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Evaluation of hydrogen sorption and permeation parameters in liquid metal membranes via Sieverts' apparatus
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Evaluation of hydrogen sorption and permeation parameters in liquid metal membranes via Sieverts' apparatus Nicholas D. Deveau, Pei-Shan Yen, Ravindra Datta* Fuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA
article info
abstract
Article history:
We have recently proposed sandwiched liquid metal membranes (SLiMM) for hydrogen
Received 16 June 2018
separation. To evaluate SLiMM, thermodynamic and kinetic parameters such as the sol-
Received in revised form
ubility, diffusion coefficient, and absorption kinetics of hydrogen in the liquid metal are
14 August 2018
needed. While there are some theoretical approaches to estimate these parameters,
Accepted 17 August 2018
including our own published recently, it is important to obtain experimental corroboration.
Available online 12 September 2018
This study utilizes the classical Sieverts' apparatus in an effort to estimate these parameters by monitoring the change in pressure with time of hydrogen introduced over a pool of
Keywords:
liquid metal within a container. The solubility was calculated from the change in pressure
Sieverts' apparatus
over the entire duration of the experiment as it attained equilibrium, while typically the
Metal-hydrogen system
diffusion coefficient could be determined from the short-time response. The theory behind
Liquid metal
the Sieverts' apparatus is extended here to provide full-time solution by linearizing Sie-
Hydrogen sorption
verts' law, along with early (very short) time solutions to determine the sorption kinetics in
Hydrogen diffusion
addition to the diffusion coefficient. In this manner, new theoretical and experimental results are obtained for hydrogen sorption and diffusion in liquid gallium and indium at different temperatures. The results corroborate the hydrogen permeability in a gallium SLiMM. © 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Due to the increasing demand for pure hydrogen in applications at different scales ranging from chemicals to fuel cells, there is significant interest in developing dense metal membranes for separation and purification of hydrogen. These metal membranes promise to be more energy efficient and selective when compared to traditional hydrogen purification technologies such as pressure swing adsorption [1]. The state-
of-the-art dense metal membrane for hydrogen uses palladium or its alloys with silver and copper, manufactured as thin foils or as thin but dense membranes deposited on a porous support. While very promising, these membranes have some distinct disadvantages. Palladium, and to a lesser extent silver, is expensive. These membranes also come up short in regard to durability, as exceedingly thin membranes (~10e20 mm) are employed to concomitantly reduce cost and increase permeance. Further, these solid but thin films can develop cracks and micropores that allow other gases to pass
* Corresponding author. E-mail address: [email protected] (R. Datta). https://doi.org/10.1016/j.ijhydene.2018.08.101 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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through and hence reduce membrane selectivity and product purity. Such defects cannot easily be repaired, requiring membrane replacement. These limitations are driving research into seeking alternative and cheaper metals with adequate hydrogen permeability [2e9], which could be fashioned into thicker, more robust, solid membranes. In this vein, we have for the first time recently proposed thicker films (~200 mm) of low-melting metals or alloys as sandwiched liquid metal membranes (SLiMMs), as shown schematically in Fig. 1 [10]. Such liquid metal membranes could potentially address some of the issues associated with solid metal membranes, e.g., sintering, hydrogen embrittlement, and thermal mismatch between support and membrane. However, liquid metals provide their own challenges. There is limited literature on liquid metal membranes due to their high reactivity with other metals [11e16], as well as wettability issues [17] with ceramic supports. There is limited data on the formation of hydrides with liquid metals and any molecular species diffusion. Although there is limited available experimental results, they indicate that the solubility and diffusivity of hydrogen of a metal increases upon melting [10]. Here, we utilize Sieverts' apparatus to determine the relevant parameters for two low melting metals of interest for SLiMM, namely, gallium (29.8 C, MP) and indium (156.6 C, MP), and compare these to our recent theoretical predictions [18]. The hydrogen permeation in a metal, as proposed by Wang and Roberts [19], consists of five sequential steps, described schematically in Fig. 2. Thus, hydrogen is first dissociatively adsorbed onto the metal surface on the feed side, and then penetrates to the sub-surface of the bulk metal. Next, the hydrogen atoms interstitially diffuse across the membrane, ultimately emerging on the opposite surface. The final step is the associative desorption of hydrogen from the second surface of the metal membrane on the permeate side. The flux of hydrogen (NH2 ) in traditional palladium-based membranes is described using what is commonly known as Sieverts' law [20], first proposed by Richardson et al. [21], NH2 ¼
ct cH$M;s KS DH 1=2 1=2 1=2 1=2 pffiffiffiffiffio pH2 ;f pH2 ;p ≡PH2 pH2 ;f pH2 ;p 2d p
(1)
which relates the hydrogen flux to the partial pressure of hydrogen (pH2 ) on the feed (f) and permeate (p) sides,
Fig. 1 e Schematic of sandwiched liquid metal membrane (SLiMM).
