Langmuir 1996, 12, 6361-6367
6361
Reaction of Lithium Hydride with Water J. H. Leckey* and L. E. Nulf Chemistry and Chemical Engineering Department, Oak Ridge Y-12 Plant, Oak Ridge, Tennessee 37831
J. R. Kirkpatrick Computational Physics and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 Received October 6, 1995. In Final Form: August 12, 1996X Lithium hydride (LiH) has been shown to react with water at a rate defined by a nonequilibrium thermodynamic-based model. The rate at any time is proportional to the surface area of the LiH multiplied by the difference in chemical activity of the reaction product, lithium hydroxide (LiOH), between the solid phase and the aqueous phase. By measuring the conductivity as a function of time for aqueous solutions resulting from the reaction of LiH with water and comparing the measurements to results from a mathematical model of the dissolution process, the geometric rate constant for the reaction was determined to be 0.0025 cm s-1, which causes the generation of 6.0 cm3 (STP) of hydrogen gas cm-2 s-1 at 35 °C. As the reaction proceeds, the resulting solution of LiOH in water becomes more concentrated, and the reaction slows. The rate has been shown to depend only slightly on temperature, and the model predicts only a small LiOH activity coefficient dependence. The derivation of the reaction rate model, experimental results, and solution to the rate equation for some simple geometries are presented in this paper.
Introduction Large quantities of lithium hydride (LiH) and lithium deuteride (LiD) are being liberated as part of the dismantlement of the national nuclear weapons stockpile. These materials are enriched in the lithium-6 isotope (6Li) but with minor variability retain the chemical properties of unenriched LiH and LiD. Since no plans currently exist for alternate uses of these materials, they will likely be stored for some time. One concern in evaluating potential storage hazard is the inadvertent reaction of these materials with water to generate hydrogen gas and heat, which could lead to a fire or an explosion and possibly the ignition of the LiH itself. In order to quantify this concern, it is necessary to determine the rate and understand the nature of this reaction. A second reason for interest in this reaction is related to the fact that processing of LiH and LiD into other compounds almost invariably involves an initial reaction/dissolution in water. In the event that it is decided to chemically convert these materials into other forms, knowledge of the reaction process is desirable in order to optimize the quantities of material and length of time required for large-scale chemical conversion operations. This paper describes both a set of experiments used to determine the reaction rate of LiH with water and the development of a nonequilibrium thermodynamicbased model for interpreting the results. The overall reaction of LiH with excess water proceeds according to the chemical equation
LiH(s) + H2O(l) f LiOH(aq) + H2(g) + heat (1) where 37 kcal of heat are generated per mole of LiH reacted and an additional 5 kcal of heat are generated per mole of lithium hydroxide (LiOH) dissolved in water as an ionic solution. * Author to whom correspondence is addressed. E-mail:
[email protected]. † Managed for the U.S. Department of Energy by Lockheed Martin Energy Systems, Inc., under Contract No. DE-AC0584OR21400. X Abstract published in Advance ACS Abstracts, December 1, 1996.
S0743-7463(95)00843-2 CCC: $12.00
In general, the reaction proceeds in two steps. The first step
LiH(s) + H2O(l) f LiOH(s) + H2(g) (fast)
(2)
is the formation of hydrogen gas and a layer of solid LiOH on the LiH surface. The second step,
LiOH(s) + H2O f Li+(aq) + OH-(aq) (slow) (3) is the dissolution in water of LiOH, reducing the thickness of the LiOH solid layer and promoting further reaction of LiH with water. To the extent that water is present in large excess, this process (reactions 2 and 3) will continue until the LiH is completely converted to an aqueous solution of LiOH. If the solubility limit of LiOH in water is reached before the LiH is completely reacted, the reaction will slow considerably as the thickness of the solid LiOH on the LiH surface increases, passivating the reaction. If this layer does not spall, the reaction will nearly stop. This treatment of the reaction process assumes that the first step, formation of hydrogen gas and a surface layer of LiOH, is fast compared with dissolution of the LiOH in water. This allows a reaction model to be developed based on the difference in chemical potential between the layer of LiOH on the LiH substrate and the bulk ionic solution of LiOH in water. The model is used with experimental data both to determine the reaction rate constant and to assess the degree of applicability of the model. Since the rate-determining step does not involve the hydride ion, the reaction rate is independent of whether the starting material is LiH or LiD. Therefore, LiH has been used exclusively in the experiments and model development. Model Development Since the second step is significantly slower than the first, it is the rate-controlling step in the sequence. For processes of this type, the rate of the reaction can be expressed by
dc/dt ) A(t) j(t)/V
(4)
where c is the concentration of LiOH in solution in moles © 1996 American Chemical Society
6362 Langmuir, Vol. 12, No. 26, 1996
Leckey et al.
