- Email: [email protected]
Quantitative contribution of non-ideal permeability under diffusion-controlled hydrogen permeation through Pd-membranes
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Quantitative contribution of non-ideal permeability under diffusion-controlled hydrogen permeation through Pd-membranes Alessio Caravella, Nobuo Hara, Hideyuki Negishi, Shigeki Hara* National Institute of Advanced Industrial Science and Technology (AIST), Research Institute for Innovation in Sustainable Chemistry, Central 5, Higashi 1-1-1, Tsukuba 305-8565, Japan
article info
abstract
Article history:
In this paper, a powerful and simple method to evaluate quantitatively the influence of the
Received 31 July 2013
non-ideal contribution of the hydrogen permeation under diffusion control is provided. For
Received in revised form
this purpose, a Non-Ideality Index (a) is defined as the ratio of the non-ideal permeability
16 September 2013
term and the overall permeability, this leading to a direct quantification of the non-ideal
Accepted 2 October 2013
effect in membrane. The used methodology is described in detail by using experimental data of a real Pd-
Available online 2 November 2013
membrane. From these data, the parameters that characterise permeability as a function Keywords:
of temperature and pressure e i.e., intrinsic permeability (b) and non-ideal factor (a) e are
Non-ideality index
evaluated, highlighting their higher suitability in describing the membrane behaviour with
Non-ideal diffusion
respect to the usual empirical parameters used by the majority of researchers (i.e., pressure
Hydrogen
exponent and corresponding permeability). It is underlined that the proposed approach is
Palladium membranes
particularly effective for modelling membrane contactors and reactors, where the empir-
Sieverts law
ical Sieverts-type law cannot be used because the large difference of pressure and temperature could lead the membrane behaviour to dramatically change even at different locations of the same device. Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
The advantages of applying Pd-based membranes to hydrogen purification and production processes do not need to be listed anymore, they being extensively reported in the literature (see, as an instance [1]). The infinite selectivity of these membranes to hydrogen has attracted the interest of a number of research groups, who have investigated several different aspects concerning the hydrogen-metal lattice interactions.
When hydrogen internal diffusion controls the overall permeation process and, at the same time, the PdeH system is close to conditions of infinite dilution (referred to as “ideal case”), the permeating flux J [molH2 m2 s1] can be satisfactorily described by the Sieverts law (Eq. (1)): i h 0:5 J ¼ pðSievÞ p0:5 f pp pðSievÞ ¼
fðSievÞ d
* Corresponding author. Tel.: þ81 29 861 9336. E-mail address: [email protected] (S. Hara). 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.10.015
(1)
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
where pf and pp indicate the hydrogen partial pressure in feed and permeate side, whereas p(Siev) [molH2 m2 s1 Pa0.5] and f(Siev) [molH2 m1 s1 Pa0.5] are the membrane permeance and permeability to hydrogen, the latter calculated from the former through the membrane thickness d. However, in most cases, ideal conditions are far from real situations, where deviations from the Sieverts law e i.e., pressure exponent n (Eq. (2)) different from 0.5 e have been found in several experimental and theoretical studies (see, for example [2e11],). i h H2 Flux ¼ pðnÞ pnf pnp
(2)
The interesting fact is that such deviations were observed not only in the presence of external mass transfer resistance and/or in the presence of slow surface phenomena (adsorption/desorption), but also for very thick membranes [2,3], where it is undeniable that the diffusion in the lattice controls the permeation. All these studies highlighted the necessity to investigate the dependence of permeability on hydrogen concentration. To do that, Flanagan and Wang described a methodology to evaluate both the Einstein hydrogen diffusivity (i.e., ideal diffusivity) and the non-ideal correction from experimental isotherms [8,10,11]. A different approach is used by Hara and co-workers, who developed a general methodology to describe any behaviour of a metal membrane in both ideal and non-ideal conditions under diffusion-controlled permeation [6,9]. Once a suitable analytical form for permeability and solubility are chosen, such powerful and relatively easy methodology allows evaluating hydrogen diffusion coefficient, solubility and mobility through a metal lattice (not necessarily restricted to just palladium alloys) as functions of temperature and pressure. However, to the Authors’ knowledge, this approach has not been used yet by researchers, who mostly prefer using Eq. (2) to characterise the membrane behaviour. The problem with Eq. (2) is that the empirical pressure exponent n changes with both pressure and temperature, and, thus, is not suitable to be used in a wide range of operating conditions. Furthermore, it is not a direct measure of any physical phenomenon related to transport in membrane, its value being, on the contrary, the overall result of the number of different elementary permeation mechanisms. Therefore, no specific information can be
4677
obtained from pressure exponent, especially concerning the non-ideal membrane behaviour. In this context, the aim of the present paper is to provide a different and systematic way to measure directly and quantitatively the non-ideal contribution to permeation under diffusion control. For this purpose, the above mentioned methodology [6,9] is applied to experimental data from permeation tests performed on a real Pd-membrane, describing in detail the scientific and technical aspects necessary for its application. By doing this, we also aim at promoting a further membrane development by spreading this approach over in order for researchers to use it for a more complete membrane characterisation.
2.
Experimental setup
The experimental system is schematically shown in Fig. 1. In particular, a self-supported20 mm-thick Pd-membrane is tested under pure hydrogen atmosphere on both membrane sides, where pressure profiles and relative flux are generated and measured using the here-introduced “alternative step method” (see Section 4.1 for details), which is performed by means of two digital back-pressure regulators. Flow rates of 100 mL min1 (20 C, 1 atm) are fed at the inlet of both feed and permeate side, and the permeating flux is measured by means of two digital mass flow metres as the difference between the flow rates of inlet and outlet divided by the membrane area (2.98 cm2). The presence of possible leaks are monitored by means of three different checks. The first check is performed outside the oven at room temperature once finished assembling the permeation cell. For this purpose, argon gas is used raising the absolute feed pressure up to 600 kPa keeping it for 5 min and the permeate side at atmospheric pressure. The second check is performed at 300 C after inserting the permeation cell in the oven and before starting the permeation test, using the same conditions as those in the first check. The third check is made after each pressure cycle performed at constant temperature by switching the hydrogen in the feed with Ar gas kept at 900 kPa for 5 min.