Fig. 2 e Schematic of the five steps of hydrogen diffusion through metal: (1) dissociative adsorption, (2) solution to the bulk, (3) diffusion, (4) dissolution to the surface, and (5) associative desorption.
membrane characteristics such as metal atom concentration (ct) and maximum hydrogen solubility (cH$M;s ), membrane thickness (d), Sieverts' equilibrium constant (KS ), and diffusion coefficient (DH ). This simple relation is derived assuming that the diffusion step is the rate-determining step (RDS) in the sequence of steps, and that the hydrogen solubility is low, so that the interstitial mole fraction can be approximated via Sieverts' law, following a square-root dependence on pffiffiffiffiffiffiffiffiffiffiffiffiffiffi hydrogen partial pressure, xH$M zKS pH2 =p , where p is standard (atmospheric) pressure. These assumptions may not necessarily remain true under all conditions for the liquid metals of interest, and indeed have even been shown [22] to be invalid under certain conditions for palladium membranes, when the adsorption/desorption steps can be the RDS. While a framework has been developed to ascertain this [22], however, the solubility and diffusion parameters needed for the model for many metals, especially liquid metals, are unknown. Finally, although we are specifically interested here in the potential application of liquid metal-membranes, there are other applications of liquid metals as well where these parameters are of interest, e.g., as coolants in fast neutron reactors, in metal casting, and perhaps even in hydrogen storage [18]. One convenient way to determine some of these kinetic and thermodynamic parameters is by monitoring hydrogen uptake, or evolution, over a liquid metal pool in what is known as the Sieverts' apparatus [23,24]. This apparatus can be set up to operate in one of two different modes: 1) a constant-volume mode, known as a pressure Sieverts' apparatus [23], or 2) a constant-pressure mode, known as a volume Sieverts' apparatus [24]. Both modes of operation follow similar set-ups and theory, having the metal of interest of known surface area and volume within a closed system under hydrogen atmosphere that monitors either the system pressure at a constant volume (pressure Sieverts' apparatus), or the total system volume at a constant pressure (volume Sieverts' apparatus), so as to determine the amount of hydrogen permeating into the metal
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as a function of time. In conjunction with theory, the results of such experiments [25e28] can provide the solubility of a gas and its diffusion coefficient. The diffusion coefficient is typically estimated from experimental data for short times only, in order to utilize the simplified mathematical analysis based on penetration theory, applicable rigorously for a metal pool of infinite depth. Further, the Sieverts' experiments can be performed for gas absorption into a metal or for metal degassing. The absorption experiments begin with the metal sample being initially devoid of hydrogen. The gas chamber is then charged with pure hydrogen gas, and the pressure, for instance, is monitored as a function of time. As hydrogen is absorbed into the metal sample, the pressure in the chamber declines until the sample becomes saturated corresponding to the final equilibrium hydrogen pressure. For degassing, or off-gassing, experiments, on the other hand, the metal sample is initially saturated with dissolved hydrogen before being exposed to an atmosphere devoid of hydrogen. As the hydrogen desorbs from the metal, the pressure increases until an equilibrium between absorbed and gaseous hydrogen is achieved corresponding to the final hydrogen pressure. For complete degassing of a sample, however, multiple such cycles are needed. The available theoretical treatment in the literature regarding analysis of Sieverts' experiments is limited. Not only is it generally limited to short times, rather than to the full duration of the experiment, but it does not account for the sorption steps (Fig. 2). Thus, Sacris and Parlee [27] studied hydrogen diffusion into liquid nickel, copper, silver, and tin, by monitoring the volume of hydrogen VH2 absorbed as a function of time, t. Following a very similar procedure, Depuydt and Parlee [26] studied hydrogen diffusion in liquid iron alloys. In both studies, the hydrogen diffusion constant was estimated from the experimental data using the relation based on penetration theory solution, applicable to short times VH2 ¼
pffiffiffiffiffiffiffiffiffiffiffi d2 rM CS pDH t 200rg
(2)
where VH2 is the volume of hydrogen absorbed, d is the metal pool diameter, rM is the density of the metal, DH is the diffusion coefficient of hydrogen atoms in the metal, and rg is the density of the gas. A goal of this study is to extend such theoretical modeling of Sieverts' experimental apparatus in order to extend its applicability to long times, and to also potentially allow evaluation of sorption kinetic parameters for hydrogen from experimental data at very short, or early, times. This goal is accomplished in two parts. First, a solution is developed that is valid over the full time of an experiment, rather than being limited to short times. Second, absorption step into the bulk material is considered and solution obtained for early times. The model is finally used to analyze experimental data for liquid Ga and In in order to estimate hydrogen sorption and transport parameters that are compared to our recent theoretical predictions [18]. If one is interested in modeling a complete set of such an experiment until the system has reached an equilibrium,
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there are two alternate approaches. In the first, Solar and Guthrie [29] extend the above penetration theory solution, which holds only for a small change in gas pressure, into an infinite series of small pressure changes. The second approach involves solution of the coupled differential equations for the mixed gas phase and the stagnant liquid metal phase. The latter approach has not been developed so far for the Sieverts' experiment, because of the non-linearity (square root dependence on pressure) of the Sieverts' solubility that precludes an analytical solution. Here, we will use the Laplace transform method to obtain such a long-term solution for the analysis of a complete data set from Sieverts' experiments [30e33], by first linearizing the Sieverts' solution step.
