. This is given in one dimension, x , by ~ a2c ~ = -ac- - aaU ax at at
where C is the atomic concentration (in mole fraction) of the diffusing species (hydrogen), U the normalized diffusion medium content (uranium content expressed as mole fraction), t is time, and a is the stoichiometric ratio for the reaction (here a = 3 mole fraction of H/mole fraction of U). The diffusion coefficient, D, is considered constant even though one would expect it to change with the change in the diffusion medium, in this case a change from pure uranium to uranium with hydride contamination. Since the solubility of hydrogen in uranium is very low and mechanical failure occurs with very low levels of hydride inclusions, D is assumed constant up to the point of metal fracture. To determine the power of the sink term, a second equation is used, that is
-a' = k,CU at
where izl is the rate constant determined previously.2aThe boundary conditions used previously for the two-sided plate problemzhwere CCc,O)
=1
0
(3)
and c(0,t )
=I
c(2,t ) =
co
(4)
Hydrogen Reaction Rates with Uranium
393
where 1 was the plate thickness. Co is the solubility of hydrogen in the metal and together with D and kl is a known function of temperature.
-‘
( k l = 10.4 exp(1592/T) sec x (mole fraction of H)-‘ (ref 2a) (5) Co = 5.58 x l o m 4exp(
- 894/T) P 1 ” x
(mole fraction of H) (with P in Pa, ref 9, 10) (6) and D = 1.9 x
exp(-5820/T) m2/Sec (ref 9, 10) (7)
In the present case, however, the second boundary condition must be modified. To take into account the loss of integrity of the uranium after a certain fraction has been transformed locally the condition is C ( X C: xC,t)
C(X 2 1
- x C ,t )
= Co
that uranium is reasonably elastic when compressed to less than 1%. The Von Mises idealized equation for elastic materials written for two-dimensional rectalinear coordinates, Y 1 and Ya, is
c = g Y12
a Y l ay2 + ayz 2
(9)
when the stresses, ay, and ay,, are written as tensile being positive. It will be noted that the two-dimensional case with two equal compressive stresses and the one-dimensional case yield the same equation. In other words, if either OYi
=;
(10)
0Y2
or uy2 = 0
(11)
c
(12)
then
(8)
where x , is defined as the point at which exactly the right amount of transformation has occurred to cause integrity loss. Computationally, this is accomplished with a simple decision (“if”) statement. Up to this point, the calculation has only one adjustable parameter. This parameter, x c , is dependent on the point in the transformation a t which stresses on the uranium are great enough to cause fracture. This could be left as adjustable; however, there are some guidelines as to approximately what its maximum value should be. It should be noted here that the criterion should be dependent upon precise sample geometry as are all fracture problems. In the following, however, a flat surface will be assumed. The criterion derivation will also use the simplest equations since use of more sophisticated equations are not justified by either the data available for the mechanics of uranium or the hydriding kinetic data. An error of approximately 50% in this parameter should be acceptable here. The x, Criterion. The parameter x, in this calculation must be defined in terms of the extent of reaction. If the extent of reaction locally is equal to a defined amount, designated a,,then the distance from the surface to that locale is xc. The problem is, therefore, to find a criterion to designate what this extent of reaction should be. Hopefully this assignment could be made from the metallurgical and physical properties of uranium saturated with hydrogen and /?-uranium hydride. Metallurgical properties such as ultimate tensile strain, reduction in area are known for this material.11-13 However, since it is moderately ductile, there is no reliable method of using this information to calculate the compressive strain in two dimensions required for fracture. Even if this point of fracture could be calculated, there is still a possibility that the loss of integrity of the uranium might occur before spalling. This would be analogous to the case of uranium oxidation where it has been deduced14 that crystallographic rearrangements and not spalling determines how closely molecular oxygen may approach the reaction front. In spite of the foregoing reservations, some crude guidelines should be possible. A maximum allowed compressive strain should be obtainable from the corrected one-dimensional fracture stress and the modulus. Since, generally, metals behave more elastic under compression than tension, the use of idealized elastic equations should provide a reasonable approximation. It will, therefore, be assumed
-
=;
ay,2
