Diffusion of H through Pd membranes: Effects of non-ideality

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Journal of Membrane Science 306 (2007) 66–74

Diffusion of H through Pd membranes: Effects of non-ideality Ted B. Flanagan a,∗ , Da Wang a , Kirk L. Shanahan b a

b

Chemistry Department, University of Vermont, Burlington, VT 05405, United States Savannah River National Laboratory, Building 999-2W, Aiken, SC 29808, United States

Received 30 March 2007; received in revised form 8 August 2007; accepted 10 August 2007 Available online 21 August 2007

Abstract H diffusion constants, DH , have been obtained from steady-state fluxes through Pd membranes with the downstream side maintained at pH2 ≈ 0. Good linearity of plots of H flux versus (1/d), where d is the thickness, attests to H permeation being bulk diffusion controlled in this temperature (423–523 K) and pH2 range (≤0.2 MPa). DH values have been determined at constant pup and also at constant H content. H fluxes through Pd membranes with three different surface treatments have been investigated (polished (unoxidized), oxidized and palladized) in order to determine the effects of these pre-treatments. The palladized and oxidized membranes give similar DH values but the polished membranes give values about 12% lower. For diffusion in a concentration gradient J = −DH∗ (cH /RT)(dμH /dx) is the proper equation for the flux, where cH is the H concentration, rather than Fick’s first law, J = −DH (dcH /dx), where DH and DH∗ are the concentration-dependent and -independent diffusion constants. DH∗ can be obtained 1/2 from DH using the thermodynamic factor, DH (cH ) = DH∗ (∂ ln pH2 /∂ ln cH )T = DH∗ f (cH ). In the commonly employed situation, where there is a large difference in concentrations between the upstream and downstream sides of a membrane, the thermodynamic factor varies with distance through the membrane and this should be allowed for in obtaining DH∗ . Procedures are given and utilized to determine DH∗ from DH (cH ) when there is a large concentration gradient through the membrane. Activation energies for diffusion, ED (cH ), have been determined. ED is found to increase with cH , which is attributed to the thermodynamic factor. Published by Elsevier B.V. Keywords: H diffusion; Pd; Non-ideality

1. Introduction It is relatively easy to obtain clean Pd surfaces since any Pd oxides are reduced by exposure to H2 at moderately high temperatures. Pd alloys are employed for H2 purification by selective permeation through the alloy membranes. There have been many investigations of H diffusion through Pd, e.g., refs. [1–7]. Studies in the literature cover a wide temperature range, and Alefeld and V¨olkl [7] have summarized these up to 1978 by an single Arrhenius plot of these DH data from about 273 to 873 K. Although a single straight line was drawn through the data, it seems that the slope, −ED /R, should vary with temperature as discussed below. Despite the large number of investigations, some aspects of diffusional behavior of H in Pd have not been examined in much o and E on H concendetail such as the dependences of DH , DH D ∗

Corresponding author. Tel.: +1 802 656 0199; fax: +1 802 656 8705. E-mail address: [email protected] (T.B. Flanagan).

0376-7388/$ – see front matter. Published by Elsevier B.V. doi:10.1016/j.memsci.2007.08.032

o exp−ED /RT and this will be done here tration where DH = DH over a temperature range of interest for H purification through Pd membranes (423–523 K). Pd and its alloys are the most suitable systems to investigate the effects of non-ideality of dissolved H because their membranes generally exhibit quite reproducible 1/2 behavior, large H2 solubilities may obtain, and pH2 –H content isotherms can be readily measured for the evaluation of the role of non-ideality. Two types of non-ideality must be considered in H permeation through membranes. The first arises from deviations from 1/2 Sieverts’ law of ideal solubility, i.e., when r = Ks pH2 , and the other is due to the concentration dependence of Fick’s diffusion constant [8], DH (r), which will be of concern here. The H/Pd, atom ratio = r, is the usual concentration unit employed for metal–H systems and it will be employed here except when it is necessary to use cH ; r = cH M/ρ, where ρ and M are the density and molar mass of the H-containing Pd. The equation relating the concentration-dependent, DH (r), ∗ , diffusion constant is given and concentration-independent, DH

T.B. Flanagan et al. / Journal of Membrane Science 306 (2007) 66–74

by [8] ∗ DH (r) = DH



1/2  ∂ ln pH2

∂ ln r 1/2

∗ = DH f (r)

(1)