Experimental apparatus and methods The experimental apparatus used in this study resembles the constant volume set-up of Small and Pehlke [28]. As shown schematically in Fig. 3, the total gas volume consists of a hightemperature “hot zone” and a room temperature “cold zone”. The volume of the hot and cold zones were estimated using helium gas and the ideal gas law relation P1 V1 ¼ P2 V2 . The cold zone was found to have a volume of 84.8 cm3 while the hot zone, accounting for the graphite crucible, had a volume of 30.69 cm3 . The cold zone (Fig. 3) comprises of tubing and fittings for controlling the gas inlet, in addition to two pressure transducers (Transducers Direct, 0e10 V DC output, 0e500 psi) at room temperature. One transducer is set up to measure the absolute pressure of the system, and is connected to a National Instruments data box (NI USB 6008) for computer acquisition of the pressure versus time data. The other transducer measures the gauge pressure of the system. The cold zone is connected to the gas supply cylinders via a buffer tank and a pressure regulator. The hot zone includes a section of tubing connecting the cold zone tubing to a chamber containing the sample. This zone is enclosed within a cylindrical ceramic heater (Watlow 120 V, 900 W) controlled by a temperature controller (Eurotherm 2116), which also monitors the temperature with a thermocouple inserted through the top of the furnace and situated just above the chamber in the middle of the furnace. Indium or gallium nuggets were placed in a graphite crucible machined from a 2.539 cm diameter, 1.126 cm long graphite rod. The cylindrical hollow of the crucible was machined to a diameter of 1.879 cm and 0.619 cm deep. As an example a sample of pre-melted indium inside the graphite crucible is shown in Fig. 4 prior to the experiments. The indium sample was 5.2550 g in mass, and the gallium sample was 5.8808 g. The copper gasket ring shown in the picture, placed on the visible knife-edge in the depression just inside the ring of bolt holes, acted as a compressive seal when the top half of the chamber was fastened to the bottom half using eight bolts. The metal (Ga or In) samples were, of course, in the molten state under experimental conditions when the temperature was raised above their respective melting points. In these experiments, hydrogen was the solute and helium was the inert gas used during the off-gassing runs.
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Fig. 3 e Schematic of Sieverts' apparatus: Gas inlet on the left, the metal sample (gray) is in a graphite crucible (black), and there are two pressure transducers for pressure monitoring. Hot zone is demarked with dashed red, cold zone with dashed blue. (For interpretation of the references to color/ in this figure legend, the reader is referred to the Web version of this article.)
Once the sample was sealed in the chamber and enclosed in the furnace, the system was purged of air and filled with the inert gas prior to heating. For each of the experiments with liquid metals, five different temperatures were studied in the range of 350e550 C of interest for SLiMM in 50 C increments. The system was then evacuated prior to the start of the absorption run, and was quickly charged with about 4 psig of hydrogen before the system inlet was again closed. The
gallium runs had initial pressures of 960e990 torr, while the indium runs had initial pressures of 880e920 torr. The pressure was monitored and recorded every few minutes until equilibrium was reached. These equilibrium pressures varied with temperature and the metal sample. Gallium runs ended between 450 torr (at 500 C) and 530 torr (at 350 C). The indium runs ended between 530 torr (at 500 C) and 580 torr (at 350 C). Between absorption experiments, the system was purged with the inert gas to completely remove any hydrogen from the metal sample. The system was initially purged of hydrogen by briefly venting to atmosphere and flushing with the inert gas. The system was then pressurized with the inert gas to around 10 psig. The hot zone valve was closed, and the cold zone was released to ambient pressure before opening the hot zone valve to reduce the pressure. Pressure reduction cycle was performed in total two to three times to reduce the pressure to values able to be monitored by the transducer. The pressure again was monitored until a new equilibrium was attained. The absorption runs were performed twice for each metal sample at each temperature.
Theory
Fig. 4 e Pre-melted (shown solidified) indium sample inside graphite crucible. The copper gasket used to seal the Sieverts' apparatus chamber is shown in place.
Our goal is to develop a theoretical model that relates the observed change in gas pressure pH2 ðtÞ in the Sieverts' device to hydrogen permeation in the liquid metal pool, based on material balance in gas and metal phases, coupled with Fick's law of diffusion and Sieverts' law of solubility in the metal phase. A schematic representation of the Sieverts' apparatus
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is shown in Fig. 5. In fact, it depicts three distinct phases in the apparatus: 1) the gas phase, 2) the metal phase, and 3) the interfacial layer. The interfacial layer is assumed to be of essentially monoatomic thickness that contains the liquid metal surface on the one side and bulk metal on the other. From the total concentration of atoms in the bulk metal, ct ¼ rM =AWM , thus, the concentration of atoms on the surface can be found using the atomic diameter of the metal as cS;t ¼ ct dM . It is further assumed that the gas phase is well-mixed, resulting in a uniform pressure and composition across the hot and cold zones, while the liquid metal pool is stagnant. The gas volumes (Vh ; VC ) and temperatures (Th ; TC ) of the hot and cold zones are assumed to be constant. Starting at time t ¼ 0 when it is introduced into the gas chamber, hydrogen dissociates on the liquid metal surface (step s1 ) and is then atomically dissolved (step s2 ) in the interfacial layer at the rate rS uniformly across the whole surface area of the metal (AM ). Consequently, the mole fraction of atomic hydrogen in the metal (xH$M ), starting at some initial value (xH$M;0 ), varies with time and depth (y) across the total metal pool of thickness (d). The coordinate z ¼ d y. The dissolved hydrogen leaves the interfacial layer via diffusion into the bulk metal at the rate of NH jd (in the schematic, the flux is shown in the positive z direction). At the very start of the experiment, for a very short time, or during the early time period, the difference between these two rates accounts for accumulation of atomic hydrogen within the interfacial layer. Thereafter, it essentially attains, for the long term, quasisteady-state (QSS) within the layer, when rS zNH jd , and any accumulation in the interfacial layer is negligible.