3.
Theoretical background [6,9]
As mentioned above, the methodology that the present study is based on was already introduced in Hara et al. (2009). It is based on the definition of an integral form for permeation flux (Eq. (3)), extending in this way a constant permeability to a pressure-dependent permeability. Zqf J ¼ 1d
fðqÞdq qp
(3)
pffiffiffi qh p
One of the practical forms for permeability is proposed in the previous reports as: Fig. 1 e Sketch of the experimental system. The set of the investigated temperatures are also reported.
fðqÞhb þ a$q2
(4)
4678
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
In Eq. (4), permeability is shown to include two additive terms: i) the intrinsic permeability b [molH2 m1 s1 Pa0.5] and ii) a non-ideal term, whose functionality has been chosen to be the product of pressure (q2 [Pa]) by the non-ideal factor a [molH2 m1 s1 Pa1.5]. Both a and b are function of temperature only. This form (polynomial in q) is particularly convenient because allows an analytical expression of hydrogen diffusion coefficient, solubility and mobility. However, it must be underlined that just permeability data are not sufficient to derive data of diffusion coefficient, solubility and mobility, because two independent pieces of information are needed to characterise the membrane behaviour. For this purpose, Hara et al. (2012) clearly showed that both permeability and solubility are sufficient to derive diffusion coefficient and mobility. Other approaches using two different independent parameters are also possible. Using Eq. (4) in Eq. (3), the flux expression is evaluated as shown in Eq. (5), where it is reported to be proportional to the difference between two third-order polynomials with the same coefficients (without the second-order term) written for feed and permeate side, respectively. J¼
1 h a a i bqf þ q3f bqp þ q3p d 3 3
(5)
Dually, the permeability expression can be determined from flux by inverting Eq. (3) (Eq. (6)): fðqÞ ¼ d
vJ vqf qp ¼q qf
4.
Results and discussion
The whole procedure of the permeability parameters evaluation can be summarised through the following steps, which have to be performed for each temperature: 1. Obtaining {flux vs. feed pressure square root} permeation data for fixed permeate pressures, which generally generates several data series (in this study, just two series, named Series 1 and Series 2, see Fig. 2). 2. Obtaining a unique overall data series by overlapping all the original series, this consisting in performing the shifting procedure described in Section 4.3. 3. Fitting the obtained overall data series by using Eq. (5) to evaluate a and b. The next sub-sections explain the details of these steps.
4.1.
Permeation data: alternative step method
(6)
The flux expression reported in Eq. (5) is important, as it is the actual functionality from which to start the membrane
Fig. 2 e Alternative step method to evaluate flux as function of feed and permeate pressure. T [ 300 C.
characterisation. In fact, the experimental data obtained from a permeation test most commonly are of the type “flux vs. feed pressure square root” for a constant permeate pressure. The next section provides the description of the calculation procedure of a and b, as well as a way to evaluate the non-ideal contribution through the Non-Ideality Index a (Section 4.5).
As said before, the first step is to obtain experimental data of flux vs. feed pressure square root for a constant reference value of permeate pressure. To do that, the here-named “Alternative Step Method” is used for each temperature considered. This method consists in measuring the permeation flux in conditions where the pressure of both membrane sides is changed alternatively according to the profiles depicted in Fig. 2a. In particular, the pressure program starts setting a feed and permeate pressure of 150 and 120 kPa, respectively. After 22 min, the permeate pressure is increased up to 200 kPa by keeping the feed pressure at 150 kPa. Then, after another 22 min, the feed pressure is increased up to 250 kPa by keeping the permeate pressure at 200 kPa. This procedure, corresponding to the Series 1 in Fig. 2a, is repeated up to a maximum feed and permeate pressure of 950 and 980 kPa, respectively. After these values, pressure starts decreasing in an analogous way (Series 2). The result of this procedure is that, for each couple of pressure values, a pressure step of 50 kPa is determined (that is the reason for the method name), except at the beginning, at the intermediate time and at the end, where also feed pressure below the atmospheric pressure is set to enlarge the pressure range investigated. The permeating flux changes accordingly (Fig. 2b), becoming alternatively positive (from feed to permeate side) and negative (from permeate to feed side). This method allows investigating the effect of feed and permeate pressure at once without the need of performing the same feed pressure profile for each permeate pressure. Furthermore, it is very useful for measurements involving very thin self-supported membranes, because it avoids using
4679
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
large pressure differences, which could damage/break the tested membranes. The next section describes how to obtain the required {qf, J} data series for a fixed reference qp from the data shown in Fig. 2 to perform the evaluation of a and b.
4.2.
Data manipulation for permeability evaluation
To evaluate the intrinsic permeability b and the non-ideal factor a, a suitable data series has to be obtained from the permeation data shown in Fig. 2. For this purpose, flux has to be evaluated as a function of the feed pressure square root at a constant reference permeate pressure. To do that, let us consider that, according to the integral form reported in Eq. (3), the flux generated by the difference between the reference qr and a generic qf can be split using an intermediate value q1 into two terms as reported in Eq. (7). Zqf J ¼ 1d 2 6 ¼ 1d 4
fðqÞdq qr
Zqf
Zq1 fðqÞdq þ qr
3
(7)
7 fðqÞdq5
q1
Generalising Eq. (7) by considering several intermediate pressures, the flux “between” qr and the generic qi can be split into the summation of several flux terms (Eq. (8)).
