GASEOUS DIFFUSION AND PORE STRUCTURE IN NUCLEAR GRAPHITES

T.J. MAYS, B. McENANEY, B.T. KELLY School of Materials Science, Bath University, XA9642926 Bath, United Kingdom

Abstract

With the incentive of providing more information for oxidation and lifecycle studies of moderators in thermal reactors, a new method has been developed to estimate pore structure in nuclear graphites. This method involves the measurement of room-temperature binary gas diffusion in graphite samples using an apparatus that was specially-built to operate at pressures below 100 Torr In these conditions diffusion in graphite pores is in the transition regime between Knudsen and bulk or molecular diffusion. Transition diffusion measurements are analysed for nuclear-grade graphites to yield pore size distributions. Results from the new method are compared with pore structure data obtained using porosimetry. pycnometry and established gas flow techniques.

1. INTRODUCTION

Synthetic graphites used as moderators in thermal nuclear reactors contain pores, mostly in the macropore size range (larger than 50 nm [1]), which arise initially during their manufacture. Subsequently these pores are developed by radiolytic oxidation in the CO2 coolant during service. This leads to a reduction in graphite density, and hence to degradations in mechanical and other properties which have implications for moderator lifecycles. It is therefore important to determine pore structure in nuclear graphites in order to understand and predict radiolytic oxidation and property changes in moderators [2]. The nuclear industry in the UK recognised the need for this approach, and consequently used and developed methods such as porosimetry and permeability for macropore structure determination in moderator graphites. However, certain problems with these methods are well-known, such as the limited information available from permeability data [3]. This paper describes a new method to estimate pore structure that might supplement established methods. The new method is based on room-temperature measurements of gas diffusion in graphite samples that are made at pressures < 100 Torr. In these conditions flow in macropores is in the transition regime between Knudsen and bulk or molecular diffusion [4]. This compares with established fluid flow methods for pore structure determination which work at higher pressures where transport is mainly in the bulk regime. The structure of this paper is as follows. First, a simple theory of gaseous diffusion in porous solids is outlined. From this theory, a generalised, transition diffusion equation is obtained which lays the ground for experimental measurements. The second part of the paper deals briefly with the apparatus that was built to measure transition diffusion in graphites. Third, results from the new apparatus - essentially pore size distributions - are reported and discussed, and compared with results from established pore structure techniques, such as porosimetry

2. THEORY OF GASEOUS DIFFUSION IN POROUS SOLIDS

A gas flow conductivity, C, may be defined as the molar flux through a solid sample per unit concentration gradient. For component i in a gas mixture, C may be written as

329 C =JlLL2_ (!) 1 Ac, / L

where J! is the molar flow rate of i through the sampie, A is the cross-section area of the sample perpendicular to the macroscopic direction of flow and Ac; is the concentration difference of i across the length, LT of the sample parallel to flow The gas flow conductivity is a measure of the ease with which a component of a gas mixture flows through a solid, and is analogous to the thermal conductivity for the flow of heat. For the isothermal, isobaric, steady-state diffusion of component i in an ideal mixture of ideal gases, Eqn (1) becomes

r =D =—* (T\

where D! is the effective diffusivity of the solid for component i. R is the gas constant, T is absolute temperature, p is pressure and Ayj is the mole fraction difference of i across L. The effective diffusivity depends on the flowing gas, the pore and surface structure of the solid and on environmental conditions (pressure, temperature and external gas composition). Experimentally, Dj can be calculated from Eqn (2) using experimental measurements of Ji, T, p and Ay; and sample dimensions. A cornerstone of this paper is the relationship between effective diffusivity and pore structure, using which experimental measurements of the former can yield estimates of the latter. This relationship involves a model of gaseous diffusion in porous solids. The model used here for nuclear graphites, which was originally proposed by Hewitt [5] and has more recently been described critically in [3], is as follows. The first part of the model refers to the pore structure. The transport porosity in graphites, i. e., that part of the accessible or open porosity through which fluid flows in the steady state, is assumed to comprise Nv non-intersecting, straight, cylindrical capillaries per unit volume. Isotropy is also assumed, so that there are Nv / 3 capillaries per face of a unit cube of material. The length, 1, of each capillary is assumed to be constant but may exceed the length, L, of the sample, by a factor q = 1 / L > 1, called the tortuosity. Finally, capillary radii, r, are assume to be distributed with a probability density function, f(r). In this model, the fractional transport pore volume, Vj, is given by