Formulation of the theoretical model Unsteady-state mass balance in the interfacial layer As discussed above and shown schematically in Fig. 2, there are a number of sequential steps involved in hydrogen
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permeation into a metal, although diffusion is often the rate determining step. Under certain conditions, however, it has been shown [22] that the adsorption (s1) and desorption (s5) steps can also have a significant impact on the hydrogen flux. On the other hand, steps s2 (dissolution from the surface into the subsurface) and s4 (evolution to the surface) are almost invariably rapid. Assuming, thus, that step s2 is rapid in comparison with adsorption step s1, the first two steps can be combined into an overall Sieverts' dissolution step (sS ) sr sr s1 : H2 þ 2S#2H$S þ1=2 s2 : H$S þ X#H$X þ S þ1 sS : ½H2 þ X#H$X
(3)
here, S represents a surface adsorption site, X represents the interfacial layer site, and sr is the stoichiometric number of step sr , such that an overall reaction (OR) is given by the linear P combination of steps, OR ¼ sr sr . As s2 is assumed to be rapid r
[6], the hence combined dissolution step (sS ) in equation (3) may be treated as an elementary step, with mass-action kinetics, i.e., u S cS;t ð1 xH$X Þ u S cS;t xH$X rS ¼ ! )
(4)
where xH$X is the mole fraction of hydrogen in the interfacial layer. For convenience, further, the kinetic parameters are lumped in the so-called step weights ! ! uS ¼ k S
rffiffiffiffiffiffiffiffi ) pH2 ) ; uS ¼ k S p
! ! where k S ¼ k 2
(5)
qffiffiffiffiffiffi qffiffiffiffiffiffiffi ) ) ) ! k 1 and k S ¼ k 2 k 1 are the dissolution step
rate constants, while those with subscripts 1 and 2 correspond to rate constants for steps 1 and 2. Further, these are related to the Sieverts' equilibrium constant via the microscopic ! ) reversibility relation KS ¼ k S = k S .
Fig. 5 e Simplified representation of the Sieverts apparatus. The hot zone, which includes the metal sample (represented as the shaded blue region), is on the left in this representation, while the cold zone is on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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The diffusive flux of atomic hydrogen in the metal phase is assumed to be given by Fick's Law of the form vxH$M NH ¼ ct DH vz
(6)
where the mole fraction of hydrogen in the bulk metal phase, xH$M , can be simplified under the assumption that the number of moles of hydrogen are much less than those of the metal, nH$M ≪nM , so that, xH$M ≡nH$M =ðnH$M þ nM ÞznH$M =nM ¼ cH$M =ct , where ct is the concentration of the metal atoms in the bulk. Using Fick's law along with the dissolution kinetics, Eq. (4), in the unsteady-state mass balance in the interfacial layer (Fig. 5), cS;t ðvxH$X =vtÞ ¼ rS þ NH jz¼d , thus yields cS;t
h i vxH$X vxH$M ) u S ð1 xH$X Þ u S xH$X ct DH ¼ cS;t ! vt vz z¼d
dnH2 ¼ dt
VC Vh þ RTC RTh
dpH2 dt
(12)
Combining this with the gas phase mass balance (dnH2 =dt ¼ AM rS =2) results in VC a dpH2 1 vxH$M jz¼d 1 vxH$M 1þ ¼ AM cS;t AM ct DH q 2 2 RTC dt vt vz z¼d
where the hot/cold volume ratio is a ¼ Vh =VC and the temperature ratio is q ¼ Th =TC . This is subject to the initial condition IC : at t ¼ 0; pH2 ¼ pH2 ;i
(14)
Unsteady-state mass balance in the metal phase (7)
where the left-hand side represents any accumulation in the interfacial layer. Further, if we assume that the solution of hydrogen in liquid metal is dilute, xH$X ≪1, and that xH$X zxH$M jz¼d , the mass balance takes the form
The mass balance of the metal phase is given by Fick's second law, vxH$M v2 xH$M ¼ DH vt vz2
(15)
subject to
8 IC : at t 0; xH$M ¼ xH$M;0 ¼ 0 > > > > > vxH$M < ¼ 0 BC1 : at z ¼ 0; vz > > > > vxH$M jz¼d vxH$M ) > : BC2 : at z ¼ d; cS;t u S u S xH$M jz¼d ¼ ct DH þ cS;t ! vt vz z¼d
vxH$M jz¼d vxH$M ) u S u S xH$M jz¼d ¼ ct DH þ cS;t ! cS;t vt vz z¼d
(8)
The forward step weight ! u S in this relation depends on the square root of hydrogen partial pressure, which is a function of time. In order to utilize the Laplace transform method for solution, consequently, we must linearize this relation. We do so about the dimensionless initial partial pressure pH2 ;i =p , and retain only the linear term, resulting in ! u S;i ! u Sz 2
pH 1þ 2 pH2 ;i
(9)
rffiffiffiffiffiffiffiffiffi pH2 ;i p
(10)
Further, we define the equilibrium bulk mole fraction corresponding to the initial hydrogen partial pressure xH$M;i ≡
! u S;i ) uS
¼ KS
rffiffiffiffiffiffiffiffiffi pH2 ;i p
Dimensionless form We recast the equations in terms of the following dimensionless variables and parameters. Thus, the dependent variables, the partial pressure and mole fraction of hydrogen, are defined as fH2
! 1 pH2 xH$M ≡ 1þ ; fH$M ≡ 2 pH2 ;i xH$M;i
The ideal gas law can be used to relate the change in the total number of moles of hydrogen in the gas phase (nH2 ) to the pressure
(17)
while the independent variables, time and position, are defined as DH z t≡ 2 t; x ≡ d d
(18)
A dimensionless parameter, Bi, similar to the Biot number, is defined along with the number of metal atom monolayers NM in the metal pool, and an experimental parameter, M
(11)
Unsteady-state mass balance in the gas phase
(16)
where the first boundary condition represents no flux at the bottom of the liquid metal pool, while the second represents mass balance at the gas-metal interface.