of LiOH per mass of water, t is time, A(t) is the surface area of the LiH as a function of time, j is the flux of LiOH into solution at any time per unit surface area of LiH, and V is the volume of solution (assumed to be constant during the reaction). In principle, the rate of reaction could be expressed for any of the species involved in the reaction; however, for comparison with experimental data, it is convenient to express the reaction rate in terms of concentration of LiOH in solution. The following derivation of the flux equation is similar to the treatment of chemical kinetics by Katchalsky and Curran1 but does not neglect the activity coefficients of LiOH. According to the theory of irreversible thermodynamics, for processes sufficiently close to equilibrium, the flux of LiOH into solution is related to its chemical potential by
j(t) ) -L∇µ
(5)
where L is the phenomenological or Onsager coefficient, µ is the chemical potential of LiOH, and ∇ is a gradient operator. In the absence of pressure and temperature gradients, as well as external fields, the gradient of the chemical potential can be expressed as
dµ/dx ) 2RT∇[ln(a()]
where a(,s is the activity of LiOH in a saturated solution. Combining eqs 4, 9, and 10 gives the rate equation
dc/dt ) (a(,sD/∆x)(A(t)/V)(1 - a((t)/a(,s)
(11)
The cluster of constants, a(,sD/∆x, represents the rate in quantity per unit area per time. It is more convenient to define a rate constant, k, which represents the rate at which LiH is reacted in terms of depth per time. This can be done by the following definition:
k ) a(,sDMw/F∆x
(12)
where Mw is the molecular weight of LiH and F is the mass density of solid LiH. The rate equation now becomes
dc/dt ) RkA(t) (1 - a((t)/a(,s)
(13)
where R is defined by
R ) F/MwV It is conventional to express the activity as the product of an activity coefficient and concentration. Thus,
(8)
This is not a rigorous definition of a diffusion coefficient in the irreversible thermodynamic sense.2 It does, however, have the correct units when the activity is expressed in terms of concentration (molarity). As will be shown later, eq 8 is completely satisfactory, because the diffusion coefficient will be included as part of the rate constant for the reaction. To the extent that the phenomenological constant, L, and the activity, a, depend on position in the interfacial region in the same manner, D may be considered a constant in this region. For geometries of interest, the extent of this interfacial region is small compared to the dimensions of the solid LiH (except for a very brief period near the end of the reaction for some conditions). Choosing D to be a constant and the interfacial region to be small allows eq 7 to be integrated from the LiOH layer to the aqueous solution bulk, in a manner similar to that used by Horne and Leckey3 for membrane transport. The result is
j(t) ) (D/∆x)[a(′ - a((t)]
(10)
(7)
where D, the diffusion coefficient, is defined by
D ) 2LRT/a(
a(,s ) a(’
(6)
where R is the gas constant, T is temperature, and a( is the mean molar chemical activity of LiOH. Combining eqs 5 and 6 gives
j(t) ) -D∇a(
ally become saturated in LiOH. At this point, the activity of LiOH attached to the LiH substrate and the activity of LiOH in solution will be equal. Because the activity of the solid LiOH does not change throughout the reaction (except for the rapid establishment of the steady-state layer),
(9)
where ∆x is the thickness of the interfacial region and a(’ is the activity of the solid LiOH. After the reaction begins, the solid LiOH region rapidly approaches a steady state with a constant thickness and composition. If enough LiH is present initially, the aqueous solution will eventu(1) Katchalsky, A.; Curran, P. F. Nonequilibrium Thermodynamics in Biophysics; Harvard University Press: Cambridge, MA, 1967; Chapter 8. (2) Hasse, R. Thermodynamics of Irreversible Processes; AddisonWesley: Reading, MA, 1963; Chapter 4. (3) Horne, F. H.; Leckey, J. H. The Physics of Superionic Conductors and Electrode Materials; Perram, J. M., Ed.; Plenum Press: New York, 1983, pp 273-277.