Jjqqir ¼ ¼ 1d
i P k¼r
i P k¼r
Jjqqkþ1 k (8)
Zqkþ1 fðT; qÞdq qk
Eq. (8) is useful because the flux data required in the summation are directly available from the experimental data obtained from alternative step method (Fig. 2). Therefore, performing this calculation for all the qi values, the wished data series e consisting of m couples of {flux vs. pressure square root} data for the fixed value qr e is obtained (Eq. (9)). n o q0 ; Jjqq0r ; .; qm ; Jjqqmr
(9)
It must be underlined that the parameters evaluation is independent of the particular choice of the reference permeate pressure, because the shift does not change the shape of the curves, but just their position in the plot. In this specific study, a reference pressure of 120 kPa is selected and the m couples depicted in Fig. 3a for the 300 C case are obtained. As anticipated above, the experimental data are divided into two series e named Series 1 and Series 2 e which correspond to increasing and decreasing pressure steps, respectively. To evaluate the permeation parameters (a and b), the original experimental data are modified by shifting Series 2 on Series 1 (Fig. 3b) according to the procedure described in the next section. The shifting procedure is necessary to maximise the calculation precision, minimising in this way the effect of the experimental error. Then, all the shifted data are fitted by using Eq. (5), obtaining in this way the desired parameters a and b, whose values are explicitly shown in Section 4.4.
4.3.
Details of the shifting strategy
Let us consider a number of curves to be overlapped. The here-adopted shifting strategy consists in setting one curve as a reference and shifting all the others on it. Because of the experimental errors, there will be a certain difference between the points of the shifted curves and those of the reference one. The aim of this procedure is minimising such a difference. For this purpose, all curves are preliminary fitted by a suitable functionality. In this work, the polynomial form is chosen to obtain a simple analytical expression of the shifting values (see later Eq. (14)). It must be underlined that, in principle, forms other than polynomials can be also selected to perform the shift. In all cases, however, the fitting determination coefficient R2 should be almost the same for all forms to assure the shifting independence of the particular form chosen. In this study, our experimental data are satisfactorily described by third order-polynomials (Eq. (10)), which, thus, have been chosen for the shifting procedure. yr ðxÞ ¼ ar0 þ ar1 x þ ar2 x2 þ ar3 x3 ¼ Fig. 3 e a) Original and b) Shifted Flux as a function of feed pressure at a fixed permeate pressure (120 kPa) evaluated from the data reported in Fig. 2 using Eq. (8). T [ 300 C.
yj ðxÞ ¼ aj0 þ aj1 x þ aj2 x2 þ aj3 x3 ¼ xhqf ;
j ¼ 1.nC
3 P
k¼0 3 P
ark xk
ajk xk
k¼0
(10)
4680
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
In these expressions, yr(x) represents the reference curve, whereas the other curves are indicated by the generic subscript ’j’. Once the polynomial coefficients are calculated by fitting, each curve has to be shifted on the reference one by a constant value cj, different for each curve. By adding the corresponding value of cj to each original curve, all the shifted curves (y*) are determined (Eq. (11)). yj ¼ yj þ cj ¼
3 X
ajk xk þ cj
(11)
k¼0
The problem consists in the evaluation of cj. For this purpose, the here-selected shifting criterion is to achieve the minimum difference between the area underlying each curve to shift (Aj*) and the area underlying the reference one (Ar, Eq. (12)). This criterion is necessary and sufficient for two curves to overlap and is equivalent to the condition of minimum distance between the corresponding points of the curves. Zxb Ar ¼
yr ðxÞdx ¼ xa Zxb
Aj ¼ xa
Zxb X 3
ark xk dx
k¼0
yj ðxÞdx
xa Zxb "
¼ xa
3 X
#
(12)
ajk xk þ cj dx
k¼0
The mathematical formulation of the minimisation problem is formalised in Eq. (13). h 2 i h 2 i v DAj min DAj 5 ¼0 vcj fcj g 2 2 DAj ¼ Ar Aj
(13)
Since in general two or more curves could be obtained within different pressure ranges, the integral extremes in Eq. (12) (xa and xb) should be selected in such a way to provide a common range to all curves. Finally, developing Eq. (13) coupled with Eq. (12), the shifting quantity cj is determined for each curve after some algebraic steps (Eq. (14)). cj ¼
3 X ark ajk ðxb xa Þk kþ1 k¼0
(14)
Let us observe that the value of c0 (trivial case of reference curve) is, as expected, zero. As already commented, the advantage of using a polynomial form lies in the simplicity of Eq. (14), which can be also easily extended to higher orders, if required. Since in this study we use just two data series (Series 1 and Series 2) to obtain the permeation parameters, just one value of cj (c1) is needed for each temperature (Eq. (15)). c1 ¼
4.4.
3 X ðark a1k Þ ðxb xa Þk kþ1 k¼0
Fig. 4 e Overall permeability as a function of the pressure square root for different temperatures. The dashed-lines indicate the ideal membrane behaviour.
(15)
Pressure-dependent permeability
The application of the above-described procedures leads to the results reported in this section. Fig. 4 shows the behaviour of the overall permeability (Eq. (4)) as a function of pressure square root. As it is possible to observe, permeability increases
with increasing pressure, indicating the effect of non-ideality. In the figure, the (constant) intrinsic permeability (flat dashedlines) is also reported for comparison, highlighting the difference between ideal and non-ideal case (coloured areas). In particular, permeability is shown to be higher at higher temperatures for all the investigated values but 250 C, whose curve approaches that of 280 C for high-pressure values. This peculiar behaviour is explained by considering that solubility is higher at lower temperatures. Since permeability is actually given by the product of diffusion coefficient and solubility, a higher value of solubility causes a permeability increase, whose effect is magnified at high pressures. Concerning the quantitative characterisation of membrane permeation, from the data shown in Fig. 4, it is possible to achieve intrinsic permeability and non-ideal factor as functions of temperature (Fig. 5). As for the former, whose calculated parameters are in line with values reported in the literature (see, for example [12]), a representation in form of Arrhenius-type plot is chosen (Pe0 [molH2 m1 s1 Pa0.5] being the pre-exponential factor and E0 [J mol1] the corresponding activation energy), since the intrinsic permeability is theoretically an activated process (Eq. (16)). E0 bðTÞ ¼ Pe0 exp RT
(16)
As for the factor a, it shows a non-monotone trend with temperature, decreasing up to a minimum value around 400 C. To explain this behaviour and link this parameter to a specific physical phenomenon, a further development of the
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
4681
Fig. 6 e Non-Ideality Index (a) calculated @ 1000 Pa0.5 and pressure exponent n as functions of temperature. Fig. 5 e Ideal permeability b and non-ideal factor a as a function of temperature. The values of the parameters based on Eq. (16) determined by regression are reported in the legend for different temperature ranges.
theory is in progress and, thus, this analysis is postponed to future works.