!
where the initial forward step weight, which is a constant, is defined as ! ! u S;i ¼ k S
(13)
Bi ≡
) 1 d2 u S d 1 nM ! u S;i ; NM ≡ ; M≡ NM DH dM 2 2nH2;i ) uS
! (19)
Interfacial Phase: The interfacial phase mass balance in dimensionless form is
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vfH$M 1 vfH$M x¼1 þ Bi fH2 fH$M x¼1 ¼ NM vx x¼1 vt
(20)
Clearly, the number of atomic monolayers in the liquid metal pool, NM is a very large number (~107), which would justify neglect of the accumulation term in the interfacial layer and hence support the QSS assumption made in the long-term analysis. In such a case, the interfacial phase mass balance simplifies to vfH$M ¼ Bi fH2 fH$M x¼1 vx x¼1
(21)
Gas Phase: The mass balance in the gas phase in dimen-
d2 f H$M sf H$M ¼ 0 dx2 8 df H$M > > BC1 : at x ¼ 0; ¼0 > > > dx < > > df H$M > > > BC2 : at x ¼ 1; ¼ Bi f H2 f H$M : x¼1 dx x¼1
Eq. (27) can be solved using the characteristic equation for a second order differential equation, in this case having roots of pffiffiffi ± s. This yields a result of C1 ex þ C2 ex . However, substituting in ex ¼ coshðxÞ þ sinhðxÞ and ex ¼ coshðxÞ sinhðxÞ and defining new constants of A1 ¼ C1 þ C2 and A2 ¼ C1 C2 gives pffiffiffi pffiffiffi f H$M ¼ A1 cosh s x þ A2 sinh s x
sionless form is dfH2 vfH$M ¼ M dt vx x¼1
(22)
IC : at t ¼ 0; fH2 ¼ 1 Metal Phase: The mass balance in the metal phase in
(27)
(28)
When applying the boundary condition BC 1, the constant A2 ¼ 0. Applying BC 2 gives the remaining constant as A1 ¼
dimensionless form is
f 2 pffiffiffi H pffiffiffi pffiffiffi cosh s þ Bi1 ssinh s
(29)
Substituting the resulting expression for the metal phase vfH$M v2 fH$M ¼ vt vx2 8 IC : at t 0; fH$M ¼ 0 > > > > > > > > < BC1 : at x ¼ 0; vfH$M ¼ 0 vx > > > > > > vfH$M > > ¼ Bi fH2 fH$M x¼1 : BC2 : at x ¼ 1; vx
into Eq. (26) for the gas phase and solving for the variable f H2 , yields (23)
lk tanðlk Þ ¼
Full solution To solve this system of the coupled differential equations, thus, we take the Laplace transform of the dependent variables, i.e., L½fH$M ðtÞ ¼ f H$M ðsÞ and L½fH2 ðtÞ ¼ f H2 ðsÞ, so that the 2nd order partial differential equation (PDE) representing mass balance in the liquid metal pool reduces to a 2nd order ordinary differential equation (ODE), while the 1st order ODE representing mass balance in the gas phase reduces to an algebraic equation. These can be solved simultaneously in the Laplace domain, and eventually inverted back into the time domain. Applying the Laplace transform to the gas phase, Eq. (22), results in df ¼ 1 M H$M dx
(30)
fraction has roots at s ¼ 0 and sk ¼ l2k , where lk represents a non-zero root of the denominator expressed as
Model solution
l2k M Bi1 l2k
(31)
From these roots, the dimensionless pressure in the time domain can be found using Cauchy's residue theorem [30,31]. fH2 ¼ Resð0Þ þ
∞ X
Resðsk Þ
(24)
(32)
k¼1
The solution is, in fact, provided by Carslaw and Jeager [31], which in rearranged form is pH2 ðtÞ ¼
1M pH2 ;i 1þM þ 4pH2 ;i MBi2
∞ X k¼1
x¼1
2
elk t MBi2 ð1 þ MÞ þ Bið1 þ Bi 2MÞl2k þ l4k (33)
Short-Time Approximation of Laplace Solution: The full
At the interface, Eq. (21) becomes df H$M dx
pffiffiffi pffiffiffi 1 þ 1 stanh s Bi 1 pffiffiffi pffiffiffi s þ M þ Bi s stanh s
To invert this expression back into the time domain, we use the Heavyside expansion theory. The denominator of the
x¼1
sf H2
f H2 ¼
¼ Bi f H2 f H$M
x¼1
(25)
x¼1
and combining this with Eq. (24) yields sf H2 ¼ 1 BiM f H2 f H$M
x¼1
(26)
For the metal phase, the Laplace transform of Eq. (23) yields
solution, Eq. (33) at short times can require an inordinately large number of terms in the summation to converge approximately to the initial condition. Therefore, it is of not a very suitable form for computation at short time. On the other hand, at larger times, the number of terms needed is quite small, and the solution provided is convenient. It is, therefore, desirable to obtain alternate closed forms solution for short times. In fact, this can be readily obtained from the Laplace domain solution as follows. Starting from Eq. (30) by noting