a((t) ) γ((c) c(t)
(14)
a(,s ) γ(,scs
(15)
and
where γ((c) is the activity coefficient of LiOH in solution at any time and γ(,s is the activity coefficient of LiOH in a saturated aqueous solution. The activity coefficient of LiOH in solution is represented as γ((c) to indicate that it is concentration dependent. Substituting eqs 14 and 15 into eq 13 gives
dc/dt ) Rk(A(t))[1 - (γ((c)/γ(,s)(c/cs)]
(16)
Thus, to the extent that the activity coefficients of LiOH are known, the reaction rate can be predicted for a given shape or range of shapes of the LiH specimen and solution volumes once k has been determined. Dimensionlessization Before solving eq 16, it is sensible to define a set of dimensionless variables. Equation 16 is transformed into the dimensionless equation
dc′/dτ ) A′(τ) [1 - γ(′(c′) c′/cs′]
(17)
by the following definitions:
c′ ) c/c∞
(18)
cs′ ) cs/c∞
(19)
A′(τ) ) A(t)/A0
(20)
Reaction of Lithium Hydride with Water
Langmuir, Vol. 12, No. 26, 1996 6363
τ ) RkA0t/c∞ ) kA0t/V0
(21)
γ(′(c′) ) γ((c)/γ(,s
(22)
and
where c∞ is the ultimate concentration of LiOH in solution, A0 is the initial surface area of LiH, and V0 is the initial volume of LiH. Presented in this manner, the differential rate equation (eq 17) is more compact and easier to manipulate. Experimental Determination of the Geometric Rate Constant k The geometric rate constant, k, is most easily determined using regularly shaped specimens of LiH, which produce a dilute solution of LiOH when the LiH has been consumed by the reaction (c′/cs′ much less than 1). At room temperature, the solubility limit of LiOH in water is 5.4 molar (M). For reactions that produce LiOH solutions less than 0.2 M, the effect of the activity coefficient terms in eq 17 is completely negligible. This provides a convenient method of determining k. The size and shape of LiH used for this determination was a cylinder with a radius of 0.80 cm and height of 1.60 cm. By choosing a cylinder with the height equal to twice the radius, the mathematical solution of the model simplifies considerably. The specimen was made by coldpressing high-purity LiH powder to a solid blank, outgassing the blank under vacuum, warm-isostatically pressing the blank, and then machining the specimen to its final dimensions. The final density of the specimen made in this manner is 99.5% of the theoretical crystalline density, with the major contaminant being about 4000 ppm oxygen. For the experiment, the specimen was completely submerged in 1800 mL of water contained in a 2-L beaker. A perforated metal basket was used to keep the specimen submerged but allow for rapid mixing of the solution. A cooling bath circulated around the outside of the beaker to minimize the temperature increase of the solution caused by the heat of the reaction. A magnetic stirring bar and the constant evolution of hydrogen served to agitate and homogenize the solution during an experimental run. The progress of the reaction was monitored by using a four-electrode conductivity probe manufactured by Quality Control Instruments, Inc., Oak Ridge, TN, that had been calibrated with a series of LiOH solutions over a range of concentrations. The solid circles in Figure 1 show the normalized LiOH concentration in solution as a function of time as determined from the conductivity measurements. The final concentration of the solution reached 0.18 M. The temperature of the solution increased from an initial value of 34 °C to a maximum value of 36 °C due to the heat of reaction and dissolution and then returned to 34 °C. For solid cylindrical specimens in a large excess of water, a closed-form solution can be derived for eq 16 or eq 17. The first step in the process is to determine A(t). Equation 9 implies that the erosion rate (that is, the rate at which the solid LiH surface recedes) of a flat surface is proportional to the difference in the activity of the solid LiOH and that of the solution. Through a combination of eqs 9 and 12, one can derive the following equation for the erosion rate at a flat surface:
dx/dt ) k[1 - (γ((c)/γ(,s)(c/cs)]
(23)
where x is the distance measured normal to the surface. For an excess of water, the c/cs term approaches zero so
Figure 1. Reaction of a 2.5-cm-radius by 5.0-cm-high cylinder of LiH with 1800 mL of water. The final concentration of the solution is 0.18 M. Solid circles are the normalized LiOH concentrations in solution, and solid squares are the solution temperature. The best least-squares fit of the data to eq 24 is given by the solid line with a value of k of 0.0025 cm/s and a standard deviation of 1.7 × 10-5 cm/s.
