J. Phys. Chem. B 2009, 113, 9795–9801
9795
Pressure-Dependent Hydrogen Permeability Extended for Metal Membranes Not Obeying the Square-Root Law Shigeki Hara,* Misaki Ishitsuka, Hiroyuki Suda, Masakazu Mukaida, and Kenji Haraya Research Institute for InnoVation in Sustainable Chemistry, National Institute of AdVanced Industrial Science and Technology (AIST), Central 5, Tsukuba 305-8565, Japan ReceiVed: March 24, 2009; ReVised Manuscript ReceiVed: May 28, 2009
Hydrogen permeability of metal membranes is generally defined by the square-root law, as the proportional coefficient of permeation flux to the square-root difference of the pressures on both sides of the membrane. However, deviation from the law has been widely reported for palladium, niobium, etc. Although n-th power instead of the square root has often been employed to determine permeability for these membranes, it has no theoretical base. These approaches do not consider concentration dependency of hydrogen diffusivity in the membrane. This study theoretically extended the definition of permeability by taking it into account, where square root of pressure was used throughout. The resultant permeability depended on pressure. This approach had the following four characteristics. First, the permeability could be qualitatively linked with pressuredependent solution and diffusion coefficients. For this purpose, the solution coefficient was also extended from Sieverts’ law. Second, the permeability could be easily evaluated from permeation flux dependent on feed-side pressure, usually measured in membrane study. Third, this approach enabled comparison of permeation ability irrespective of obeying permeation law. Fourth, permeation flux could be estimated for any pressure conditions visually and analytically. Thus, analytically estimated values were more precise than those using the conventional square-root law. These characteristics are successfully demonstrated using experimental results obtained not only for a palladium membrane in this study but also for palladium and niobium membranes in the literature. 1. Introduction Dense palladium membranes are permeable only to hydrogen. This feature is attractive in hydrogen separation and purification processes not only in industries requiring hydrogen such as semiconductor manufacturing but also for power generation using fuel cells. From this point of view, various researches and developments on metal membranes have been intensively conducted to reduce the necessary amount of palladium, expensive and limited in resources.1-4 Hydrogen permeation flux of metal membranes, J (mol H2/ (m2 s)), is generally written by the square-root law:
J)
φ f (√p - √pp) l
DK 2
C ) K√p
(2)
* To whom correspondence should be addressed. Tel.: +81-29-861-9336. Fax: +81-29-861-4726. E-mail:
[email protected].
(3)
where C (mol H/m3) and p are hydrogen-atom concentration in themetal,andequilibriumhydrogenpressureinthehydrogen-metal system, respectively. This equation is equivalent to Henry’s law in low concentrations concerning the reaction, H2(gas) ) 2H(in metal):1
(γC)2 ) Keq p/p0
(1)
where pf and pp (Pa) are hydrogen pressures on both sides of the membrane, that is, feed side and permeation side, respectively, and l (m) is membrane thickness. The φ (mol H2/(m s Pa0.5)) is hydrogen permeability, widely used as a measure indicating hydrogen-permeation ability of the metal. This permeability is known to have the following relation with diffusion coefficient, D (m2/s), and solution coefficient, K (mol H/(m3 Pa0.5)):2
φ)
The solution coefficient, so-called Sieverts’ constant, is defined as the following Sieverts’ law:
(4)
where p0, γ, and Keq are standard pressure (101 325 Pa), activity coefficient, and equilibrium constant, respectively. The above discussion is based on constant D, K, and φ, independent of hydrogen pressure. However, concentration does not completely obey Sieverts’ law especially at high hydrogen activities because Henry’s law is a kind of approximation in the low-concentration region.5-9 It causes deviation from the square-root law for permeation. Hurlbert and Konecny10 reported that hydrogen permeation for palladium membranes thicker than 20 µm could be well described using the following power law with n ) 0.68:
J)
φ′ f n {(p ) - (pp)n} l
10.1021/jp9026767 CCC: $40.75 2009 American Chemical Society Published on Web 06/25/2009
(5)
9796
J. Phys. Chem. B, Vol. 113, No. 29, 2009
Hara et al.