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pffiffiffi pffiffiffi s 3, which results in tanhð s Þ/1. As a pffiffiffi result, and using a change of variable h ¼ s, the Laplace domain solution can be reduced to
that when t/∞,
h þ Bi h þ Bi ¼ f H2 z 3 h þ ðBiÞh2 þ ðMBiÞh hðh þ bÞðh þ gÞ
(34)
b¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bi2 4MBi Bi þ Bi2 4MBi ;g ¼ 2 2
(35)
Eq. (34) can be rewritten using partial fraction expansion as f H2
at q' 0 fH$X ¼ 0
(39)
The system can then be modeled with a single partial differential equation for the metal phase, with a change in the coordinate based on y ¼ z d (or x' ¼
y d
¼ x 1). Further, the
mass balance in the bulk metal takes the form
where Bi
IC :
1 A2 B2 C2 þ þ ¼ bgðg bÞ h hþb hþg
(36)
where the constants A2 ¼ Biðg bÞ, B2 ¼ ðBi bÞð gÞ, and C2 ¼ ðBi gÞðbÞ. The terms in the brackets can each be inverted using pffiffiffi Laplace Transform Inversion Tables using h ¼ s , followed by combination and simplification finally provides pH2 ¼pH2 ;i
pffiffiffi
pffiffiffi 2 bexp g2 t erfc g t gexp b2 t erfc b t 1 bg (37)
subject to the conditions 8 < IC : BC1 : : BC2 :
at q' 0 fH$M ¼ 0 at x' ¼ 0 fH$M ¼ fH$X ðq'Þ at x'/∞ fH$M ¼ 0
for 0 x' ∞ for all q' > 0 for all q' > 0
(41)
Interfacial Phase: The mass balance for the interfacial phase can be solved independently of the metal phase. The result of this is fH$X ¼
1 exp Bi xH$M;i þ 1 q' xH$M;i þ 1
(42)
Metal Phase: The metal phase equations can be solved using the expression for fH$X in Eq. (42) and a composite vari'
2
The results obtained above, including those for short times, are based on the QSS assumption for the interfacial layer, which may alternately be viewed as the “outer solution” in terms of the singular perturbation theory [34], in which the simplified differential equations were first obtained by setting the singular perturbation parameter, 1=NM ¼ 0. It may be of interest to also obtain the “inner solution,” i.e., for the early time, when the QSS assumption for the interfacial layer does not apply, i.e., accumulation term in it is not negligible, from which additional information on the interfacial solution kinetics might be gleaned. For this case, we wish to further consider the full Sieverts' kinetics, i.e., without the dilute solution assumption that was made for the above QSS analysis. The scaling of dimensionless time t ¼ ðDH =d2 Þt used for the outer solution is further inappropriate for the inner solution. Therefore, we rescale time by defining a new dimensionless time as follows [34]: q'≡ðNM Þt. Using this in the dimensionless mass balance equations, and again setting 1=NM ¼ 0, results in dfH2 =dq' ¼ 0 and vfH$M =vq' ¼ 0. In other words, i.e., both gasphase and bulk metal phase concentrations remain virtually unchanged corresponding to their initial values over the very small period of interest for the inner solution, while the mass balance for the interfacial layer remains unchanged. In other words, pH2 /pH2 ;i , or fH2 /1, and xH$M /0, i.e., fH$M /0. Further, H$M since fH$M /0, one might argue that vfvx /0, so that the x¼1
diffusion term in the interfacial mass balance may be dropped at early times, which decouples this from bulk metal balance. Dropping the diffusion term, the mass balance equation for the interfacial layer reduces to
which is subject to the initial condition
(40)
x able h ¼ pffiffiffiffiffiffiffiffiffi , as '
Inner solution: early-time approximation
vfH$X zBi 1 xH$M;i þ 1 fH$X vq'
vfH$M 1 v2 fH$M ¼ NM vx'2 vq'
(38)
fH$M ¼
q =NM
1 exp Bi xH$M;i þ 1 q' erfcðhÞ xH$M;i þ 1
(43)
Thus, the flux of interstitial hydrogen at the metal/gas interphase is vxH$M ct xH$M;0 DH vfH$M vh ¼ vy y¼0 d vh h¼0 vx' x' ¼0 rffiffiffiffiffiffiffi
ct xH$M;0 DH NM 1 exp Bi xH$M;i þ 1 q' ¼ ' xH$M;i þ 1 d pq
NH jy¼0 ¼ ct DH
(44)
Using the relation in gas-phase mass balance, Eq. (12), along with AM NH jy¼0 =2 ¼ dnH2 =dt, results in rffiffiffiffiffiffiffi AM ct RTC DH xH$M;i pH2 ¼ pH2 ;i 2ð1 þ a=qÞ VC p xH$M;i þ 1 8 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = < pffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ) 2 t ) u S xH$M;i þ 1 t erf : ; u S xH$M;i þ 1
(45)
Below we use the theoretical results obtained above to estimate sorption and diffusion parameters for hydrogen diffusion in liquid metals.