that the erosion rate equals k. If one further makes the approximation that the erosion rate on the outside of the cylinder equals that on a flat surface, then one can integrate to find expressions for the radius (R) and length (Z) of the cylinder. Those expressions are as follows:
R ) R0 - kt
(24)
Z ) Z0 - 2kt
(25)
and
where R0 is the initial radius of the cylinder and Z0 is the initial length. The factor of 2 in eq 25 comes from the fact that the cylinder is being eroded at both ends. Equations 24 and 25 can be combined into an equation for A(t). The final solution for concentration is
c(t) ) c∞[1 - (1 - kt/R0)2(1 - 2kt/Z0)]
(26)
Equation 26 can be changed to a dimensionless form, but the result is not particularly compact. The result is
c′(τ) ) 1 - {1 - Z0τ/[2(R0 + Z0)]}2{1 - 2R0τ/[2(R0 + Z0)]} (27) However, for a cylinder whose length is exactly twice its radius, eqs 26 and 27 have convenient, compact forms that are as follows:
c(t) ) c∞[1 - (kt/R0)3]
(28)
c′(τ) ) 1 - (1 - τ/3)3
(29)
and
The specimen used in the experiment was twice as long as its radius so that these last two solutions are applicable to matching the theory to the experimental results. The best least-squares fit of the data shown in Figure 1 to eq 28 gives a value of k of 0.0025 cm s-1 and a standard deviation of 1.7 × 10-5 cm s-1 at the mean reaction temperature of 35 °C. Using the stoichiometry of the chemical reaction equation (eq 2), it can be shown that this reaction rate corresponds to the production of 6.0 cm3
6364 Langmuir, Vol. 12, No. 26, 1996
Leckey et al.
(STP) of hydrogen gas cm-2 s-1. Similar experiments on a variety of sizes of LiH cylinders produced identical results for the rate constant, k. This type of experiment was also repeated at a constant 60 °C with a value of k determined to be slightly smaller, resulting in an activation energy of -3.0 kcal mol-1. Using the Arrhenius equation with the value of k determined at 35 °C and the activation energy allows the temperature dependence of the rate constant to be predicted at any temperature over the range studied. Having determined k, one can use eq 17 to predict the reaction kinetics of LiH with water for a variety of LiH geometries, solution volumes, and temperatures. Solution of the Rate Equation for Flat Disks Activity Coefficient Equal to One In general, it is not possible to obtain an analytical solution to eq 17 because the concentration dependence of the activity coefficient of LiOH in water makes the equation nonlinear. Because the activity coefficient of aqueous LiOH solutions is less than 0.8 for concentrations greater than 0.05 M, a precise prediction as to the rate of the reaction as the concentration of the aqueous solution approaches the LiOH solubility limit is not possible. It is useful, however, to examine the solution for the ideal case, to compare it with the numerical solution necessary for the more realistic case, and to compare it with experimental results. The solution presented here is for flat disks (radius significantly greater than the height) because that is a convenient geometry for experiments. The solution to eq 17 for this case with the activity coefficient of LiOH in water equal to 1 is
c′(τ) ) cs′[1 - exp(-τ/cs′)]
(30)
In terms of dimensioned variables, this becomes
c(t) ) cs[1 - exp(-2kc∞t/csZ0)]
(31)
where Z0 is the initial height of the disk and c(t) is less than the smaller of cs and c∞. Equation 31 can be expressed alternatively as
c(t) ) cs[1 - exp(-2πR02Rkt/cs)]
(32)
Note that the functional form of these solutions is intuitively correct. When c(t) is much less than cs, eq 32 is linear in time, as expected for a flat disk that does not change surface area as it reacts. As c(t) approaches cs (if the disk is large enough), the reaction slows to an exponential decay. Activity Coefficients of LiOH In order to solve eq 17 more precisely, it is necessary to know the activity coefficients of LiOH in water as a function of concentration and preferably temperature. The activity coefficients of LiOH in water at 25 °C as determined by Harned and Swindells4 from electromotive force measurements are listed in Table 1, column 4. Although these activity coefficients are based on the molality scale, the molality and molarity are essentially identical for aqueous LiOH solutions over the range of interest (as can be seen from columns 1 and 2 of the table). The activity coefficients in Table 1 can be used to estimate the activity coefficients at other temperatures (4) Harned, H. S.; Swindells, F. W. J. Am. Chem. Soc. 1926, 126, 48. (5) International Critical Tables; Washburn, E. W., Ed.; McGraw Hill Book Co.: New York, 1930.