where φ′ (mol H2/(m s Pan)) is a constant, known as permeability for the power law. Morreale et al.11 investigated palladium membranes 1 mm thick at high temperatures and pressures. They concluded that hydrogen permeation flux obeyed the power law with n ) 0.62. Zhang et al.12 gave n ) 0.52-0.59 for palladium membranes 51 µm or thicker. Itoh et al.13 analyzed the permeation behavior of amorphous Pd-Si membranes using n ) 0.229-0.468. Nambu et al.7 and Zhang et al.8 reported deviation from the square-root law for niobium membranes. As reviewed above, many studies used the power law to describe the permeation behavior not obeying the square-root law. However, the power law with n * 0.5 was based on no theory except for amorphous alloy. In other words, this law gave no qualitatiVe information on the relation among the permeability, solubility, and diffusivity. Therefore another approach was needed. Wang et al.14 proposed to approximate solution behavior in a certain pressure range by
C ) K' √p + R
(6)
where K′ and R are constants for approximation. The solution coefficient K′ could be related to diffusion coefficient and permeability evaluated using the square-root law. Nevertheless, each constant represented an average, depending on how to choose the pressure range. Therefore, this approach was still insufficient for qualitative comparison among membrane materials. Zhang et al.8 insisted that hydrogen permeability was not useful any more for such membranes, so that permeation flux along with feed- and permeation-side pressures should be provided as a measure indicating the membrane performance. Nevertheless, this approach could not estimate the flux for any other pressure conditions while the power law could do it. All the above approaches did not consider concentration dependency of hydrogen diffusivity in the membrane. In this study, therefore, another approach was proposed by taking it into account: the definition of permeability was extended using concentration-dependent or equilibrium-hydrogen-pressure-dependent diffusivity. Square-root of pressure was used throughout this approach. It provided permeability as a function of the square root of pressure. The pressure-dependent permeability had four characteristics. First, it could be qualitatiVely linked with solution and diffusion behavior. Second, it could be easily evaluated from permeation measurement. Third, it was comparable with those evaluated in different manners in literature. Fourth, permeation flux could be estimated from it Visually and precisely. These characteristics are derived from a theoretical point of view in sections 2.1, 2.2, 2.3, and 2.4, respectively. In the subsequent sections, they were demonstrated using experimental data: in the Results, the comparability, easy evaluation, and visual precise flux estimation were demonstrated in this order, using data obtained in this study; in the Discussion, the comparability and qualitative linkage were proved using data reported in literature. 2. Theory 2.1. Permeability as a Function of Hydrogen Pressure. The basic assumptions in this study are (i) a flat uniform membrane, (ii) steady state, and (iii) hydrogen-atom-diffusion controlling for permeation. Hydrogen-atom flux inside the membrane, N (mol H/(m2 s)), is given by one-dimensional Fick’s first law under the assumption (i):
N ) -D(C)
dC dx
(7)
where concentration dependency is considered for hydrogen diffusivity. The x (m) is located inside the membrane measured from the feed-side surface. The feed-side and permeation-side surfaces respectively correspond to x ) 0 and l. In the steady state, assumption (ii), the atom flux inside the membrane is independent of x. Therefore, the integral of eq 7 from x ) 0 to l becomes15
N)
1 l
∫CC D(C) dC f
p
(8)
where Cf and Cp are hydrogen-atom concentrations just under the surfaces of the feed and permeation sides, respectively. Because one hydrogen molecule is composed of two atoms, hydrogen-molecule flux through the membrane is equivalent to the half of hydrogen-atom flux, that is16
J)
1 2l
∫CC D(C) dC f
p
(9)
According to the diffusion controlling, assumption (iii), hydrogen atoms just under the membrane surface can be regarded to be in the thermodynamical equilibrium state with hydrogen molecules in the adjacent gas phase. Therefore, the concentrations, Cf and Cp, can be given as a function of hydrogen pressure, pf and pp, respectively. This equilibrium relation is well-known as a pressure-concentration-temperature (PCT) isotherm, experimentally determined by solubility measurement. To evolve this theory further, a variable q (Pa0.5) is introduced:
q ) √p
(10)
Equation 9 can be converted as follows:
J)
1 2l
dq ∫qq D(q) dC dq f
p
(11)
Now, the pressure-dependent solution coefficient, K(q) (mol H/(m3 Pa0.5)), is proposed (the “pressure-dependent” is equivalent to “q-dependent” throughout the text because pressure is one-to-one correspondent to q.). It is defined by the following form:
K(q) )
dC dq
(12)
This form is a natural extension of Sieverts’ law, eq 3, and Wang’s approximation, eq 6. Introducing differential overcomes the difficulty to meet both precise description and comparable permeability for membranes not obeying square-root law. Pressure-dependent permeability, φ(q), is defined similar to eq 2:
φ(q) )
D(q)K(q) 2
(13)
Hydrogen Permeability of Metal Membranes
J. Phys. Chem. B, Vol. 113, No. 29, 2009 9797
1 J ) (φqf - φqp) l
Using these definitions of the pressure-dependent K and φ, eq 11 can be simplified into the following form:
J)
1 l
∫qq φ(q) dq f
p
(14)
This equation is the key formula connecting the pressuredependent permeability to the permeation flux. 2.2. Evaluation of Pressure-Dependent Permeability from Permeation Tests. The above definition of permeability uses solution and diffusion behaviors. However, they are not always available. This section explains how to evaluate the permeability in practice from permeation tests. First, a primitive function, Φ(q), is introduced:
This form is exactly the same as eq 1, showing that the definition of pressure-dependent permeability is consistent with the conventional square-root law. The consistency is important in membrane material development to compare the pressuredependent permeability with constant permeability reported in literature. In other words, the resultant pressure-dependent permeability is useful as a measure indicating permeation ability. Meanwhile, substitution of eq 5 into eq 18 gives
φ(q) ) 2nφ′q2n-1 dΦ(q) φ(q) ) dq
(16)
Therefore
lim
qffqp
(20)
(15)
Now, the right side of eq 14 can be rewritten in the following form:
1 J ) {Φ(q)q)qf - Φ(q)q)qp} l
(19)
φ(q)q)qp J 1 Φ(q)q)qf - Φ(q)q)qp ) lim ) p f p f p l q fq l q -q q -q (17)
Using this formula, the permeability for the power law with n * 0.5 can be converted into pressure-dependent permeability, comparable to those evaluated in the square-root law. 2.4. Permeation Flux Estimation Using Pressure-Dependent Permeability. Permeation flux is practically more important than permeability. Equation 14 teaches that the permeation flux multiplied by l is given as the integrated area in a graph showing φ(q) against q, that is, the area surrounded by q ) qp and qf; φ(q) and the abscissa. If φ(q) is given as a polynomial of q, the flux can be estimated as a function of qf and qp, which is practically of great importance. Detailed procedure is described in section 4.3.
f
This equation shows how to practically evaluate the pressuredependent permeability: as the feed-side pressure approaches to the permeation-side pressure, the limit of the permeability conventionally defined by the square-root law gives the permeability value at this permeation-side pressure. Nevertheless, this procedure needs a series of permeation tests changing feed-side pressure for a constant permeation-side pressure; much work is necessary to obtain the permeability as a function of pressure. Now, eq 16 is differentiated with respect to qf taking into account that Φ(q)q)qp is independent of qf. The result is as follows:
φ(q)q)qf dJ ) f l dq
(18)
This form teaches that the pressure-dependent permeability can be evaluated by differentiating the permeation flux with respect to qf. This procedure saves much experimental effort because it requires just one series of permeation tests changing feed-side pressure for a certain constant permeation-side pressure. The consistency between two procedures, eqs 17 and 18, is experimentally proved in section 4.2. 2.3. Comparison with Other Definitions of Permeability. The above formulas about pressure-dependent permeability have been derived as a natural extension of the conventional relation among permeability, solubility, and diffusivity. However, is the resultant permeability comparable to those in the literature? If D and K are constant or independent of q, φ becomes constant according to eq 13. In this case, the right of eq 14 can be analytically integrated as follows:
3. Experimental Section Pure palladium foil 50 µm thick, purchased from Nilaco Co. (Japan), was used as a hydrogen permeable membrane. The membrane was annealed at 973 K for 5 h in an Ar-5% H2 atmosphere to reduce dislocations and point defects in the crystal structure. Then it was mounted in a hydrogen permeation cell. Operating conditions for hydrogen permeation tests are summarized in Table 1. Pure hydrogen was introduced on both sides of the membrane: one was feed side and the other was permeation side. Feed-side pressure was controlled in the range of 0.1-1.0 MPa (absolute pressure) while permeation-side pressure was fixed to a constant pressure during each series of permeation tests. Several series were carried out for different permeation-side pressures. Hydrogen flow rate in the outlet of the permeation side was measured by a mass flow meter, calibrated in prior using a soap flow meter. Permeation rate was estimated as the flow rate difference between the inlet and outlet of the permeation side. After these permeation tests, a 23% CO2-2% CO-75% H2 gas mixture was introduced in the feed side and the effluent from the permeation side was analyzed by gas chromatography to confirm no defect in the membrane. TABLE 1: Operating Conditions for Hydrogen Permeation Tests effective membrane area introducing gas pressure hydrogen introducing rate temperature
feed side permeation side feed side permeation side
2.98 cm2 pure hydrogen (99.99999%) 0.1-1.0 MPa (abs) 0.1-0.9 MPa (abs) 100 µmol/s 0-70 µmol/s 513-773 K
9798
J. Phys. Chem. B, Vol. 113, No. 29, 2009
Hara et al.
Figure 1. Hydrogen permeation flux as a function of square-root pressure difference for pp ) 0.116 MPa at different temperatures. Figure 3. Pressure-dependent hydrogen permeability. Horizontal lines are estimated in the conventional square-root law.
According to eq 17, permeability at a certain permeationside pressure, pp, can be read as the slope at the cross point to the horizontal line indicating no flux, where pf is coincident with pp. For example, permeability at a permeation-side pressure of 0.502 MPa, or qp ) 709 Pa0.5, is determined as follows: the flux data for a permeation-side pressure of 0.502 MPa can be fitted by the following third-order polynomial of the square root of feed-side pressure:
Figure 2. Hydrogen permeation flux as a function of the square root of feed-side pressure for different constant pp.
4. Results 4.1. Comparability with the Permeability in the SquareRoot Law. Hydrogen permeation flux of the palladium membrane is shown in Figure 1. The solid lines are linear through the origin. The flux is nearly proportional to the difference of the square roots of hydrogen pressures on both sides. For example, permeability derived in the conventional square-root law is 9.59 nmol H2/(m s Pa0.5) for 573 K. However, the data align not completely on the linear lines. The flux for a constant temperature tends to be concave, which is clearer at lower temperatures. Similar trends can be seen also for thin membranes where surface reaction kinetics is supposed to control the overall permeation process.17-21 Nevertheless, the surface kinetics is not considered in this study for the following reasons. First, the membrane in this study is 50 µm thick. The surface kinetics is thought to become rate-limiting for several micrometers or thinner.10,17-21 Second, this trend is widely reported also for thicker membranes as mentioned in the Introduction. Third, reevaluated permeability is consistent with those for 1 mm thick palladium membranes reported by Morreale et al. revealed in section 5.1. Finally, the theory developed under the diffusion controlling, assumption (iii), can describe the permeation behavior consistently as demonstrated below. To investigate the pressure dependency in detail, permeation flux at 573 K is depicted as a function of the square root of feed-side pressure for different permeation-side pressures from 0.116-0.902 MPa in Figure 2. It should be noted that this figure uses not the square root difference as the abscissa. The plotted data include those measured at various feed pressures from 1.0 to 0.1 MPa not only in the descending order but also in the ascending order. Data measured in the descending order are almost agreement with those in the ascending order, suggesting the high reproducibility and reliability of the measurement.