4.5.
n ¼ 0:5729a2 þ 0:3782a þ 0:5 R2 ¼ 0:9986
Evaluation of non-ideality contribution
In order to quantify the non-ideal effect, a more convenient Non-Ideality Index (a) is introduced here as: a$q2 1 ¼ aðqÞh b b þ a$q2 1 þ a$q 2
As a quite remarkable fact, the shape of the two data series is very similar, this indicating a strong correlation between a and n under diffusion-controlled permeation. To provide the level of this correlation, the {n vs. a} data are fitted using a second-order polynomial by setting the intercept to 0.5, because a has to be null in ideal case. The fitting result is reported in Eq. (18), where it is possible to observe the high correlation degree.
(17)
According to its definition, the index a represents the contribution of the non-ideal part of the overall permeability with respect to the overall permeability itself and, thus, is a direct measure of the PdeH system non-ideality. To quantify the maximum non-ideal effect in the conditions considered in this study, a is calculated at 1000 kPa, corresponding to a q value of 1000 Pa0.5. By doing so, the plot depicted in Fig. 6 is obtained, where the trend of the pressure exponent n (Eq. (2)) with temperature is also shown to provide a different view of the non-ideality effect. The behaviour of the pressure exponent indicates that the PdeH system is gradually approaching ideal conditions, as it continuously decreases with increasing temperature towards the ideal value of 0.5. However, as already remarked above, the value of the pressure exponent does not give any quantitative information on the non-ideality degree.
(18)
However, it must be remarked that the reported correlation e which cannot be extrapolated to values of a other than those used for its evaluation e is not an attempt to justify the use of pressure exponent, but rather an empirical expression that could help researchers to better analyse results from permeation tests.
5.
Conclusions
In this work, a method to evaluate the effect of the non-ideal effect under diffusion-controlled hydrogen permeation was introduced. For this purpose, a Non-Idealily Index (a) was defined to provide the quantitative contribution of the nonideal permeability as a function of pressure and temperature. Interestingly, the index was found to be highly correlated to the pressure exponent, this opening a possible way to study the non-ideality from the pressure exponent value which in general does not have any physical meaning. The non-ideality index was obtained by a further development of a methodology already introduced in the literature,
4682
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
which was described in detail in the present investigation by using experimental data from permeation tests performed on a real Pd-membrane. This led to the evaluation of intrinsic permeability (b) and non-ideal factor (a), from which the index a was appropriately defined. The proposed approach allowed a complete characterisation of membrane behaviour, providing quantitative expressions for permeability and flux that can be directly used in simulations of Pd-membrane devices (reactors and contactors) over a wide range of temperature and pressure. This opens the way to avoid using the pressure exponent, which is not generally constant with operating conditions and does not provide any scientific information on membrane.
Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research in the framework of a research fellowship funded by the Japan Society for the Promotion of Science (JSPS), which is gratefully acknowledged.
references
[1] Gallucci F, Fernandez E, Corengia P, van Sint Annaland M. Recent advances on membranes and membrane reactors for hydrogen production. Chem Eng Sci 2013;92:40e66. [2] Morreale BD, Ciocco MV, Enick RM, Morsi BI, Howard BH, Cugini AV, et al. The permeability of hydrogen in bulk palladium at elevated temperatures and pressures. J Memb Sci 2003;212:87e97.
[3] Bustamante F, Enick RM, Cugini AV, Killmeyer RP, Howard BH, Rothenberger KS, et al. High-temperature kinetics of the homogeneous reverse wateregas shift reaction. AIChE J 2004;50:1028e40. [4] Guazzone F. Engineering of substrate surface for the synthesis of ultra-thin composite Pd and PdeCu membranes for H2 separation [PhD Thesis]. Worcester, UK: Worcester Polytechnic Institute; 2005. [5] Gielens FC, Tong HD, Vorstman MAG, Keurentjes JTF. Measurement and modeling of hydrogen transport through high-flux Pd membranes. J Memb Sci 2007;289:15e25. [6] Hara S, Ishitsuka M, Suda H, Mukaida M, Haraya K. Pressuredependent hydrogen permeability extended for metal membranes not obeying the square-root law. J Phys Chem B 2009;113:9795e801. [7] Caravella A, Scura F, Barbieri G, Drioli E. Sieverts law empirical exponent for Pd-based membranes: critical analysis in pure H2 permeation. J Phys Chem B 2010;114:6033e47. [8] Flanagan TB, Wang D. Exponents for the pressure dependence of hydrogen permeation through Pd and PdeAg alloy membranes. J Phys Chem C 2010;114:14482e8. [9] Hara S, Caravella A, Ishitsuka M, Suda H, Mukaida M, Haraya K, et al. Hydrogen diffusion coefficient and mobility in palladium as a function of equilibrium pressure evaluated by permeation measurement. J Memb Sci 2012;421e422:355e60. [10] Flanagan TB, Wang D. Temperature dependence of H permeation through Pd and Pd alloy membranes. J Phys Chem A 2012;116:185e92. [11] Wang D, Flanagan TB. H2 isotherms and diffusion parameters of H in fcc PdeMg alloys. J Phys Chem C 2013;117:1071e80. [12] Yun S, Ko JH, Oyama ST. Ultrathin palladium membranes prepared by a novel electric field assisted activation. J Memb Sci 2011;369:482e9.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Quantitative contribution of non-ideal permeability under diffusion-controlled hydrogen permeation through Pd-membranes Alessio Caravella, Nobuo Hara, Hideyuki Negishi, Shigeki Hara* National Institute of Advanced Industrial Science and Technology (AIST), Research Institute for Innovation in Sustainable Chemistry, Central 5, Higashi 1-1-1, Tsukuba 305-8565, Japan
article info
abstract
Article history:
In this paper, a powerful and simple method to evaluate quantitatively the influence of the
Received 31 July 2013
non-ideal contribution of the hydrogen permeation under diffusion control is provided. For
Received in revised form
this purpose, a Non-Ideality Index (a) is defined as the ratio of the non-ideal permeability
16 September 2013