Results and discussion Due to the complexity of Eqs. (33) and (37), a simple leastsquares regression of the experimental data was not able to provide the best sorption and diffusion parameters. Therefore, the following approach was adopted to determine the parameters: 1) first, the steady-state final pressure was used with the first term of Eq. (33), i.e. pH2 ðt/∞Þ ¼ 1M 1þM pH2;i , to calculate the
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parameter M, and hence the solubility xH$M;i and the Sieverts' constant KS ; 2) next, the inner solution, Eq. (45), was used with a leastsquares regression over the first ~1000 s of experimental ) data to determine initial estimates of u S and DH ; and 3) finally, the full solution, Eq. (33), is plotted against the ) experimental data and the values of u S and DH and manually adjusted to provide the best fit.
Steady-state data
nH$M nH$M 2nH2 ;∞ z ¼ nH$M þ nM nM nM
(46)
At a pressure of 18.7 psia, thus, the hydrogen solubility in gallium ranges from xH$M ¼ 0.042 at 350 C to 0.063 at 500 C, while solubility in indium is slightly higher, ranging from xH$M ¼ 0.055 at 350 C to 0.077 at 500 C. The calculated solubilities are plotted in Fig. 6. In both cases, there is a general trend of increasing solubility with increasing temperature, as would be expected because of the endothermic heat of solution of hydrogen in these metals as noted below. Linear trendlines are included in Fig. 6 simply as a visual aide. Substituting Eq. (46) into the Sieverts' solubility relation mentioned in the Introduction allows the Sieverts' constant to be calculated as
with the pre-exponential factor is defined as DSS A ≡ exp R
(49)
Gallium was found to have a pre-exponential factor of A ¼ 0:5192 and an enthalpy of solution of DHS ¼ 15:8 kJ/mol. For indium, the pre-exponential factor was found to be A ¼ 0:3267
and the enthalpy of solution DHS ¼ 11:9 kJ/mol. A van't Hoff plot of the Sieverts' constants is provided in Fig. 7. The hence determined thermodynamic parameters of hydrogen solubility in indium and gallium are presented in Table 1 along with those for palladium [22], simply for the purpose of comparison. These values differ somewhat from those predicted from the Pauling Bond Valence e Modified Morse Potential (PBVMMP) model developed in our lab [18]. Thus, the measured enthalpy changes are smaller than those predicted by the PBV-MMP model, while the entropy change is in good agreement. This results in a larger Sieverts' constants than those predicted from the thermodynamic estimates provided by the PBV-MMP framework.
Inner solution The steady-state pressure was used to calculate the dimensionless parameter M from the first term of Eq. (33), i.e. pH2 ðt/∞Þ ¼ 1M 1þMpH2;i . With this and the solubility and Sieverts'
sffiffiffiffiffiffiffiffi 2nH2 ;∞ p0 KS ¼ nM pH2
(47)
These were used then to calculate the thermodynamic
(48)
The solubility of hydrogen in the metal sample can be calculated from the total moles of hydrogen absorbed nH2 ;∞ . The solubility, expressed as a mole fraction xH$M;S can be expressed with a simplification, assuming the moles of hydrogen in the metal are much less than the moles of metal present, as xH$M;S ¼
DHS KS ¼ A exp RT
parameters, DHS and DSS , using the van't Hoff relation for temperature dependence
constant, we next utilize the inner solution, Eq. (45), for data )
fitting over the first ~1000 s to estimate the parameters u S and DH . Fig. 8 depicts an example regression over the first 1000 s of an experiment run with indium at 450 C, along with a theoretical prediction using parameters predicted from the PBV-
Fig. 6 e Solubility plot versus inverse temperature (18.7 psia). Gallium data are red circles, indium data are blue squares. Linear trendlines included only as a visual aide. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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Fig. 7 e Arrhenius plot of Sieverts' constants. Gallium data are red circles, indium data are blue squares. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
MMP framework [18]. The theoretical results closely mirror the regression. Both also provide a general mirror of the trend of the experimental data, however the models have a more pronounced initial curvature than the data. The diffusion coefficient is commonly represented via an Arrhenius relation
Table 1 e Comparison of entropy and enthalpy of solution for palladium, indium, and gallium. PBV-MMP estimates for indium and gallium also provided. Pd
DHS (kJ/mol) DSS (J/mol K)
12:1 63
Ga 15:8 5:4
Ga (PBV-MMP)
In
27 5:5
11:9 9:3
In (PBV-MMP) 50.9 9:7
ED DH ¼ DH;0 exp RT
(50)
while the pre-exponential factor is [21]. DH;0 ¼ aa2 zk
z kB T hn0 DSD 2 sinh exp h 2kB T R
(51)
Using the diffusivities derived from the regressions and the Arrhenius relation of Eq. (50), the gallium pre-exponential factor was calculated as DH;0 ¼ 8:63 104 cm2 =s while the activation energy was calculated as ED ¼ 20:0 kJ/mol. For indium, the pre-exponential factor was found to be DH;0 ¼ 1:32 103 cm2 =s while the activation energy was ED ¼ 25:7 kJ/mol. The pre-exponential factors are about three times smaller compared to the predictions from the PBV-MMP framework
Fig. 8 e Inner solution regression for an indium 450 C run, zoomed to the relevant area (~1000 s) of the data. The data points are blue, the experimental regression is red, and the theoretical prediction using PBV-MMP parameters is dashed red. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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Fig. 9 e Arrhenius plot of diffusivities. Gallium data are red circles, indium data are blue squares. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
Table 2 e Comparison of entropy and activation energy of diffusion for palladium, indium, and gallium. PBV-MMP estimates for indium and gallium also provided.