J ) 2.98 × 10-11(qf)3 - 1.48 × 10-8(qf)2 + 1.75 × 10-4qf - 1.27 × 10-1
(21)
Permeability is calculated as follows:
lJ q - 709 ) lim [1.49 × 10-15(709 + ∆q)3 - 7.40 × 10-13 ×
φ(709) ) lim
f
qff709
∆qf0
(709 + ∆q)2 + 8.75 × 10-9(709 + ∆q) ) 4.47 × 10
-15
-9
) 9.97 × 10
6.35 × 10-6]/∆q × 709 - 1.48 × 10 × 709 + -12
2
0.5)
8.75 × 10-9
mol H2 / (m s Pa
(22) In the same manner, permeability values for different pressures and temperatures are evaluated. The results are indicated as various keys in Figure 3. It is worthwhile mentioning the abscissa, which is square root of pressure instead of square root of feed pressure. Permeability determined by eq 22 is not only for qf but also for qp because they are equivalent (709 Pa0.5) at the limit. Therefore, the resultant permeability can be written as a function of q irrespective of feed side or permeation side. This description is consistent with the form of eq 13, where permeability is defined using q-dependent diffusion and solution coefficients. As a result, permeability has to be written as a function of q not designating feed or permeation side as well. Permeability is clearly shown to increase with pressure. The horizontal solid lines represent conventional permeability estimated from the linear lines in Figure 1. The conventional permeability is found to be a kind of average of the pressuredependent permeability, showing the comparability between two permeabilities evaluated in two different manners.
Hydrogen Permeability of Metal Membranes
J. Phys. Chem. B, Vol. 113, No. 29, 2009 9799
φ(q) ) aq2 + bq + c
(23)
The constants determined in this manner, a, b, and c, are summarized in Table 2. The resultant pressure-dependent permeability is drawn by solid curves in Figure 3. The curves represent the permeability at any pressure in the pressure range focused in this study. To determine these constants, all the available data are employed for higher reliability. However, just one series for a given permeation-side pressure is usually sufficient because every series has the same shape as explained above, resulting in the same permeability. Thus, the necessary experiments can be reduced. In the Discussion, just one series is used. 4.3. Visual and Precise Estimation of Permeation Flux. Permeation flux can be estimated using pressure-dependent permeability according to eq 14. Once permeability has been determined in the form of a second-order polynomial, the flux can be analytically obtained: Figure 4. Hydrogen permeation flux (a) shifted vertically and (b) shifted horizontally from Figure 2. Key shape represents permeationside pressure as shown in Figure 2.
TABLE 2: Constants in Eq 23 of Pressure-Dependent Permeability of Palladium for Different Temperatures temp (K)
-15
a (10 mol H2/(m s Pa1.5))
-12
-9
b (10 mol H2/(m s Pa))
c (10 mol H2/(m s Pa0.5))
513 533 553 573 623 673 723 773
29.7 5.99 2.97 3.81 0.236 2.63 6.00 4.37
-2.81 -2.10 0.877 -0.383 4.01 0.0825 -5.06 -2.39
13.1 7.08 7.17 8.40 9.23 13.6 18.5 20.6
4.2. Easy Evaluation of Pressure-Dependent Permeability. This section explains that the permeability can be easily evaluated as a function of pressure. That is, the consistency between two forms to determine φ(q), eqs 17 and 18, is experimentally demonstrated. Once the consistency is proved, eq 18 may be used for the evaluation to reduce the necessary experiments. Five series of the permeation flux in Figure 2 have a common feature in shape. To show it clearly, each series is shifted vertically or horizontally, depicted in Figure 4, a and b, respectively. All the flux data shifted vertically in Figure 4a are on a certain curve, meaning that the slope of each series, (dJ)/(dqf), is independent of permeation-side pressure: eq 17 tells that the slope at qf ) qp gives permeability divided by membrane thickness, but qf ) qp is not essential. That is why eq 18 using a constant permeation-side pressure gives permeability also for pressures different from the permeation-side pressure. The consistency between eqs 17 and 18 has been thus proved experimentally. These series of flux data do not agree with any other shift. For example, these flux data are not on a line in Figure 4b, indicating that the slopes at the same permeation flux have no direct relation each other. Because all the shifted flux data in Figure 4a align on a curve with a slight curvature, they can be well fitted by a polynomial, e.g., a third-order polynomial. By differentiating it, permeability is derived in the following second-order polynomial:
J)
1 a f3 b {(q ) - (qp)3} + {(qf)2 - (qp)2} + c(qf - qp) l 3 2 (24)
[
]
This formula enables estimation of the flux for any set of feedside and permeation-side pressures around pressures examined to determine these coefficients, a, b, and c. Meanwhile, permeation flux is visualized as mentioned in the Theory section: it is proportional to the integrated area in Figure 3, depicting pressure-dependent permeability against the square root of pressure. As examples, two areas are hatched: one corresponds to the flux from 0.301 MPa feed side to 0.101 MPa permeation side, and the other from 0.902 to 0.608 MPa. The area is wider for the former than for the latter though the pressure difference is smaller for the former, 0.200 MPa, than for the latter, 0.294 MPa. Thus, it is visually found that permeation flux is higher for the former condition. The areas can be estimated as 41.4 and 37.2 mmol H2/(m2 s) using eq 24. These values agree within a few percent with those experimentally obtained, indicating the validity of this approach. The comparison is summarized in Table 3. Permeation flux can be conventionally anticipated using the constant permeability. However, the resultant values differ from the experimental results by around 10%. Although pressure-dependent permeability might be apparently complicated, the flux can be visually recognized and precisely estimated. 5. Discussion The approach described above is easily applicable to other membranes because it requires only permeation flux as a TABLE 3: Comparison of Hydrogen Permeation Flux Obtained Experimentally, Estimated Using Constant Permeability (9.59 nmol/(m s Pa0.5)), and Estimated Using Pressure-Dependent Permeability with the Values in Table 2 (573 K, Two Pressure Conditions) permeation flux mmol H2/(m2 s) (diff from the exptl value)
experimental using constant permeability using pressuredependent permeability
from 0.301 to 0.101 MPa
from 0.902 to 0.608 MPa
40.6 44.3 (+9.1%)
37.0 32.6 (-11.9%)
41.4 (+2.0%)
37.2 (+0.5%)
9800
J. Phys. Chem. B, Vol. 113, No. 29, 2009
Figure 5. Comparison of hydrogen permeability among Hurlbert and Konecny10 (squares), Morreale et al.11 (triangles), and this study (circles). Gray, white, and black symbols are respectively estimated using the power law with n * 0.5 in the literature, reevaluated using polynomials for 0.1 MPa, and converted using eq 20 for 0.1 MPa.
Hara et al.
Figure 7. Hydrogen permeation flux of a niobium membrane against the square root of feed-side pressure (closed squares), and hydrogen concentration in niobium against the square root of equilibrium pressure (open squares).8 Solid lines are polynomials fitted to these experimental data.
to eq 20. For example, φ(q) ) 1.24φ′q0.24 is given for n ) 0.62. The permeation behavior can be written explicitly using pressure itself by the form
J≈
Figure 6. Fitted third-order polynomials as well as permeation flux data for 1 mm thick palladium membranes reported by Morreale et al.11
function of feed-side pressure for a constant permeation-side pressure, usually examined in membrane study. Some applications to the data in the literature are presented below. 