T

where f (r) = (∂ ln pH2 /∂ ln r)T and is referred to as the thermodynamic factor. In the 1960s, Wicke and his students carried out groundbreaking research in this area using Pd–H and Pd alloy–H ∗ values were derived from systems as reviewed in ref. [8]. DH DH (r) for Pd–H and for some Pd–Ag alloy–H systems using Eq. (1) which is appropriate when the H concentration in the membrane can be assumed to be nearly constant. Their measurements on Pd–H were made in the very dilute range where f(r) = 1 or else in the concentrated hydride phase where f(r) > 1.0. The latter caused DH (r) to increase by a factor of ∼10 compared to the dilute region. Since these workers generally investigated diffusion near room temperature, where the dilute phase in Pd–H is limited to small values, the concentration dependence was only investigated at the two extremes of very dilute or very concentrated. Z¨uchner and co-workers [9,10] and also K¨ussner [11], who were Wicke’s students, employed time-lag techniques where a pulse of H is deposited, generally electrochemically, at the upstream side of an H-containing Pd or Pd-alloy membrane. The time for the perturbation to reach the downstream side can be used to calculate DH . Since the H concentration is initially uniform and the perturbations do not change the uniformity significantly, Eq. (1) can be employed using an average membrane concentration in f(r). Both K¨ussner [11] and Z¨uchner and co-workers [9,10] determined the concentration dependence of DH for Pd0.77 Ag0.23 and Pd0.6 Ag0.4 alloy membranes, which were chosen because they do not have two-phase, plateaux at T ≥ 298 K allowing DH to be measured as a continuous function of r. For the former, DH decreases from about 3 × 10−7 (r = 0) to 0.4 × 10−7 cm2 /s (r = 0.18) at 303 K and for the latter, DH is constant at 3 × 10−7 cm2 /s up to about r = 0.09 and then increases to 13.0 × 10−7 cm2 /s at r = 0.23. Salomons et al. [12] measured the dependence of DH upon r for a Pd foil at four temperatures from 473 to 603 K and from r = 0 to about 0.07 using time constants for the relaxation of pressure perturbations followed gravimetrically; their DH values were somewhat smaller than those in the reported in the literature. Isotherms given by Blaurock [13] were employed to determine DH which decreased almost linearly with r except for an initial region from r = 0 to about 0.015 and then appeared to be nearly independent of r [12]. They did not measure the ED as a function of r. In the present work DH and ED will be measured over a lower temperature range than used by Griessen et al., but still at temperatures where significant H contents can be achieved before hydride phase formation takes place. A complication in the present permeation experiments is that the H concentrations are not uniform but vary markedly from rup at the upstream side to 0 at the downstream side of the membrane and therefore, it seems that Eq. (1) cannot be employed directly. Recently, a procedure has been given

67

for determining concentration-independent diffusion constants from the concentration-dependent ones for this steady-state situation [14]. In an earlier study of H diffusion through Pd by the authors [15], which employed the same experimental set-up, the desirability of oxidizing the Pd before measurements and the effect of CO on the Pd were the main topics. There was no investigation of the dependence of the diffusion parameters the H concentration. 2. Experimental In the present experiments, fluxes are determined from the decrease of pup on the upstream side of the Pd membrane while the downstream side is kept at pH2 ≈ 0. Changes were followed using a sensitive electronic diaphragm gauge (MKS), which could be read to ±13 Pa. Steady-state fluxes are established very rapidly for these Pd membranes at the temperatures employed (423–523 K). The decrease of pup during measurements is small because the upstream volume is large and, in any case, the fluxes can be corrected for the small pup decreases resulting from permeation. The decrease of pup during the permeation measurements depended upon the thickness and temperature of the membrane and, e.g., for a 380 ␮m membrane at 423 K, the flux decreased over the first 5 min by 0.9% and for the subsequent 5 min period, it decreased by 1%. For a 95 ␮m membrane (523 K), the flux decreased 5% in the first 5 min period, i.e., the greater J, the greater the decrease of pup . It should be stressed that the fluxes can be corrected for the pup decrease, e.g., if the second 5 m period is multiplied by a factor of (p1,av /pi,av )1/2 where p1,av and pi,av refer to the average pup during the initial 5 m period and to any other 5 m period, respectively. After this correction, the first few 5 m periods were closely equal. Generally, however, the fluxes were taken as the initial ones based on the average flux over the first 5 m period and if there was a significant fall in pup , e.g., 5%, the data were corrected for the pup fall. Usually two or three successive determinations were made for the same conditions and it was found that the results either did not differ or the differed insignificantly. For determinations of the dependence of DH on r, isotherms were employed to determine the appropriate pup needed to obtain the desired r. It should be noted that isotherms for Pd–H are quite well established and, e.g., the present isotherm at 473 K is very close to the data given by Blaurock [13]. The changes of DH (r) with rup should be quite reliable for a specific Pd membrane of a given thickness because the errors are introduced only by the flux measurements which depend on the values of the flux. For example, for a 84-␮m membrane at 523 K, and r = 0.005, the error in DH based on the 13 Pa variation of the pressure gauge was 0.9% and for the same membrane at r = 0.04 at 423 K, the error was 0.7%; these are not significant enough to obscure the trends of DH (r) with r. The areas of the membranes exposed to the H2 in the diffusion apparatus were all the same, 1.77 cm2 , but a large range of membrane thickness was employed. Temperatures were controlled to about ±1 ◦ C; temperature controller settings were verified by a thermocouple placed in contact with the membranes within the