2 VT =7tNv < r >q (3)

where < > denotes a mean value. It should be recognised that this model is a necessary simplification of the complicated pore structure in graphites. The second part of the model of gaseous diffusion in porous solids refers to flow in capillaries. From simple kinetic theory arguments, see [4], the isothermal, steady-state molar flow rate, ji, of a component i in an inert, ideal, binary mixture of ideal gases diffusing through a single, straight capillary of radius r and length 1» r is given by

= 7trpDm rinyl2mKil Jl amRTl U-ainyli+Dm/DKJ

where D^ is the bulk or molecular binary diffusion coefficient for the mixture (i = 1, n = 2 or i = 2, n =1). The sub-subscripts 1 and 2 for mole fraction, y, refer to opposite ends of the capillary. The flow rate ratio a is given by

330 (5) where M is molecular weight. Assuming completely diffuse (random) reflection of molecules after collision with pore walls, the Knudsen diffusion coefficient DK is given by

DK, = | r < v, > (6) where the mean molecular speed, < v; >, of gas molecules is given by

< v, > = ^RT/TIM,) (7)

It is useful at this stage to note that bulk or molecular diffusion refers to flow that involves only intermolecular collisions in the gas phase (i. e., for high pressures and large pores) and Knudsen diffusion refers to flow that only involves molecule-pore wall collisions (low pressures, small pores). Transition diffusion refers to the case where both gas-gas and gas-solid collisions are important in flow. It should also be noted that capillary diffusion, Eq (4), depends in a complex way on pore radius, mainly due to the radius-dependence of the Knudsen diffusion coefficient in the logarithmic term. The total molar flow rate, Ji, through an area A of material is simply the sum of flows through each of the (Nv A / 3) capillaries in this area. Accounting for the distribution of pore radii, f(r), this gives

(8)

Combining Eqns (2), (4) and (8), remembering that q = 1 / L and for convenience defining the diffusivity ratio X, = D i / D in, gives two equations S() (9) J +D ainAy1 Q U /D J which in the model diffusivity ratio equation, and

JRTL (10) ApAyiD m which is the experimental diffusivity ratio equation. A detailed derivation of these equations in contained in [6], together with a commentary on the properties of the generalised transition diffusion equation, Eqn (9). But here it is sufficient to note that, in principle, it appears to be possible to estimate the unknown, characteristic pore structure function [(Ny / q) f(r)] from the use of Eqn (9) as a model for experimental values of diffusivity ratio that are calculated using Eqn (10). The following sections summarise how this estimation is carried out.

331 3 EXPERIMENTAL

Experimentally the aim is to measure the factors on the right hand side of Eq (10) and hence to calculate the diffusivity ratio as a function of pressure. In this work the only unknown is the molar flow rate since all other factors are fixed (R = 8.314 J K"1 mol" , T = 300 K, L / A « 0.012, Ay; « 1 and Din can be calculated for binary gases from empirical functions of p and T [6]). For He-Ar diffusion, it turns out for this work that

X, /1000 = 3.914 (V^imols"1), i = HeorAr (11)