Table 1. Activity Coefficients of LiOH in Water at 25 °C concn, ma
concn, M
density,b g/cm3
activity coefficient
0.050 0.100 0.200 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000
0.050 0.100 0.200 0.501 1.002 1.501 2.002 2.505 3.012 3.515 4.022
1.0012 1.0024 1.0048 1.0110 1.0234 1.0355 1.0474 1.0592 1.0700 1.0805 1.0960
0.803 0.760 0.702 0.616 0.554 0.528 0.513 0.501 0.494 0.487 0.481
a An extra significant figure has been added to the original work (ref 4) for comparison with the calculated molarity values. b The density values have been interpolated from the value published in ref 5.
using the equation
[∂ ln(γ()/∂T]p,c ) ∆H2/(νRT2)
(33)
where the partial derivative is at constant pressure and composition, T is temperature, ∆H2 is the heat of solution of LiOH in water, ν, the number of ions per molecule, is 2 for LiOH, and R is the gas constant. The temperature dependence of ∆H2 can be approximated by6
∆H2(T) ) ∆H2(298 K) + Scpc1/2(T - 298)
(34)
where ∆H2(298 K) is about -4.9 kcal mol-1, the parameter Scp, as given in ref 7, is 0.0188 kcal kg1/2 K-1 mol-3/2, and c is the molality of the solution. Thus, eq 33 can be integrated using eq 34 to predict the activity coefficients as a function of temperature for any concentration. For concentrations greater than 4 M, the activity coefficients are projected linearly. General Solution of the Rate Equation for Flat Disks Because of the activity coefficient dependence on concentration, it is, in general, necessary to determine the solution to the rate equation by numerical means. The procedure involves determining the exposed surface area as a function of LiOH concentration in solution and then numerically integrating eq 17 as an initial value problem, with the activity coefficient data given in the last section. The time step size used was 1 s. Figure 2 shows a plot (curve labeled “Real”) of the solution for the reaction model at 25 °C of 2.0 L of water with a disk having a radius of 7.0 cm and a height of 0.69 cm, using the activity coefficients in Table 1. The ultimate concentration of the solution, c∞, was 5.3 M. Also shown on the graph is the result with the activity coefficients set to 1 (labeled “Ideal”). Plotted against the right-hand axis is the difference between the two results. Interestingly enough, the numerical solution results deviate by no more than 2%, even though the activity coefficients range from 1.0 to 0.4. When the reaction starts, the driving force for the reaction is overwhelmingly to dissolve LiOH in solution, and the value of the activity of the solid LiOH matters very little. As the reaction nears completion, the ratio of the activity coefficient of the LiOH in solution to that of the solid (6) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions; Reinhold Publishing Co.: New York, 1958; Chapters 8 and 13. (7) Gucker, F. T., Jr.; Schminke, K. H. J. Am. Chem. Soc. 1933, 1013, 55.