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permeation system and these temperatures agreed very closely with that measured by the controller. The Pd membranes were prepared by arc-melting Pd (Engelhard, 99.97%) into buttons and then rolling to the desired thickness; the membranes were then briefly annealed. The thickness of the membranes was taken as the average of several places using a micrometer and was subject to about 5% error. Some Pd membranes were cleaned carefully and employed directly and these will be referred to as polished. Some were oxidized in the atmosphere for ∼30 min. at 953 K before employing because this procedure has been shown [15] to improve membrane reproducibility. Since oxides formed by Pd do not penetrate into the alloy appreciably and are rapidly reduced in the presence of H2 , the thickness of the membranes will be taken as those before oxidation or palladization. Palladized membranes were prepared by coating them electrolytically with Pd black in a weakly acidic solution of PdCl2 . The isotherms were measured with Pd foils prepared from the same Pd as that used for the membranes. The measurements were made with a conventional Sieverts’ volumetric apparatus where the H contents are determined from the pH2 changes as described elsewhere [16]. 3. Results and Discussion In this research, DH values will be determined in the dilute phase of Pd over the temperature range from 423 to 523 K. An advantage of using dilute solution is that simple models such as the mean field, lattice gas model, which is conveniently referred to as a regular interstitial solution (RIS) model [17], describe the thermodynamics relatively well. In addition, measurement at low H contents avoids problems connected with site blocking which occurs at high H contents [8]. For the present experiments, where the downstream side is maintained at pH2 ≈ 0, the flux and specific permeability of H2 , are given by   −DH cup dc (2) J = −DH ≈ d dx P = −DH Ks p1/2 up

(3)

where J is the flux, P the specific permeability, d the membrane 1/2 thickness and Ks a temperature-dependent constant relating pH2 to cH via Sieverts’ law, cH = Ks p1/2 . In the present experiments, fluxes and specific permeabilities have been measured and the diffusion constants were derived from J using H2 isotherms measured here. This paper will be organized as follows: (i) isotherms (423–523 K), (ii) demonstration of the slow step as bulk diffusion rather than surface control, (iii) effect of surface treatments of the Pd membranes on DH , (iv) temperature dependence of ED , (v) theory of the effect of non-ideality arising from f(r) = 1.0 ∗ from experimental D , (vii) on DH , (vi) determination of DH H dependence of ED on H concentration, (viii) effect of nono , (ix) concentration profiles during ideality (f(r) = 1.0) on DH permeation under non-ideal conditions.

Fig. 1. H2 solubilities for Pd. The dashed straight line shows ideal behavior where the slope, i.e., thermodynamic factor = 1.0.

3.1. Dilute phase isotherms for Pd–H Dilute phase H2 absorption isotherms were measured using Pd foil at the same temperatures as the diffusion constants were determined (423–523 K) and they are given as plots of 1/2 ln pH2 against ln r in Fig. 1, where the ideal slope of 1.0 is shown by the dashed straight line. It can be seen that the slopes are all ≤1.0 and deviations from 1.0 increase with r and with decrease of temperature. Thermodynamic factors, i.e., slopes, 1/2 (∂ ln pH2 /∂ ln r)T = f (r) (Eq. (1)), have been evaluated from these isotherm data. 3.2. Dependence of J upon 1/d at constant pup and T These Pd membranes were oxidized as described above. In our earlier studies [15], it was shown that there is a linear relation 1/2 between the permeabilities and pup as r → 0 and, if allowance is made for deviations from Sieverts’ law of ideal solubility, the relationship becomes linear over a large range of r at 473 K. This linearity demonstrates that the upstream membrane surface is in equilibrium with the upstream H2 at pup . A possible role of the thermodynamic factor was not considered in this earlier permeation test for bulk diffusion and this could be a factor at the higher pup . In view of this, it is desirable to demonstrate that bulk diffusion is the controlling step under conditions where the thermodynamic factor does not play a role and for temperatures other than 473 K. A better test for establishing bulk diffusion as the ratecontrolling step is whether or not there is a linear relation between J and (1/d) because any non-ideality due to f(r) = 1.0 will not affect the plot since rup and the temperature are held constant in these flux measurements and therefore, f(r) will be constant with (1/d). Such experiments were carried out here with a large number of different membranes of different thickness (91–480 ␮m). Results are shown in Fig. 2 at both 423 and 523 K. Both plots extrapolate to the origin as they should and are linear over the whole range even at the higher fluxes found

T.B. Flanagan et al. / Journal of Membrane Science 306 (2007) 66–74

69

Table 1 Apparent DH (r) in units of cm2 /s for the three different surface treatments of Pd membranes: oxidized (122 ␮m), palladized (104 ␮m), polished (119 ␮m) r

Fig. 2. H2 flux plotted against 1/d for Pd membranes at 423 and 523 K with pup = 50.65 kPa.