term and the overall permeability, this leading to a direct quantification of the non-ideal
Accepted 2 October 2013
effect in membrane. The used methodology is described in detail by using experimental data of a real Pd-
Available online 2 November 2013
membrane. From these data, the parameters that characterise permeability as a function Keywords:
of temperature and pressure e i.e., intrinsic permeability (b) and non-ideal factor (a) e are
Non-ideality index
evaluated, highlighting their higher suitability in describing the membrane behaviour with
Non-ideal diffusion
respect to the usual empirical parameters used by the majority of researchers (i.e., pressure
Hydrogen
exponent and corresponding permeability). It is underlined that the proposed approach is
Palladium membranes
particularly effective for modelling membrane contactors and reactors, where the empir-
Sieverts law
ical Sieverts-type law cannot be used because the large difference of pressure and temperature could lead the membrane behaviour to dramatically change even at different locations of the same device. Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
The advantages of applying Pd-based membranes to hydrogen purification and production processes do not need to be listed anymore, they being extensively reported in the literature (see, as an instance [1]). The infinite selectivity of these membranes to hydrogen has attracted the interest of a number of research groups, who have investigated several different aspects concerning the hydrogen-metal lattice interactions.
When hydrogen internal diffusion controls the overall permeation process and, at the same time, the PdeH system is close to conditions of infinite dilution (referred to as “ideal case”), the permeating flux J [molH2 m2 s1] can be satisfactorily described by the Sieverts law (Eq. (1)): i h 0:5 J ¼ pðSievÞ p0:5 f pp pðSievÞ ¼
fðSievÞ d
* Corresponding author. Tel.: þ81 29 861 9336. E-mail address: [email protected] (S. Hara). 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.10.015
(1)
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
where pf and pp indicate the hydrogen partial pressure in feed and permeate side, whereas p(Siev) [molH2 m2 s1 Pa0.5] and f(Siev) [molH2 m1 s1 Pa0.5] are the membrane permeance and permeability to hydrogen, the latter calculated from the former through the membrane thickness d. However, in most cases, ideal conditions are far from real situations, where deviations from the Sieverts law e i.e., pressure exponent n (Eq. (2)) different from 0.5 e have been found in several experimental and theoretical studies (see, for example [2e11],). i h H2 Flux ¼ pðnÞ pnf pnp
(2)
The interesting fact is that such deviations were observed not only in the presence of external mass transfer resistance and/or in the presence of slow surface phenomena (adsorption/desorption), but also for very thick membranes [2,3], where it is undeniable that the diffusion in the lattice controls the permeation. All these studies highlighted the necessity to investigate the dependence of permeability on hydrogen concentration. To do that, Flanagan and Wang described a methodology to evaluate both the Einstein hydrogen diffusivity (i.e., ideal diffusivity) and the non-ideal correction from experimental isotherms [8,10,11]. A different approach is used by Hara and co-workers, who developed a general methodology to describe any behaviour of a metal membrane in both ideal and non-ideal conditions under diffusion-controlled permeation [6,9]. Once a suitable analytical form for permeability and solubility are chosen, such powerful and relatively easy methodology allows evaluating hydrogen diffusion coefficient, solubility and mobility through a metal lattice (not necessarily restricted to just palladium alloys) as functions of temperature and pressure. However, to the Authors’ knowledge, this approach has not been used yet by researchers, who mostly prefer using Eq. (2) to characterise the membrane behaviour. The problem with Eq. (2) is that the empirical pressure exponent n changes with both pressure and temperature, and, thus, is not suitable to be used in a wide range of operating conditions. Furthermore, it is not a direct measure of any physical phenomenon related to transport in membrane, its value being, on the contrary, the overall result of the number of different elementary permeation mechanisms. Therefore, no specific information can be
4677
obtained from pressure exponent, especially concerning the non-ideal membrane behaviour. In this context, the aim of the present paper is to provide a different and systematic way to measure directly and quantitatively the non-ideal contribution to permeation under diffusion control. For this purpose, the above mentioned methodology [6,9] is applied to experimental data from permeation tests performed on a real Pd-membrane, describing in detail the scientific and technical aspects necessary for its application. By doing this, we also aim at promoting a further membrane development by spreading this approach over in order for researchers to use it for a more complete membrane characterisation.
2.
Experimental setup
The experimental system is schematically shown in Fig. 1. In particular, a self-supported20 mm-thick Pd-membrane is tested under pure hydrogen atmosphere on both membrane sides, where pressure profiles and relative flux are generated and measured using the here-introduced “alternative step method” (see Section 4.1 for details), which is performed by means of two digital back-pressure regulators. Flow rates of 100 mL min1 (20 C, 1 atm) are fed at the inlet of both feed and permeate side, and the permeating flux is measured by means of two digital mass flow metres as the difference between the flow rates of inlet and outlet divided by the membrane area (2.98 cm2). The presence of possible leaks are monitored by means of three different checks. The first check is performed outside the oven at room temperature once finished assembling the permeation cell. For this purpose, argon gas is used raising the absolute feed pressure up to 600 kPa keeping it for 5 min and the permeate side at atmospheric pressure. The second check is performed at 300 C after inserting the permeation cell in the oven and before starting the permeation test, using the same conditions as those in the first check. The third check is made after each pressure cycle performed at constant temperature by switching the hydrogen in the feed with Ar gas kept at 900 kPa for 5 min.