Reaction of Lithium Hydride with Water
Figure 2. Plot of the solution to the rate equation for the reaction of 2.0 L of water with a LiH disk with a radius of 7.0 cm and a height of 0.69 cm. This reaction produces an aqueous solution with an ultimate concentration, c∞, of 5.3 M. At the completion of the reaction c∞ and cs are equal. The curve labeled “Real” was generated using the activity coefficients listed in Table 1, whereas the curve labeled “Ideal” was generated with the activity coefficients set to 1. The difference between the two solutions is plotted on the right-hand axis.
approached 1, and the solution to the rate equation is nearly the same as if the activity coefficients had been neglected. Solutions for solid cylinders under the assumption of excess water have been shown. Also, solutions have been shown for a disk under the assumption that activity is exactly 1. Solutions can be derived under the assumption of excess water for hollow cylinders, spheres, hemispheres, and hemishells. The general method is that one integrates eq 16 together with additional equations in the form of eq 26 for the erosion of the surfaces. With numerical integration, one can extend the general method to account for the effects of finite amounts of water as well as activities that are different from 1. Algorithms have been written and tested that solve the general method for all of the shapes mentioned in this paragraph. The authors believe that the general method can be extended to solve for the erosion of objects of almost any shape. However, it seems likely that irregular shapes will require an algorithm for which the surface is divided into a grid of surface elements and the motion of each element calculated for each time step. While the authors have done some preliminary work along this line, such a general algorithm has not yet been needed and thus has not yet been developed.
Langmuir, Vol. 12, No. 26, 1996 6365
Figure 3. Sketch of the rounded corner geometry.
Using the rate constant determined in this work, along with model described in this paper will allow the hazards associated with the inadvertent reaction of LiH and LiD with water to be delineated as accurately as possible. Additionally, the information can be used to help optimize large-scale LiH chemical conversion operations. Acknowledgment. This work was funded by the Department of Energy under Contract DE-AC0584OR21400 with Lockhead Martin Energy Systems, Inc. The authors thank W. K. Duerksen and R. H. Reiner for many helpful suggestions in preparing this paper. Appendix A: The Rounded Corner and Its Effect on LiH Dissolution The derivations of analytical solutions shown in the body of this paper contain the tacit assumption that the edges and corners of the LiH pieces are perfectly sharp and that this perfect sharpness is maintained throughout the dissolution process. Of course, there is no such thing as an object with a zero radius of curvature along an edge or at a corner. Intuition suggests that the erosion process at the edges and corners should lead to an increase in the radius of curvature with time. The authors have derived a model for the time-dependent radius of curvature, added this model to a numerical calculation of the erosion of solid bodies, and evaluated the effect of the time-dependent radius of curvature on the erosion rates. Derivation of the Rounded Corner Model
Conclusions LiH reacts with water in a predictable manner to generate hydrogen gas and an aqueous LiOH solution. The thermodynamic-based reaction model performs well in predicting the reaction rate for reactants that produce dilute and moderately concentrated solutions, but it remains untested for the situations that produce solutions near the saturation limit of LiOH in water. The small dependence of the reaction on the LiOH activity coefficients allows the rate equation to be solved analytically for a number of shapes and sizes of LiH specimens, with a high degree of accuracy. Where this analytical solution is not possible, a numerical solution can be used. Because the rate of the reaction is only weakly temperature dependent (activation energy of -3 kcal mol-1), the rate as a function of temperature can also be determined with reasonably high confidence. The reason for the negative activation energy for the reaction is not well understood. It may relate to an interaction between the hydrogen gas formed in the first step of the reaction and the interface region between the solid and the solution.
Consider Figure 3. The figure represents a twodimensional edge with a nonzero radius of curvature. The objective is to estimate the change in the radius of curvature as the surface erodes. At some time t, the radius of curvature is r. The surface of the part near the corner is composed of the quarter circle 2-3-4 of radius r with a center at point 9. It is tangent to two straight line segments 1-2 and 4-5. At some later time t + ∆t, the surface has moved and is composed of the quarter circle 6-7-8 which is tangent to straight lines at points 6 and 8. The new quarter circle has a radius of curvature r2 and a center at point 10. From time t to time t + ∆t, the 45° point of the arc has moved a distance dr as it goes from point 3 to point 7. During this period, the straight lines tangent to the arcs have moved a distance dx. In particular, points 1 and 5 have moved a distance dx to become points 6 and 8. The center of curvature has moved a distance z in both the horizontal and vertical directions. The points where the arcs are tangent (points 6 and 8) are a distance z from the previous tangent points (points 2 and 4). Given r, dr, and dx, one can find r2 and z.