at 523 K. It can therefore, be concluded that bulk diffusion is the rate-controlling step in the present experiments. 3.3. The effect of surface treatments of Pd membranes on apparent DH values Permeation measurements were carried out with polished Pd membranes (∼120 ␮m thick), i.e., they had not been oxidized, with membranes which had been oxidized and which had been palladized, i.e., coated with Pd black. The polished membrane was carefully cleaned and appeared quite shiny. Measurements were made with these membranes at a series of r values and temperatures. The data were quite reproducible for the oxidized and palladized membranes but less so for the polished one. Results are shown in Table 1 at 423, 473 and 523 K. Similar data were obtained at 453 and 503 K but are not shown. At a given r, e.g., 0.020, DH for the oxidized and palladized surfaces are greater than that for the polished membrane at each temperature. The differences appeared to decrease with increase of temperature, for instance, the decrease from oxidized to polished is 11% at 473–523 K and 14 and 18%, at 453 and 423 K, respectively. DH for the palladized and oxidized membranes can be considered to be the same in view of the errors in measurements of d. Due to surface roughness and to Pd black the surface areas of the oxidized [18] and palladized membranes are larger than that of the polished membrane and therefore, more H2 dissociation will take place on these. However, the areas appropriate for Eqs. (2) and (3) are the inner ones because the rate-controlling diffusion step takes place within the bulk phase just below the outer surfaces and therefore, the relevant areas for diffusion should be the same for all three membranes, however, the unoxidized surface appeared to have a slightly smaller effective area. Arrhenius’ plots of ln DH against 1/T for the three different surface treatments are shown in Fig. 3 at r = 0.02, where it can be seen that the polished membrane has smaller DH values over the whole temperature range as compared to the other two. From the slopes, the activation energies, ED , are found to be 24.7, 24.5 and 24.9 kJ/mol H for the palladized, oxidized and pol-

DH × 10−6 (oxidized)

DH × 10−6 (palladized)

DH × 10−6 (polished)

423 K 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045

– 6.14 6.08 6.03 5.96 5.69 5.49 5.33

– – 6.11 6.10 5.95 5.72 5.54 –

– – 5.24 5.28 5.15 4.91 4.85 –

473 K 0.010 0.015 0.020 0.025

13.08 12.81 12.79 12.58

– 13.41 13.32 13.26

– 11.50 11.48 11.27

523 K 0.010 0.015 0.020 0.025

22.89 22.91 22.92 22.92

23.52 23.35 23.16 23.22

20.55 20.49 20.46 20.30

ished membranes, respectively. Similar results were obtained for r = 0.025. The purpose of the figure is to illustrate the differences between the three surface treatments at the various temperatures and not to obtain concentration-independent activation energies ∗ , which will be derived below. for diffusion, ED The differences between the various Pd membrane surfaces may partially explain some results in the literature. Examination of these in Table 2 suggests that generally DH for palladized and oxidized membranes are greater than those for the polished membranes. In the work of Jost and Widmann [4,5], Pd spheres were coated with Pd black and they reported 13.7 × 10−6 cm2 /s at 473 K, which is certainly relatively fast and close to that in Table 1 for the other palladized or oxidized membranes. The value of DH at 473 K from Bohmholdt and Wicke [6], who also employed palladized Pd tubes, is somewhat low perhaps due to

Fig. 3. ln DH plotted against 1/T for several different surface treatments of Pd at r = 0.02. (♦) unoxidized; ( ) oxidized (Ioed); (䊉) palladized.

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Table 2 o DH , ED and DH values for pure Pd from literature References

o DH (10−3 cm2 /s)

ED (kJ/mol H)

DH (473 K) (10−6 cm2 /s)

T (K-range)

Surface

[4,5] [2] [3] [6] [19] [1] [24] [7] [8] [25] [15]

6.0 6.0 4.3 3.7 4.9 2.9 2.5 2.9 5.3 3.1 5.6

23.9 23.6 23.4 24.0 24.0 22.0 21.8 22.2 22.8 22.3 23.2

13.7 14.6 11.1 9.7 11.0 10.9 9.8 10.3 16.0 10.6 15.5

466–576 443–563 486–652 293–373 300–709 533–913 230–1000 – 273–373 423–723 423–523

Palladized Polished Polished Palladized Polished Palladized All surfaces – Palladized Polished Oxidized

extrapolation to 473 K from significantly lower temperatures. Another apparent exception to this correlation between the different surface treatments and DH values appears to be the results of Holleck [1] who employed palladized membranes but the DH values found by him were smaller than those for the other palladized or oxidized ones. There is, however, a possibility that some sintering of the Pd black took place because he carried out measurements up to 913 K. Aside from these results, larger DH values generally appear to be found for palladized or oxidized Pd membranes. ∗ on temperature 3.4. Dependence of ED