5.1. Comparison of Pressure-Dependent Permeability with the Literature. Morreale et al.11 examined 1 mm thick palladium membranes, where hydrogen atom diffusion in the membranes should control permeation rate. They concluded that the permeation flux was described using the power law with n ) 0.62. Although the power could reproduce the experimental permeation flux, the resultant permeability, φ′ (mol H2/(m s Pa0.62)), was low and not comparable with conventional square-root permeability determined with n ) 0.5 in most literature. Meanwhile, Hurbert and Konecny10 studied membranes with different thicknesses. The results showed that hydrogen permeation through membranes thicker than 20 µm was controlled by hydrogen atom diffusion in the membranes and the permeation could be described using the power law with n ) 0.68. These permeabilities using n * 0.5 are portrayed with gray symbols in Figure 5, which are nearly 1 order of magnitude lower than those evaluated in this study. Therefore, permeability is reevaluated according to the approach in this study using the data reported in the literature. As an example, the fitted polynomials as well as the data reported by Morreale et al. are shown in Figure 6. Thus, reevaluated permeability for 0.1 MPa is indicated with white symbols in Figure 5. All the data evaluated by the new approach are comparable, suggesting its wide applicability. Furthermore, pressure-dependent permeability for the membranes obeying the power law can be also evaluated according
1.24φ′p0.12 f (√p - √pp) l
(25)
where p is a value between pf and pp. The permeability at 0.1 MPa evaluated using eq 20 is depicted with black symbols in Figure 5. The consistency with those evaluated using polynomials is proved. As explained above, the comparability has again been demonstrated: the pressure-dependent permeability proposed in this study enables comparison with different permeation behaviors obeying not only the conventional square-root law but also the power law. 5.2. Qualitative Linkage among Permeability, Solubility, and Diffusivity. Zhang et al.8 investigated a palladium-coated niobium membrane around 0.5 mm thick. They showed that hydrogen permeation flux was not proportional to the squareroot difference of the hydrogen pressures on both sides of the membrane because of deviation from Sieverts’ law. Therefore, they estimated hydrogen diffusion coefficient not using Sieverts’ constant but using equilibrium concentrations for 0.03 and 0.01 MPa, and permeation flux from 0.03 MPa feed side to 0.01 MPa permeation side. In this section, the diffusion coefficient is reevaluated by applying the new approach to demonstrate the remaining characteristic or qualitative linkage using the experimental data. Hydrogen permeation and solution data in the literature are portrayed in Figure 7, where the units of the variables have been modified for the subsequent procedure from the original. Unfortunately, only three permeation-flux data are available, so that a second-order polynomial is adopted to represent the permeation behavior as the simplest function describing the nonlinearity. The function passing through these data is determined as follows:
J ) 2.68 × 10-6(qf)2 - 3.67 × 10-4qf + 9.90 × 10-3 (26) Hydrogen concentration is fitted by the following third-order polynomial:
C(q) ) -2.22 × 10-3q3 + 3.70 × 10-1q2 + 44.8q
(27)
Hydrogen Permeability of Metal Membranes
J. Phys. Chem. B, Vol. 113, No. 29, 2009 9801
These two functions are drawn by solid lines in the figure. Either reproduces the experimental data well. Applying these functions to eqs 18 and 12, respectively, pressure-dependent permeability and solution coefficient are deduced as shown below:
φ(q) ) 2.68 × 10-9q - 1.84 × 10-7 K(q) ) -6.66 × 10-4q2 + 7.40 × 10-1q + 44.8
(28)
(29)
Diffusion coefficient can be now calculated using the relation among permeability, diffusivity, and solubility, that is, eq 13. The following form is derived as a result:
D(q) ) -8.05 × 10-6
(
)
q - 68.7 q2 - 1.11 × 103q - 6.73 × 104 (30)