3.
Theoretical background [6,9]
As mentioned above, the methodology that the present study is based on was already introduced in Hara et al. (2009). It is based on the definition of an integral form for permeation flux (Eq. (3)), extending in this way a constant permeability to a pressure-dependent permeability. Zqf J ¼ 1d
fðqÞdq qp
(3)
pffiffiffi qh p
One of the practical forms for permeability is proposed in the previous reports as: Fig. 1 e Sketch of the experimental system. The set of the investigated temperatures are also reported.
fðqÞhb þ a$q2
(4)
4678
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
In Eq. (4), permeability is shown to include two additive terms: i) the intrinsic permeability b [molH2 m1 s1 Pa0.5] and ii) a non-ideal term, whose functionality has been chosen to be the product of pressure (q2 [Pa]) by the non-ideal factor a [molH2 m1 s1 Pa1.5]. Both a and b are function of temperature only. This form (polynomial in q) is particularly convenient because allows an analytical expression of hydrogen diffusion coefficient, solubility and mobility. However, it must be underlined that just permeability data are not sufficient to derive data of diffusion coefficient, solubility and mobility, because two independent pieces of information are needed to characterise the membrane behaviour. For this purpose, Hara et al. (2012) clearly showed that both permeability and solubility are sufficient to derive diffusion coefficient and mobility. Other approaches using two different independent parameters are also possible. Using Eq. (4) in Eq. (3), the flux expression is evaluated as shown in Eq. (5), where it is reported to be proportional to the difference between two third-order polynomials with the same coefficients (without the second-order term) written for feed and permeate side, respectively. J¼
1 h a a i bqf þ q3f bqp þ q3p d 3 3
(5)
Dually, the permeability expression can be determined from flux by inverting Eq. (3) (Eq. (6)): fðqÞ ¼ d
vJ vqf qp ¼q qf
4.
Results and discussion
The whole procedure of the permeability parameters evaluation can be summarised through the following steps, which have to be performed for each temperature: 1. Obtaining {flux vs. feed pressure square root} permeation data for fixed permeate pressures, which generally generates several data series (in this study, just two series, named Series 1 and Series 2, see Fig. 2). 2. Obtaining a unique overall data series by overlapping all the original series, this consisting in performing the shifting procedure described in Section 4.3. 3. Fitting the obtained overall data series by using Eq. (5) to evaluate a and b. The next sub-sections explain the details of these steps.
4.1.
Permeation data: alternative step method
(6)
The flux expression reported in Eq. (5) is important, as it is the actual functionality from which to start the membrane
Fig. 2 e Alternative step method to evaluate flux as function of feed and permeate pressure. T [ 300 C.
characterisation. In fact, the experimental data obtained from a permeation test most commonly are of the type “flux vs. feed pressure square root” for a constant permeate pressure. The next section provides the description of the calculation procedure of a and b, as well as a way to evaluate the non-ideal contribution through the Non-Ideality Index a (Section 4.5).
As said before, the first step is to obtain experimental data of flux vs. feed pressure square root for a constant reference value of permeate pressure. To do that, the here-named “Alternative Step Method” is used for each temperature considered. This method consists in measuring the permeation flux in conditions where the pressure of both membrane sides is changed alternatively according to the profiles depicted in Fig. 2a. In particular, the pressure program starts setting a feed and permeate pressure of 150 and 120 kPa, respectively. After 22 min, the permeate pressure is increased up to 200 kPa by keeping the feed pressure at 150 kPa. Then, after another 22 min, the feed pressure is increased up to 250 kPa by keeping the permeate pressure at 200 kPa. This procedure, corresponding to the Series 1 in Fig. 2a, is repeated up to a maximum feed and permeate pressure of 950 and 980 kPa, respectively. After these values, pressure starts decreasing in an analogous way (Series 2). The result of this procedure is that, for each couple of pressure values, a pressure step of 50 kPa is determined (that is the reason for the method name), except at the beginning, at the intermediate time and at the end, where also feed pressure below the atmospheric pressure is set to enlarge the pressure range investigated. The permeating flux changes accordingly (Fig. 2b), becoming alternatively positive (from feed to permeate side) and negative (from permeate to feed side). This method allows investigating the effect of feed and permeate pressure at once without the need of performing the same feed pressure profile for each permeate pressure. Furthermore, it is very useful for measurements involving very thin self-supported membranes, because it avoids using
4679
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
large pressure differences, which could damage/break the tested membranes. The next section describes how to obtain the required {qf, J} data series for a fixed reference qp from the data shown in Fig. 2 to perform the evaluation of a and b.
4.2.
Data manipulation for permeability evaluation
To evaluate the intrinsic permeability b and the non-ideal factor a, a suitable data series has to be obtained from the permeation data shown in Fig. 2. For this purpose, flux has to be evaluated as a function of the feed pressure square root at a constant reference permeate pressure. To do that, let us consider that, according to the integral form reported in Eq. (3), the flux generated by the difference between the reference qr and a generic qf can be split using an intermediate value q1 into two terms as reported in Eq. (7). Zqf J ¼ 1d 2 6 ¼ 1d 4
fðqÞdq qr
Zqf
Zq1 fðqÞdq þ qr
3
(7)
7 fðqÞdq5
q1
Generalising Eq. (7) by considering several intermediate pressures, the flux “between” qr and the generic qi can be split into the summation of several flux terms (Eq. (8)).