The pup employed in the gas phase permeation studies of Koffler et al. [19] was very low, i.e., 0.4–0.6 Pa, and consequently rup was very small and in the ideal range where, with reference to Eq. (1), f(r) = 1.0. Their Arrhenius plot for the permeability of H through Pd (their Fig. 4 [19]) can be seen to be very linear from 300 to 709 K with a reported EP∗ = 15.67 kJ/mol H [19]. Recently, Hirscher and co-workers [20] also determined the permeability through Pd at low pH2 and found good agreement with the results of Koffler et al. The following relation holds between the activation energies for diffusion and permeability [19], ∗ EP∗ = ED + HHo ,

Table 3 ∗ from E = 15.67 kJ/mol H [19] and H o from ref. [21] Derived values of ED P H T (K)

EP∗ (kJ/mol H)

HHo (kJ/mol H)

273 373 473 573 673

15.67 15.67 15.67 15.67 15.67

−10.05 −9.37 −8.46 −7.47 −6.45

∗ (kJ/mol H) ED

25.7 25.1 24.1 23.1 22.1

These HHo values can be used, together with the constant EP∗ ∗ at different temperatures as shown value from [19], to derive ED in Table 3. On this basis, there is an appreciable temperature ∗. dependence of ED ∗ from the literature plotted against Fig. 4 shows values of ED T, including a data point from the present work at an average temperature of 473 K, compared with values predicted from Eq. (4) and it is assumed that the literature values are for infinite ∗ must be deterdilution which may not be the case. Since ED mined over a temperature range and the average temperature of each investigation has been used in Fig. 4. The experimental values from the literature suggest a temperature dependence of ∗ for Pd–H (Fig. 4) and, as noted, this has not been generally ED appreciated.

(4)

where HHo is the enthalpy of (1/2)H2 (g) solution as r → 0 and the other two quantities should be for similar conditions. Since ∗ must it is known that HHo varies with temperature [21–23], ED ∗ also vary with temperature in order for EP to be constant as ∗ is positive found by Koffler et al. [19]. HHo is negative while ED and, since the former becomes less exothermic with increase of ∗ must decrease with temperature increase. temperature, ED ∗ must decrease with Holleck [1] first pointed out that ED temperature and, in support of this, he cited values of ∗ = 23.4 kJ/mol H at 333 K (av) [6] and 22.0 kJ/mol H at 673 K ED (av) [1]. As far as we are aware, there have been no other dis∗ for Pd–H. Since cussions of the temperature dependence of ED o the work of Holleck, HH values have been measured over a wide temperature range by L¨asser and Powell for Pd–H [21,22] and their results are supported by earlier work of Kuji et al. [23].

∗ as a function of temperature. (), calculated from E∗ = E∗ − H o ; Fig. 4. ED H D P () from the literature (see Table 2); ( ) present data.

T.B. Flanagan et al. / Journal of Membrane Science 306 (2007) 66–74

3.5. Theory for the determination of ∗ , from concentration-independent diffusion constants, DH concentration-dependent ones, DH (r) For H concentrations larger than the very dilute, it is inap∗ at moderate temperatures propriate to equate DH (r) with DH because non-ideality due to the thermodynamic factor can be significant (Eq. (1)). For conditions where rup rdown , the H concentration varies markedly through the membrane, and the degree of non-ideality will also vary. Recently Eq. (5) has been proposed for this situation [14]. ∗ F (r ) DH up , (5) rup r where F (rup ) = 0 up f (r) dr. ∗ will be determined from D (r) in In the present work, DH H four different ways: (a) directly from F(r) (Eq. (5)) using f(r) from the isotherms at these temperatures (Fig. 1), (b) from Eq. (5) using the lattice gas, mean field (regular interstitial solution (RIS)) model [17] for evaluation of F(r), (c) extrapolation of ln DH (r) to r = 0 and (d) use of an average H concentration, i.e., rup /2 = rav , in Eq. (1).

DH (r) =

(a) Direct use of Eq. (5) Eq. (5) can be employed directly using the integrals,  rup 0 f (r) dr = F (rup ), i.e., the areas under plots of the 1/2 ∗ is then slopes, (∂ ln pH2 /∂ ln r)T = f (r), against r. DH given by DH rup /F(r). A plot of f(r) against r is illustrated in Fig. 5 at 423 K up to rup = 0.055. (b) Mean field, lattice gas (regular interstitial solution model (RIS)) This simple model for metal–H systems [17] is given by Eq. (6) using the approximation, μEH ≈ g1 × r which will, for convenience, be referred to as the RIS model   r 1/2 o + μEH (r) RT ln pH2 = μH + RT ln 1−r

Fig. 5. Plot of the thermodynamic factor against r for Pd at 423 K.