For instance, diffusion coefficients 1.50, 2.87, and 3.67 × 10-9 m2/s are calculated for 0.01, 0.02, and 0.03 MPa, respectively. Their average is practically consistent with 2.95 × 10-9 m2/s reported in the literature, suggesting that the relation among three variables, eq 13, is reasonable. Thus, the first characteristic or qualitative linkage has been proved using experimental data. Furthermore, it should be noted that this approach gives the diffusion coefficient as a function of pressure while Zhang et al. determined just one value for the diffusion coefficient. 6. Conclusions This study extended the definition of hydrogen permeability for metal membranes not obeying the square-root law, where the square root was fixed for hydrogen pressure and concentration dependency was considered for diffusion coefficient instead. The extended definition was written in the three mutually consistent forms, eqs 13, 17, and 18. The resultant permeability depended on pressure. This approach had the following four characteristics, which were derived theoretically and proved also using experimental data. First, the permeability could be qualitatiVely linked with pressure-dependent solution and diffusion coefficients. In other words, this approach was useful to obtain diffusion coefficient as a function of equilibrium hydrogen pressure from solution and permeation behaviors, as presented in section 5.2. For this purpose, the definition of solution coefficient was also extended from Sieverts’ law. Second, the pressure-dependent permeability could be easily evaluated from permeation flux dependent on
feed-side pressure for a constant permeation-side pressure, usually measured in membrane study, as demonstrated in section 4.2. Third, the resultant permeability was comparable to those evaluated according to the conventional square-root law demonstrated in section 4.1. Furthermore, this approach enabled comparison of permeability determined by the power law with different powers as described in section 5.1. Fourth, permeation flux could be Visually and analytically estimated for any pressure conditions using the pressure-dependent permeability as explained in section 4.3. Thus, estimated flux was more precise than that using the conventional square-root law. This study has assumed a flat uniform membrane. However, most state-of-the-art membranes are asymmetric membranes including thin palladium membranes on porous ceramic supports. To analyze their permeation behavior, this approach is insufficient. Improved approaches will be presented elsewhere in the future. References and Notes (1) Shu, J.; Grandjean, B. P. A.; Van Neste, A.; Kaliaguine, S. Can. J. Chem. Eng. 1991, 69, 1036. (2) Buxbaum, R. E.; Marker, T. L. J. Membr. Sci. 1993, 85, 29. (3) Dolan, M. D.; Dave, N. C.; Ilyushechkin, A. Y.; Morpeth, L. D.; McLennan, K. G. J. Membr. Sci. 2006, 285, 30. (4) Phair, J. W.; Donelson, R. Ind. Eng. Chem. Res. 2006, 45, 5657. (5) de Rosset, A. J. Ind. Eng. Chem. 1960, 52, 525. (6) Itoh, N.; Xu, W.-C.; Hara, S.; Kimura, H.-M.; Masumoto, T. J. Membr. Sci. 1997, 126, 41. (7) Nambu, T.; Shimizu, N.; Ezaki, H.; Yukawa, H.; Morinaga, M. J. Jpn. Inst. Metals 2005, 69, 841. (8) Zhang, G. X.; Yukawa, H.; Watanabe, N.; Saito, Y.; Fukaya, H.; Morinaga, M.; Nambu, T.; Matsumoto, Y. Int. J. Hydrogen Energy 2008, 33, 4419. (9) Hara, S.; Huang, H.-X.; Ishitsuka, M.; Mukaida, M.; Haraya, K.; Itoh, N.; Kita, K.; Kato, K. J. Alloys Compd. 2008, 458, 307. (10) Hurlbert, R. C.; Konecny, J. O. J. Chem. Phys. 1961, 34, 655. (11) Morreale, B. D.; Ciocco, M. V.; Enick, R. M.; Morsi, B. I.; Howard, B. H.; Cugini, A. V.; Rothenberger, K. S. J. Membr. Sci. 2003, 212, 87. (12) Zhang, Y.; Lu, J.; Maeda, R.; Nishimura, C. Mater. Trans. 2008, 49, 754. (13) Itoh, N; Xu, W.-C.; Hara, S.; Kimura, H.-M.; Masumoto, T. J. Membr. Sci. 1998, 139, 29. (14) Wang, W.-L.; Ishikawa, K.; Aoki, K. Collected Abstracts of the 2009 Spring Meeting of the Japan Institute of Metals (Japanese) 2009; p 78. (15) Crank, J. The mathematics of diffusion; Oxford University press: London, 1975; Chapter 4, p 44. (16) Hara, S.; Huang, H.-X.; Ishitsuka, M.; Itoh, N.; Kita, K.; Kato, K. Proc. ICIM 8 2004, 463. (17) Morooka, S.; Yan, S.; Yokoyama, S.; Kusakabe, K. Sep. Sci. Technol. 1995, 30, 2877. (18) Li, A.; Liang, W.; Hughes, R. Catal. Today 2000, 56, 45. (19) Dittmeyer, R.; Ho¨llein, V.; Daub, K. J. Mol. Catal. A: Chem. 2001, 173, 135. (20) Thoen, P. M.; Roa, F.; Way, J. D. Desalination 2006, 193, 224. (21) Li, H.; Xu, H.-Y.; Li, W.-Z. J. Membr. Sci. 2008, 324, 44.
JP9026767