Jjqqir ¼ ¼ 1d
i P k¼r
i P k¼r
Jjqqkþ1 k (8)
Zqkþ1 fðT; qÞdq qk
Eq. (8) is useful because the flux data required in the summation are directly available from the experimental data obtained from alternative step method (Fig. 2). Therefore, performing this calculation for all the qi values, the wished data series e consisting of m couples of {flux vs. pressure square root} data for the fixed value qr e is obtained (Eq. (9)). n o q0 ; Jjqq0r ; .; qm ; Jjqqmr
(9)
It must be underlined that the parameters evaluation is independent of the particular choice of the reference permeate pressure, because the shift does not change the shape of the curves, but just their position in the plot. In this specific study, a reference pressure of 120 kPa is selected and the m couples depicted in Fig. 3a for the 300 C case are obtained. As anticipated above, the experimental data are divided into two series e named Series 1 and Series 2 e which correspond to increasing and decreasing pressure steps, respectively. To evaluate the permeation parameters (a and b), the original experimental data are modified by shifting Series 2 on Series 1 (Fig. 3b) according to the procedure described in the next section. The shifting procedure is necessary to maximise the calculation precision, minimising in this way the effect of the experimental error. Then, all the shifted data are fitted by using Eq. (5), obtaining in this way the desired parameters a and b, whose values are explicitly shown in Section 4.4.
4.3.
Details of the shifting strategy
Let us consider a number of curves to be overlapped. The here-adopted shifting strategy consists in setting one curve as a reference and shifting all the others on it. Because of the experimental errors, there will be a certain difference between the points of the shifted curves and those of the reference one. The aim of this procedure is minimising such a difference. For this purpose, all curves are preliminary fitted by a suitable functionality. In this work, the polynomial form is chosen to obtain a simple analytical expression of the shifting values (see later Eq. (14)). It must be underlined that, in principle, forms other than polynomials can be also selected to perform the shift. In all cases, however, the fitting determination coefficient R2 should be almost the same for all forms to assure the shifting independence of the particular form chosen. In this study, our experimental data are satisfactorily described by third order-polynomials (Eq. (10)), which, thus, have been chosen for the shifting procedure. yr ðxÞ ¼ ar0 þ ar1 x þ ar2 x2 þ ar3 x3 ¼ Fig. 3 e a) Original and b) Shifted Flux as a function of feed pressure at a fixed permeate pressure (120 kPa) evaluated from the data reported in Fig. 2 using Eq. (8). T [ 300 C.
yj ðxÞ ¼ aj0 þ aj1 x þ aj2 x2 þ aj3 x3 ¼ xhqf ;
j ¼ 1.nC
3 P
k¼0 3 P
ark xk
ajk xk
k¼0
(10)
4680
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
In these expressions, yr(x) represents the reference curve, whereas the other curves are indicated by the generic subscript ’j’. Once the polynomial coefficients are calculated by fitting, each curve has to be shifted on the reference one by a constant value cj, different for each curve. By adding the corresponding value of cj to each original curve, all the shifted curves (y*) are determined (Eq. (11)). yj ¼ yj þ cj ¼
3 X
ajk xk þ cj
(11)
k¼0
The problem consists in the evaluation of cj. For this purpose, the here-selected shifting criterion is to achieve the minimum difference between the area underlying each curve to shift (Aj*) and the area underlying the reference one (Ar, Eq. (12)). This criterion is necessary and sufficient for two curves to overlap and is equivalent to the condition of minimum distance between the corresponding points of the curves. Zxb Ar ¼
yr ðxÞdx ¼ xa Zxb
Aj ¼ xa
Zxb X 3
ark xk dx
k¼0
yj ðxÞdx
xa Zxb "
¼ xa
3 X
#
(12)
ajk xk þ cj dx
k¼0
The mathematical formulation of the minimisation problem is formalised in Eq. (13). h 2 i h 2 i v DAj min DAj 5 ¼0 vcj fcj g 2 2 DAj ¼ Ar Aj
(13)
Since in general two or more curves could be obtained within different pressure ranges, the integral extremes in Eq. (12) (xa and xb) should be selected in such a way to provide a common range to all curves. Finally, developing Eq. (13) coupled with Eq. (12), the shifting quantity cj is determined for each curve after some algebraic steps (Eq. (14)). cj ¼
3 X ark ajk ðxb xa Þk kþ1 k¼0
(14)
Let us observe that the value of c0 (trivial case of reference curve) is, as expected, zero. As already commented, the advantage of using a polynomial form lies in the simplicity of Eq. (14), which can be also easily extended to higher orders, if required. Since in this study we use just two data series (Series 1 and Series 2) to obtain the permeation parameters, just one value of cj (c1) is needed for each temperature (Eq. (15)). c1 ¼
4.4.
3 X ðark a1k Þ ðxb xa Þk kþ1 k¼0
Fig. 4 e Overall permeability as a function of the pressure square root for different temperatures. The dashed-lines indicate the ideal membrane behaviour.
(15)
Pressure-dependent permeability
The application of the above-described procedures leads to the results reported in this section. Fig. 4 shows the behaviour of the overall permeability (Eq. (4)) as a function of pressure square root. As it is possible to observe, permeability increases
with increasing pressure, indicating the effect of non-ideality. In the figure, the (constant) intrinsic permeability (flat dashedlines) is also reported for comparison, highlighting the difference between ideal and non-ideal case (coloured areas). In particular, permeability is shown to be higher at higher temperatures for all the investigated values but 250 C, whose curve approaches that of 280 C for high-pressure values. This peculiar behaviour is explained by considering that solubility is higher at lower temperatures. Since permeability is actually given by the product of diffusion coefficient and solubility, a higher value of solubility causes a permeability increase, whose effect is magnified at high pressures. Concerning the quantitative characterisation of membrane permeation, from the data shown in Fig. 4, it is possible to achieve intrinsic permeability and non-ideal factor as functions of temperature (Fig. 5). As for the former, whose calculated parameters are in line with values reported in the literature (see, for example [12]), a representation in form of Arrhenius-type plot is chosen (Pe0 [molH2 m1 s1 Pa0.5] being the pre-exponential factor and E0 [J mol1] the corresponding activation energy), since the intrinsic permeability is theoretically an activated process (Eq. (16)). E0 bðTÞ ¼ Pe0 exp RT
(16)
As for the factor a, it shows a non-monotone trend with temperature, decreasing up to a minimum value around 400 C. To explain this behaviour and link this parameter to a specific physical phenomenon, a further development of the
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
4681
Fig. 6 e Non-Ideality Index (a) calculated @ 1000 Pa0.5 and pressure exponent n as functions of temperature. Fig. 5 e Ideal permeability b and non-ideal factor a as a function of temperature. The values of the parameters based on Eq. (16) determined by regression are reported in the legend for different temperature ranges.
theory is in progress and, thus, this analysis is postponed to future works.