71

 ≈ μoH + RT ln

r 1−r

 + g1 × r

(6)

where g1 is a constant at a given temperature and represents the first order term in a polynomial expansion in r of μEH (r). The partial free energy, g1 , has enthalpic and entropic components, i.e., g1 = h1 –Ts1 . Experimental isotherms are needed to evaluate g1 and they have been determined for Pd–H [23], and re-determined here from isotherms (Fig. 1) measured at the same temperatures as those of the diffusion studies. r From F (r) = 0 up f (r) dr and Eq. (6), we obtain F (rup ) = − ln(1 − rup ) +

2 g1 × rup

, (7) 2RT and using this result and the approximations that (−ln (1 − rup )/rup ) ≈ 1 and ln (x − 1) ≈ x (where x is small) in Eq. (5) gives ∗ ln DH = ln DH +

ln F (rup ) g1 × rup ∗ ≈ ln DH + . rup 2RT

(8)

∗ from extrapolation of ln D (r) to r = 0 (c) ln DH H Eq. (8) predicts a linear relation between DH (r) and rup at least up to rup ≈ 0.04 and therefore, ln DH can be extrap∗ . This method does not require olated to rup = 0 to obtain DH non-ideality values for the calculation of DH (r). ∗ employing an average r = (r /2) in Eq. (1) (d) DH av up An average value of rav = (rup /2) can be used in Eq. (1) which works quite well at small rup as seen from Eq. (7) because at small r where the −ln(1 − rup ) term can be neglected. At high r values, the approximation is not as good.

3.6. Evaluation of the various methods ∗ from the varThere is considerable scatter in the values of DH ious approaches. Method (a) should be accurate over the whole range of r values provided that accurate f(r) are known. Method (b) is not very accurate at relatively high r because the linear approximation of μEH (r), i.e., g1 × r, is no longer appropriate. Method (c), extrapolation of ln DH (r → 0), is only viable if reliable DH (r) values are available at small r. The use of an average r = (rup /2) in Eq. (1), method (d) is a reasonable approximation for Pd–H at small r and is certainly the easiest method to employ. ∗ from experimental 3.6.1. Concentration-independent DH DH (r) values Both the concentration-dependent and -independent H diffusion constants are important. The former, DH (r) values are needed to estimate actual permeabilities under operating condi∗ has a tions of a membrane where non-ideality is a factor. DH unique value at each temperature and is of fundamental interest because of its direct relation to the H mobility. H fluxes were measured at 423, 453, 473, 503 and 523 K as a function of rup for several oxidized Pd membranes. The DH (r) values always decreased or else remained constant with increase of rup in the range employed, r = 0–0.08. The decreases are due

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Table 4 ∗ (10−6 cm2 /s) for oxidized Pd (76 ␮m) at 423 K as DH (r) (10−6 cm2 /s) and DH a function of r and corrected values from procedures (a), (b) and (d) rup

DH (r)

∗ (a) DH

∗ (b) DH

∗ (d) DH

0.020 0.025 0.030 0.035 0.040 0.045 0.050

5.96 5.81 5.68 5.40 5.20 5.06 4.86

6.45 6.43 6.43 6.27 6.22 6.22 6.15

6.56 6.56 6.59 6.43 6.37 6.38 6.31

6.48 6.46 6.45 6.31 6.25 6.27 6.24

to the thermodynamic factor and its importance is expected to increase with rup and to decrease with increase of temperature. Some results using methods (a), (b) and (d) are shown in Table 4 for 423 K up to rup = 0.050. Fig. 6 shows a plot of these ln DH (r) values against rup where extrapolation of ln DH (r) to rup → 0 gives 6.65 × 10−6 cm2 /s which is in good agreement with the ∗ . The D∗ values are nearly constant at the derived values of DH H lower r values and then decrease at higher r. The results using the average r = (rup /2) are quite close to those from method (a) (Table 4) but, at higher r, the former are slightly greater due to deviations from the linear dependence on r (Eq. (8)). These results show an effect of the thermodynamic factor even at quite low r values at 423 K, e.g., at rup = 0.02 or rav ≈ 0.01 where there is a significant effect. When comparing different Pd membranes, however, errors are introduced by the thickness and area measurements and when comparing with results in the literature these become important. ∗ obtained by these methods were not as conThe values of DH stant at 503 and 523 K as those at the lower temperatures, e.g., ∗ derived from D (r) using methods (a) or (b) Table 4. The DH H for T ≥ 473 K increased with r, i.e., the thermodynamic factor was too large, which is opposite of the trend at the lower temperatures. It is proposed that at the high temperatures f(r) ≈ 1.0 at low H contents, i.e., r = 0 to ≈0.015 and then, when r > 0.015, f(r) < 1.0. There is some evidence for this from the isotherms at higher temperatures. Salomons et al. [12], who studied the effect

Fig. 6. Plots of ln DH against H/Pd at 423 K for Pd (76 ␮m). () experimental data; () corrected by method (a); ( ) corrected by method (b); (♦) corrected by method (d).