Measurements of molar flow rate for each gas were made using a Wicke-Kallenbach cell [3], that operated at pressures < 100 Torr (13.3 kPa), in which conditions mean free path calculations indicated that transition diffusion would occur in the graphites. In this cell, pure He was directed through a mass flowmeter to one side of a sample, pure Ar was directed to the other side through another mass flowmeter, and the amounts of each gas diffusing through the sample into the other stream were determined using a mass spectrometer. Details of the apparatus that was built specially for this work are in [7]. The three nuclear-grade graphites selected for study are: AXF (a moulded, petroleum coke filler-coal tar pitch binder graphite made by POCO Graphite Inc., Decateur, TX, USA); Isograph-50 or IG-50 (an isostatically-pressed graphite made in Osaka, Japan by Toyo Tanso Co. Ltd.) and ATJ (similar to AXP but made by Union Carbide in Sheffield, UK).

4. RESULTS AND DISCUSSION

Fig. 1 is a plot of He and Ar difrusivity ratios, X I 1000, as a function of pressure, p / kPa, for the three graphites. A number of initial observations may be made from this figure.

(i) He diffbsivity ratios are larger than Ar diffusivity ratios by a factor of about 3.2 for = all values of p for each graphite; the theoretical ratio from Eq (9) is V(MAT / Mne) 3.162, where M is molecular weight. This close agreement gives some early confidence in the validity of the model and experimental data.

(ii) The plots for each gas and graphite are curved, in general. This indicates that, as anticipated, flow is in the transition regime between Knudsen diffusion (where X should increase linearly with p) and bulk diffusion (where \ should be constant).

(iii) There are systematic differences between the plots for each graphite. ATJ data show the least curvature while ATJ data show the most curvature, while in general AXF seems to offer the least resistance to diffusion (high X), and IG-50 the most resistance.

To elaborate on these observations, statistical analysis of the transition diffusion data, using Eq (9) as a model, yielded estimates of the ratio (Nv / q) and of the function f(r), the probability density of pore radii. The method used to estimate these quantities is, in fact, non-trivial on account of the properties of Fredholm integral equations of the first kind, like Eq (9). Details of the method are beyond the scope of this paper, but may be found in [7] and [8].

332 o o o

(0 U

n

0 2 4 6 8 10 12 14 16 pressure /kPa

o o * 1 " i • i • i ' i ' i o Argon

•p

- i i i . t . i . i , • , i 0 2 4 6 8 10 12 14 16 pressure / kPa

FIG.l He and Ar difiusivity ratios as a function of pressure for each graphite, as measured using the new method.

Fig. 2 contains plots of the probability density function f(r) that were estimated for each gas in each graphite. It may be noted that the functions for each graphite for either He or Ar are similar. This indicates that both gases 'see' the same pore structure during flow. Thus quoted values for transition diffusion pore structure need not refer to a particular gas. 3 Estimates of the factor (Nv / q) / mm for each graphite are as follows: AXF - 33 0- IG-50 - 1.04, and ATJ -0.0896. These results may be summarised as follows:

(a) The probability density functions, f(r), for each graphite are unimodal.

(b) The graphites ordered in increasing modal pore size are: AXF (0.55 urn), IG-50 (1.04 urn) and ATJ (4.71 um). The spread of pore sizes also increases in this order.

(c) The graphites ordered in decreasing values of Ny are: AXF, IG-50 and ATJ.

These results may be compared with observations from other methods. Hg porosimetry data also indicate that pore size distributions in the graphites are unimodal or 'unipore1. However, while porosimetry yields the same order of pore sizes for the graphites,