Due to plasticity, eq 12 cannot be used directly to find the maximum stress to failure on tensile specimens. The conversion to true fracture stress should, however, eliminate the largest inaccuracy. The stress-strain data for uranium saturated with hydrogen have been recently obtained by Powell, et al.13 The modulus is approximately 138 (a 30) GPa and the true fracture stress is 0.917 (a PZ 0.030) GPa. These values are strain-rate dependent but are reliable to within 50%. The required stress level in both the Y1 and Yz directions on a surface to cause fracture is the same as the true fracture stress under one-dimensional tension or approximately 0.92 GPa. This will give a strain of about 0.66% if the modulus under compression is the same as that in tensioh. To translate this information into extent of reaction one may conceive of the following steps. First, an increment of the reaction occurs in a small volume and an expansion of the material takes place. The plane of the surface, however, is constrained such that a compressive stress is applied to make the material conform to the original surface area. If this strain is greater than the above 0.66%)then the material fractures and the hydrogen gas can advance past this volume increment. If the strain is less than the 0.66%) the hydrogen advances into and past this volume only by diffusion. In the reaction step one assumes a small volume expansion fraction, f, where
-
f=
Vafter reactiQn
-1
(13)
Vwre metal
-
Under hydrostatic expansion each linear dimension will increase by Yjf (since (1 + %f13 1 f wit$ f small). To conform to the original area, a strain of 1/3f must therefore be effected to both dimensions of the surface plane. Thus fracture occurs when %f equals or exceeds 0.0066. Since the density of /?-UH3is 11.1and uranium is 19.07, the extent of reaction, a,corresponding to the above fraction expansion can be calculated by
+
f 0.74a (14) which will give the extent of reaction where fracture will occur as being CY, = (3 x 0.0066)/0.74 0.027 (15) For pure uranium the modulus is a little higher, -170 GPa, according to a summary by Holden.12 The ultimate tensile
The Journal of Physical Chemistry, Voi. 79, No. 4, 1975
394
J. B. Condon
strength, however, is considerably lower. It is down to about 0.40-0.60 GPa for randomized material. From the foregoing discussion, it appears that the loss of metal integrity must occur at rather low values for ac.That is from the point of view of fracture alone, ac should be in the neighborhood of 2%. This parameter is so poorly defined, however, that it must be considered an adjustable. The model thus has one adjustable parameter. For calculation purposes, the upper and lower limit of this parameter is taken as 3.0 and 1.6%. It will be shown later that the choice of criterion does not affect the observed (resultant) activation energy, and the 1.6-3.0% value is within the scatter of the best available data. Performance of Calculations. The numerical calculations were made on a variety of IBM 360 computers. A fully implicit solution scheme was used to generate a tridiagonal system of simultaneous e q ~ a t i 0 n s .The l ~ system was solved and checked for convergence with an allowance for up to five iterations. The error was of the order Ax2 At and each of the step sizes had to be continuously adjusted to make certain of proper control. The errors were such that the linear rates determined would be low; however, continued down adjustments of Ax and At could be made to check the appropriateness of the step-size choice. A semiinfinite solid problem iterated to a converging linear rate would have been more efficient. The two-sided plate problem was chosen, however, in order to determine the behavior of this reaction problem at the beginning and end of the reaction. This sometimes presented machine storage problems when Ax was very small and the thickness large; therefore, some of the linear rates were calculated at thicknesses not reported in the literature or used here.
+
Experimental Section
Justification. Hydriding experiments were undertaken to check the literature values available and to obtain values a t temperatures not reported for a higher statistical check. With respect to the first reason, there are some minor inconsistencies in the literature. The inconsistencies are outside the limits of error imposed by the 2.2 f 0.8% extent parameter described above. Samples and Method. The samples used were nominally &mil thick (-0.2 mm), rolled foils which, after electropolishing and pickling, were reduced to a 0.15-0.17 mm thickness. The surface area, counting both sides, was 24.4 cm2. Chemical analysis revealed that carbon at 98 wppm, iron at 85 wppm, silicon at 60 wppm, aluminum at 38 wppm, and nitrogen at 10 wppm were the principal contaminants. The oxygen analyzed at 660 wppm which is undoubtedly due to surface oxide since metallurgical examination did not justify such a high value. Estimates from this latter method produced a value of