of r on DH in Pd–H at elevated temperatures, also found that DH did not change very much with r from 0 to ≈0.015, however, they made no comments about this. 3.7. Effect of non-ideality from the thermodynamic factor on ED The activation energy for H diffusion in Pd is also affected by the thermodynamic factor because it must be the main cause of the decrease in DH (r) with r. An expression for the activation energy for diffusion, ED (r), determined at constant rup can be obtained by differentiation of Eq. (5), i.e., ∗ ∂ ln F (rup ) ∂ ln rup ∂ ln DH ∂ ln DH = + − ∂(1/T ) ∂(1/T ) ∂(1/T ) ∂(1/T )

and from this equation,   ∂ ln F (rup ) ∗ ED = ED − R ∂(1/T ) rup

(9)

(10)

∗ will be consince at constant rup , the ln rup term drops out ED sidered to be constant over the small interval from 423 to 523 K. ∗ obtained from Eq. (11) is an approximate expression for ED Eqs. (7) and (10) based on the RIS model [17],  ∗   (h1 rup /(2 ln(1 − rup )) ED ED = + . (11) RT rup − (g1 rup /RT )

Since s1 is negative at small r [23], h1 < g1 , e.g., g1 = −32.7 kJ/mol H and h1 = −64 kJ/H from the present isotherm data at 473 K. Since h1 is negative for Pd–H and ∗ . For example, 2 ln{(1–rup )/rup } < −(g1 × rup )/RT, ED (r) > ED ∗ for rup = 0.05, ED (rup = 0.05) − ED = 1.9 kJ/mol H which is an ∗ appreciable difference. Differences between ED (rup ) and ED occur even at quite low r, e.g., at rav ≈ (rup /2) = 0.01, ED is 24.6 ∗ = 23.9 kJ/mol H. This effect of the thermodycompared to ED namic factor may help to explain variations in the ED values reported (Table 2). ∗ from E using Eq. (10) directly Attempts to determine ED D were unsuccessful because values of (∂F (r)/∂(1/T ))rup determined from the present isotherms (Fig. 1) are not sufficiently ∗ from Eq. (10) accurate. And therefore, any evaluation of ED using the temperature dependence of rav = (rup /2) would also be unsuccessful. Until more accurate F(r) values are avail∗ can be obtained by extrapolation to r = 0 or by able, ED up ∗ is directly using Eq. (11). Another way to determine ED ∗ . This has been done using D∗ from an Arrhenius plot of DH H obtained from experimental DH values (at rup = 0.025) which o,∗ = 6.1 × 10−3 cm2 /s. gives a value of 24.1 kJ/mol H and DH ∗ Other values of ED calculated using different sets of experimental DH values and different correction procedures gave values from 23.5 to 24.3 kJ/mol H over the present temperature range. The extrapolated value from Fig. 7 is 23.9 kJ/mol H. It seems that it is difficult to give a more precise value than 23.9 ± 0.4 kJ/mol H (423–523 K). Fig. 7 shows ED values as a function of r for a 76 ␮m Pd membrane and it can be seen that they increase in a non-linear way at higher r values. If these are corrected using Eq. (11),

T.B. Flanagan et al. / Journal of Membrane Science 306 (2007) 66–74

73

If the RIS model is employed to evaluate ln [F(rup )/(rup )], we obtain  o,∗  

  g1 rup (1 − rup ) DH ED ln = (13) − ln − ln o DH RT 2RT rup o,∗ and the results in Table 5 for DH have been obtained from this equation (423 K) using experimental ED values t the rup shown. When rup increases, the increase of ED dominates over o leading to the observed decrease of D (r) the increase of DH H o increase significantly with rup . It is noteworthy that ED and ln DH with rup while ln DH itself decreases less.

∗ against H/Pd for Pd (76 ␮m) slopes. () experimental Fig. 7. Plot of ED and ED ∗. ED data; ( ) experimental data corrected using Eq. (11) to give ED

∗ values are obtained as shown. The average nearly constant ED ∗ value of these ED is 23.9 kJ/mol H, which is in good agreement with that predicted from constant EP (Fig. 4). Bohmholdt and Wicke [6] determined ED values at high and at low H contents (293–373 K) and found that at r → 0, ED = 23.4 kJ/mol H and at r = 0.74, ED = 24.1 kJ/mol H, i.e., the change is in the same direction as found here. o DH

3.7.1. Effect of the thermodynamic factor on o. Besides affecting ED , the thermodynamic factor affects DH o Bohmholdt and Wicke [6] found that DH is greater in the concentrated phase (r = 0.70) than in the very dilute phase, which is in the same direction as the trend found here. It o values range from can be seen from Table 2 that the DH o −3 about 3 to 6 × 10 . If the ln DH values are extrapolated o,∗ = 5.25 × 10−3 cm2 /s which agrees with the to rup → 0, DH −3 2 4.9 × 10 cm /s given by Koffler et al. [19] quite well and o,∗ the latter should correspond to DH because of the very low o,∗ r employed. The values of DH obtained from the Arrhenius ∗ to determine E∗ are from 5.1 to 6.3 × 10−3 cm2 /s, plots of DH D which are in reasonable agreement with those in Table 5. Eq. (12) can be derived from Eqs. (5) and (10)  o,∗  