333 n §

•rl 5 O u 0, 0 2 4 6 8 transport pore radius /

•H O

ft 0 2 4 6 8 10 transport pore radius /|im

FIG. 2 Plots of transport pore radius probability density functions, f(r), for He and Ar for each graphite, estimated from the transition diffusivity data in Fig. 1. modal pore sizes from that method are on average 77 % lower than those estimated from the transition diffiasivity data. This suggests that the latter are not as sensitive to pore constrictions as Hg penetration. The same ranking order of pore sizes for the graphites is also found, qualitatively, using optical microscopy. These results are, of course, consistent with Fig. 1, notably that the graphite with the smallest pores (AXF) has the least curved diffusivity plot - i. e., flow is near to Knudsen diffusion - and the graphite with the largest pores (ATJ) has the most curved plot - i. e., flow approaches bulk diffusion. Open pore volumes for the graphites, measured using an He pycnometer, were: AXF - 17.4 %; IG-50 - 9.0 % and ATJ - 16.1 %. From Eq (3) and the values for (Nv / q) and f(r) estimated using the new method, minimum transport pore volumes (i. e., for q = 1) were estimated as: AXF - 2.70 %; IG-50 - 0.67 % and ATJ - 0.63 %. Thus the relatively large transport pore volume for AXF appears to be the reason why diffusion in this material is easiest of the three graphites. However, the restricted diffusion in IG-50 compared with ATJ appears to be due to the larger pore sizes in the latter graphite, for their transport pore volumes are similar.

334 A powerful way to test a model is to use it to predict quantities not related directly to the measurements that were made to estimate parameters in the model. Here, (Nv / q) and f(r) were used to predict gas transport parameters that were measured using established methods, viz., N2 permeability and Ar-C02 difrusivity. Predicted permeability parameters (the viscous flow coefficient, Bo, and the slip flow coefficient, Ko) were on average 20 % higher than measured, while predicted Ar-CO2 diffusivities (determined at 1 atmosphere pressure, where flow is almost entirely due to bulk diffusion) were within 5 % of those measured. This suggests that there is a fundamental difference between the ways that permeable and diffusive flows 'see' pore structures. The nature of this difference is as yet unknown; exploring it may be an interesting area of future work. Finally, it should be noted that the reverse procedure of predicting transition diffusion data from permeability or bulk molecular diffusion pore structures is impossible, for the latter comprise only mean values of pore sizes, rather than the whole pore size distribution which can be estimated using the new method.

5. CONCLUDING REMARKS

A new method has been developed to characterise pore structure in nuclear graphites. The method involves an analysis of the isothermal (room temperature) diffusion of He and Ar in samples as a function of pressure less than 100 Torr, where flow in macropores is in the transition regime between Knudsen and bulk molecular diffusion. Pore structures estimated from the new method agree qualitatively, or are broadly consistent with results from Hg porosimetry, optical microscopy, He pycnometry, N2 permeability and Ar-CC»2 bulk diffusivity. However, some quantitative differences were observed. Pore sizes from porosimetry were lower than those from the new method, due it was presumed to the well- known sensitivity of Hg penetration to pore constrictions. Also, while pore structures determined using the new method accurately predicted bulk molecular Ar-CC>2 diffusion data, they over-predicted N2 permeability parameters. Further work should be directed to understanding the difference between diffusion and permeation pore structures, and to exploring the utility of the method in oxidation and lifetime studies of moderator graphites.

REFERENCES

[1] SING, K. S. W, et al., Pure Appl. Chem. 57 (1985) 603. [2] KELLY, B. T., In: Porosity in Carbons, (PATRICK, J. W., Ed), Edward Arnold, London (1995) 151. [3] McENANEY, B., MAYS, T. J., ibid. 93. [4] MASON, E. A., MALINAUSKAS, A. P., Gas Transport in Porous Media: the Dusty- gas Model, Elsevier, Amsterdam (1983). [5] HEWITT, G. F., Chem. Phys. Carbon 1 (1965) 73. [6] FULLER, E. N, SCHETTLER, P. D, GIDDINGS, J. C, Ind. Eng. Chem. 58 (1966) 19. [7] MAYS, T. J, Gaseous Diffusion and Pore Structure in Nuclear Graphites, PhD Thesis. University of Bath, UK (1989). [8] MILLER, G. F., In: Numerical Solution of Integral Equations (DELVES, L. M., WALSH, J., Eds), Clarendon Press, Oxford (1974) 175.

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