6366 Langmuir, Vol. 12, No. 26, 1996
Leckey et al.
The first thing needed is an expression for dr. Of course, dx is simply
dx ) k∆t
(A.2)
where m ˘ is the mass rate of change, A is the surface area, C0 is the surface concentration, D is the diffusivity, and ∆ is the thickness of the diffusion layer. For a cylinder,
m ˘ /L (cylinder) ) 2DC0/ln[(r + ∆)/r]
x3 - x7 ) y3 - y7 ) dx (1 + φ)
(A.1)
One can develop a value for the erosion rate on the curved surface using diffusion from a cylinder compared to a flat plate. The flat plate solution is
m ˘ /A (flat) ) C0D/∆
It is necessary to force x3 - x7 g dx. If one requires that
(A.15)
the following is the result for dr:
dr ) dx(1 + φ)/x2
(A.16)
The next step is to do some geometry to solve for r2 and z. If one examines points 3, 7, and 10 in Figure 3, one gets the following:
x7 - x10 ()y7 - y10) ) x3 - x9 + [x9 - x10] [x3 - x10] ) r/x2 + [z] - [dx (1 + φ)] (A.17)
(A.3) However, from points 7 and 10, one gets
Converting from diffusion per unit length to diffusion per unit area,
r2 ) [x7 - x10]x2
(A.18)
Using the first two terms of the Taylor series for ln(1 + x), one gets
r2 ) zx2 + r - [dx (1 + φ)]x2
(A.19)
m ˘ /A (cylinder) ≈ DC0/[r∆(1 - ∆/2r)/r]
If one examines points 5, 8, and 10 in Figure 3, one gets the following:
m ˘ /A (cylinder) ) DC0/[r ln[(r + ∆)/r]]
(A.4)
(A.5)
Continuing the approximations,
m ˘ /A (cylinder) ≈ DC0(1 + ∆/2r)/∆
(A.6)
Therefore,
x8 - x10 ()y6 - y10) ) x5 - x10 - [x5 - x8] ) z + r - [dx] (A.20) However, from points 8 and 10, one gets
Comparing the flat plate and cylindrical fluxes, one finds that
m ˘ /A (cylinder) ≈ m ˘ /A (flat) (1 + ∆/2r)
(A.7)
(A.8)
If one makes that assumption, then the relationship between dx/dr and dr/dt is as follows:
dr/dt ≈ (dx/dt) (1 + ∆/2r)
(A.9)
(A.10)
It seems slightly more convenient to write
dr/dt ≈ (dx/dt)(1 + φ)
(A.11)
where, of course, φ is defined by
φ ) (∆/D)(D/2r) ) ∆/2r
dr ∼ dx(1 + φ)
(A.13)
z ) dx [1 + x2φ/(x2 - 1)]
(A.23)
r2 ) r + dx [x2φ/(x2 - 1)]
(A.24)
z ) dx
(A.25)
r2 ) r
(A.26)
and
What these mean is that, in the limit of infinite radius of curvature (i.e., a flat surface), the results for z and r2 reach the flat plate values. Effect of the Rounded Corner Model on Dissolution of LiH Cylinders
which leads to
x3 - x7 ) y3 - y7 ∼ dx (1 + φ)/x2
(A.22)
At this point, it seems appropriate to examine the consequences of the solutions a bit more. As one may see from eq A.24, as long as both dx and φ >0, then r2 > r. In other words, r2 does not reach an asymptote but rather grows without bound. It seems worth demonstrating that the solution satisfies certain limits. If φ f 0 (which can come about from, among other things, r f ∞), then
(A.12)
Note that φ is not a constant but is rather a function of r, which is, in turn, a function of t. Next, one needs to determine dr, which is the distance point 3 travels to become point 7. If one blindly applies eq A.11, one gets
r2 ) z + r - dx
Equations A.19 and A.22 constitute a pair of linear simultaneous equations for r2 and z. Skipping several lines of algebra, the answers are as follows:
or
dr/dt ≈ (dx/dt) [1 + (∆/D)(D/2r)]
(A.21)
so that
One assumes that
surface velocity ∝ m ˘ /A
r2 ) x8 - x10
(A.14)
The catch with that form is that, unless 1 + φ > x2, one gets x3 - x7 < dx, which is not consistent with an increase in the radius of curvature. In fact, one can show that, for dx large enough, x7 > x8, which is a nonsense condition.