4.5.
n ¼ 0:5729a2 þ 0:3782a þ 0:5 R2 ¼ 0:9986
Evaluation of non-ideality contribution
In order to quantify the non-ideal effect, a more convenient Non-Ideality Index (a) is introduced here as: a$q2 1 ¼ aðqÞh b b þ a$q2 1 þ a$q 2
As a quite remarkable fact, the shape of the two data series is very similar, this indicating a strong correlation between a and n under diffusion-controlled permeation. To provide the level of this correlation, the {n vs. a} data are fitted using a second-order polynomial by setting the intercept to 0.5, because a has to be null in ideal case. The fitting result is reported in Eq. (18), where it is possible to observe the high correlation degree.
(17)
According to its definition, the index a represents the contribution of the non-ideal part of the overall permeability with respect to the overall permeability itself and, thus, is a direct measure of the PdeH system non-ideality. To quantify the maximum non-ideal effect in the conditions considered in this study, a is calculated at 1000 kPa, corresponding to a q value of 1000 Pa0.5. By doing so, the plot depicted in Fig. 6 is obtained, where the trend of the pressure exponent n (Eq. (2)) with temperature is also shown to provide a different view of the non-ideality effect. The behaviour of the pressure exponent indicates that the PdeH system is gradually approaching ideal conditions, as it continuously decreases with increasing temperature towards the ideal value of 0.5. However, as already remarked above, the value of the pressure exponent does not give any quantitative information on the non-ideality degree.
(18)
However, it must be remarked that the reported correlation e which cannot be extrapolated to values of a other than those used for its evaluation e is not an attempt to justify the use of pressure exponent, but rather an empirical expression that could help researchers to better analyse results from permeation tests.
5.
Conclusions
In this work, a method to evaluate the effect of the non-ideal effect under diffusion-controlled hydrogen permeation was introduced. For this purpose, a Non-Idealily Index (a) was defined to provide the quantitative contribution of the nonideal permeability as a function of pressure and temperature. Interestingly, the index was found to be highly correlated to the pressure exponent, this opening a possible way to study the non-ideality from the pressure exponent value which in general does not have any physical meaning. The non-ideality index was obtained by a further development of a methodology already introduced in the literature,
4682
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 4 6 7 6 e4 6 8 2
which was described in detail in the present investigation by using experimental data from permeation tests performed on a real Pd-membrane. This led to the evaluation of intrinsic permeability (b) and non-ideal factor (a), from which the index a was appropriately defined. The proposed approach allowed a complete characterisation of membrane behaviour, providing quantitative expressions for permeability and flux that can be directly used in simulations of Pd-membrane devices (reactors and contactors) over a wide range of temperature and pressure. This opens the way to avoid using the pressure exponent, which is not generally constant with operating conditions and does not provide any scientific information on membrane.
Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research in the framework of a research fellowship funded by the Japan Society for the Promotion of Science (JSPS), which is gratefully acknowledged.
references
[1] Gallucci F, Fernandez E, Corengia P, van Sint Annaland M. Recent advances on membranes and membrane reactors for hydrogen production. Chem Eng Sci 2013;92:40e66. [2] Morreale BD, Ciocco MV, Enick RM, Morsi BI, Howard BH, Cugini AV, et al. The permeability of hydrogen in bulk palladium at elevated temperatures and pressures. J Memb Sci 2003;212:87e97.
[3] Bustamante F, Enick RM, Cugini AV, Killmeyer RP, Howard BH, Rothenberger KS, et al. High-temperature kinetics of the homogeneous reverse wateregas shift reaction. AIChE J 2004;50:1028e40. [4] Guazzone F. Engineering of substrate surface for the synthesis of ultra-thin composite Pd and PdeCu membranes for H2 separation [PhD Thesis]. Worcester, UK: Worcester Polytechnic Institute; 2005. [5] Gielens FC, Tong HD, Vorstman MAG, Keurentjes JTF. Measurement and modeling of hydrogen transport through high-flux Pd membranes. J Memb Sci 2007;289:15e25. [6] Hara S, Ishitsuka M, Suda H, Mukaida M, Haraya K. Pressuredependent hydrogen permeability extended for metal membranes not obeying the square-root law. J Phys Chem B 2009;113:9795e801. [7] Caravella A, Scura F, Barbieri G, Drioli E. Sieverts law empirical exponent for Pd-based membranes: critical analysis in pure H2 permeation. J Phys Chem B 2010;114:6033e47. [8] Flanagan TB, Wang D. Exponents for the pressure dependence of hydrogen permeation through Pd and PdeAg alloy membranes. J Phys Chem C 2010;114:14482e8. [9] Hara S, Caravella A, Ishitsuka M, Suda H, Mukaida M, Haraya K, et al. Hydrogen diffusion coefficient and mobility in palladium as a function of equilibrium pressure evaluated by permeation measurement. J Memb Sci 2012;421e422:355e60. [10] Flanagan TB, Wang D. Temperature dependence of H permeation through Pd and Pd alloy membranes. J Phys Chem A 2012;116:185e92. [11] Wang D, Flanagan TB. H2 isotherms and diffusion parameters of H in fcc PdeMg alloys. J Phys Chem C 2013;117:1071e80. [12] Yun S, Ko JH, Oyama ST. Ultrathin palladium membranes prepared by a novel electric field assisted activation. J Memb Sci 2011;369:482e9.