 F (rup ) DH ED ln = (12) + ln o DH RT rup

3.7.2. H concentration profiles The role of solute non-ideality on experimental DH values obtained by fluxes through membranes has been considered by Barrer [26], Crank [27] and Jost [28] where the concentration profile is assumed to be known. They give procedures for the ∗. calculation of the thermodynamic factor and DH In the present work, the concentration profile is unknown but ∗ can be obtained by the thermodynamic factor is known and DH the methods given above. The concentration profile can then be determined from the following equation, which was derived from Eq. (6) and given in [14], F (rup ) − F (r) x = d F (rup )

(14)

where x is the distance through the membrane and d is the thickness. F(r) can be determined at each r from integration of plots of f(r) against r. Results are shown in Fig. 8 for rup = 0.050 (F(rup ) = 0.0375) and at 423 K. It can be seen that at all (x/d) except 0 and 1.0, the H concentration is smaller than the ideal value given by the linear relation between r and (x/d). The concentration profile depends upon rup ; for rup at large H contents where f(r) > 1.0, the deviations from the linear, ideal profile will be in the opposite direction from that in Fig. 8. In the steady-state the flux is the same at any x throughout the membrane, i.e., J = −DH (cH )(∂cH /∂x) must be constant at all x and corresponding r. These J values must also be equal to

∗ −E . where ED = ED D

Table 5 o DH and ED as a function of r for Pd (423 K) r

o DH (10−3 cm2 /s)

o,∗ DH (10−3 cm2 /s)

ED (kJ/mol H)

0.010 0.015 0.020 0.025 0.030

5.83 6.04 6.43 6.70 7.12

5.61 5.64 5.83 5.90 5.93

23.6 24.0 24.2 24.5 25.4

Fig. 8. Plot of H/Pd at various degrees of penetration across a Pd membrane at 423 K and rup = 0.050. () data calculated from Eq. (14); solid straight line, ideal behavior where f(r) = 1.0.

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T.B. Flanagan et al. / Journal of Membrane Science 306 (2007) 66–74

Table 6 Fluxes as a function of x for Pd at 423 K, rup = 0.05, d = 76 ␮m and ∗ = 6.4 × 10−6 (cm2 /s) where −J(exp) = 3.6 × 10−6 (mol H/s)cm−2 DH (x/d)

f(r) at x

(dcH /dx)/(mol H/cm−3 /cm)

−J (10−6 (mol H/s) cm−2 )

0 0.07 0.325 0.52 0.75

0.56 0.61 0.75 0.85 0.92

1.02 0.78 0.75 0.79 0.60

3.7 3.7 3.9 3.7 3.5

the measured steady-state value, −DH (exp) (cH,up )/d. The flux ∗ (c /RT )(∂μ (c )/∂x) which must is also given by J = −DH H H H equal –DH × cH (∂cH /∂x) and at each x, it must be the same as the experimental flux. The flux at each x is also given by   dcH ∗ J = −DH × f (r) . (15) dx The calculated fluxes are shown as a function of x (Table 6) are given to show the consistency of the results and the fluxes are generally close to the experimental J = –DH (exp) (cH,up )/d and reasonably constant demonstrating the validity of the analysis. 4. Conclusions The permeation of H2 through these Pd membranes has been shown from the linearity of J versus 1/d plots to be bulk diffusion controlled. It has been shown that oxidized and palladized Pd membranes give similar DH values but the unoxidized (polished) ones give smaller values. In this range of dilute concentrations DH was found to either not change appreciably or else decrease with r. The observed dependence of DH upon r in the dilute phase of Pd–H has been determined and explained in terms of the thermodynamic factor, which can be significant even at low H concentrations at, e.g., 423 K. Qualitatively, the present results for the change of DH with r agrees with the results of Salomons et al. [12]. ∗ from D (r) when There are several ways to determine DH H there is a large concentration profile through the membrane, i.e., cup cdown . This is a very common situation and pertinent to practical H2 purification. The results do not differ appreciably ∗ values determined from in the range examined here between DH Eq. (5) and those employing Eq. (1) with an average H content in the membrane rav = (rup /2). At higher H contents and for Pdrich alloys such as Pd0.77 Ag0.23 this will no longer be a good approximation. These appear to be the first systematic determination of the o,∗ changes of ED and DH with r for Pd–H and these increases

can be accounted for by the thermodynamic factor using the RIS model. The change of ED with r is significant even at low H contents. The concentration profile has been calculated for a given rup at 423 K and the H concentrations within the membrane are smaller than the ideal ones corresponding to a uniform gradient over the membrane for the value of rup selected in the dilute phase region where f(r) < 1.0. Acknowledgement We gratefully acknowledge support for this work from the Washington Savannah River Company under DOE Contract No. DE-AC09-96SR18500. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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