Calculations have been done to estimate the effect of having a time-dependent, rounded corner in the LiH dissolution. For all the cases, the cylinder radius was 8.0 cm. Values of the cylinder length to radius ratio varied from 0.1 to 40. The volume of the water was adjusted so that c∞/cs ) 0.001. The initial corner radius for cases with
Reaction of Lithium Hydride with Water
Langmuir, Vol. 12, No. 26, 1996 6367
a nonzero corner was 0.0008 cm. Calculations compared the curves of LiOH concentration in the solute vs time. For convenience in comparing different cylinder sizes, the LiOH concentration was plotted as c/c∞ and the time as t/t∞ where t∞ is defined as the time for which the cylinder would be exactly dissolved for corners with zero radius and for an infinite amount of water. A form for this time scale is as follows:
t∞ ) k-1 max
{
cylinder radius 1/2 cylinder length
}
(A.27)
where k is the rate constant (with units depth per unit time) from the main body of this paper. The rounded corner model can be incorporated into an algorithm that estimates the erosion of a solid piece of LiH. The authors have alluded to algorithms for simple shapes such as solid cylinders in the body of this paper. The addition of the rounded corner model is straightforward. One approximates a regular object (such as a solid cylinder) as a set of regular surfaces with rounded corners located where the line segments of the regular surfaces meet. The equation for erosion of flat surfaces is used to calculate the erosion of the regular surfaces. This gives the motion of the line segments tangent to the quarter circles that represent the rounded corners. The changes in the line segments can be used to calculate the changes in the rounded corners as is shown earlier in this appendix. If one has the dimensions of the regular object and the radii of curvature for the rounded corners, one can calculate the volume of the LiH piece. The change in volume of the LiH piece during a time step is used to calculate the change in the LiOH concentration in solution. Figure 4 shows the comparison between cases with a zero radius of curvature corner and one with a timedependent, rounded corner. The case is that for which the largest difference in curves of normalized LiOH concentration was seen. The cylinder length was 1/2 of the radius. The maximum difference between the curves was ∼0.019. For cylinders with either larger or smaller length to radius ratios, the maximum difference was less than that. Calculations were also run with the initial radius of the corner raised by a factor of 10. The difference between results from those and from the “normal” radius of curvature could not be seen on the plots. Cases were
Figure 4. LiH dissolution results comparison for rounded and sharp corner models for cylinders of aspect ratio z/r ) 0.5 in a large tank (c∞/csolid ) 0.001).
also run with the water volume reduced so that c∞/cs ) 0.326. There was a significant difference between the LiOH concentration curves for this water volume and that with the much larger water volume. A least-squares fitting operation was done for the results from the cases with the largest differences in curves of LiOH concentration. The objective was to see what would be the difference in values of k inferred from the two different LiOH curves given by the sharp corner and the time-dependent, rounded corner. This is an estimate of the error caused by assuming a sharp corner. The k inferred for the rounded corner was ∼1.8% higher than that found for the sharp corner. However, the maximum differences between the curves for the zero radius of curvature corners and the rounded corners were almost the same as those for the calculations with the larger water volume. The conclusion that can be drawn from the results reported in this appendix is that the maximum effect of having a time-dependent, rounded corner is less than 2% of the LiOH concentration. There are other approximations in the development of the models described in the rest of this paper that are at least that large. Therefore, the effect of the time-dependent, rounded corner may be neglected. LA950843H