ED (kJ/mol) DSzD (J/mol K) DH;0 (cm2/s)
Pd
Ga
Ga (PBV-MMP)
In
In (PBV-MMP)
27.0 2.3 2:83 102
20.0 29 8:63 104
10 15 2:5 103
25.7 20 1:32 103
8.8 5:1 1:2 103
[18], while the activation energies of diffusion are roughly twice as large as the PBV-MMP predictions. These diffusion coefficient results are slightly lower than hydrogen diffusion in solid palladium, which typically ranges from 104 to 1 cm2/ s. An Arrhenius plot of the diffusion coefficients is provided in Fig. 9. The thermodynamic parameters for indium and gallium are presented in Table 2 along with those for palladium [22]. )
While the reverse sorption weight constant u S was determined from the least-squares regression of the short-time
experimental data, the forward weight constant ! u S was calculated from the reverse constant and the Sieverts' constant. The forward sorption step weights follow an Arrhenius relation, with the results for indium plotted in Fig. 10 below. The forward sorption rate constant is found from the linear regression in Fig. 10, and after factoring out the pressure dependence the van't Hoff form of the equation is ! 29100 k S ¼ 2:64 1015 e RT
Fig. 10 e Plot of adsorption step weight versus 1=T for indium.
(52)
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Fig. 11 e Arrhenius plot of desorption step weight for indium.
The reverse sorption step weight can also be modeled with the Arrhenius relation. A plot of the desorption step weight for indium along with the Arrhenius regression is shown in Fig. 11. The resulting Arrhenius form of the reverse sorption rate constant is )
16900 RT
k S ¼ 1:50 1017 e
(53)
Comparing these relations to the PBV-MMP predictions is more difficult, however taking the ratio of the forward to reverse rate constants gives the Sieverts' constant, which is modeled via the van't Hoff relation to give the change in enthalpy and change in entropy of solution according to Eqs. (48) and (49). For indium, the experimental change in enthalpy is 12.2 kJ/mol, which is significantly lower than the PBV-MMP framework prediction of 50:9 kJ/mol [18]. The entropy change
is 33:6 J/mol K, which is larger than the PBV-MMP prediction of 9:7 J/mol K.
Full solution Due to the complexity of Eq. (33), a least-squares regression was not able to be utilized to determine the sorption and diffusion parameters. Instead, the results from the short time with sorption analysis were used as starting points for a guess-and-check estimation, changing the parameters to generate the best fitting plot. Also, while the solution calls for an infinite summation especially as t/0, a finite number of terms must be used. The model was, thus, split over two timeperiods of the experiment: the initial short-time section (0e25,000 s) and the long-time portion (~75,000 s to the end of the experiment). For the short-time portion, the
Fig. 12 e Long-time solution for indium at 450 C, using the first two terms of the summation only. The data points are blue, the long-time solution is the solid red line, and the short-time approximation is the dashed red line. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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 4 3 ( 2 0 1 8 ) 1 9 0 7 5 e1 9 0 9 0
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Fig. 13 e Plot of hydrogen flux through a SLiMM [10] versus inverse temperature with various Sieverts' law estimates. The Sieverts' apparatus parameters (dashed blue) is a worse fit than the PBV-MMP parameters [18] (dashed red), while both predict lower flux compared to the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
approximation developed as Eq. (37) was used. The long-time portion used two terms in the summation from Eq. (33). The previously calculated Sieverts' constant from the equilibrium pressure was utilized here. Next, the diffusion coefficient was varied until reasonable agreement with the mid-time portion of the experiment was modeled. Due to their relation to the Sieverts' constant, only one of the sorption rate constants needs to be determined. However, it was found that the value of the rate constants does not have an impact on the model predictions. An example, for an indium run at 450 C, is shown
in Fig. 12. The short-time approximation is plotted from 0 to 75,000 s while the long-time solution is plotted from 25,000 s to the end of the run. While the short-time approximation has a good fit from 0 to 25,000 s and the long-time solution fits the data well from ~75,000 s on, there is a region from 25,000e75,000 s, when neither equation provides a reasonable fit to the data. The diffusion coefficients found from the long-time model were found to be two orders of magnitude smaller than the corresponding coefficients from the short-time approximation
Fig. 14 e Plot of hydrogen flux through a SLiMM [10] versus the difference of partial pressure square roots with various Sieverts' law estimates. The Sieverts' apparatus parameters (dashed blue) is worse than the PBV-MMP parameters [18] (dashed red), while both predict lower flux than the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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above. It is not known why the diffusion coefficients would differ so drastically between the theoretical models. The preexponential factor is around 200 times smaller compared to the predictions from the PBV-MMP framework [18] for each liquid metal, while the activation energy of diffusion is only slightly larger than the PBV-MMP predictions. These diffusion coefficient results are lower than hydrogen diffusion in solid palladium.
Application to membrane permeation The diffusion parameters identified, along with the Sieverts' constant, can be used in Eq. (1) to model the hydrogen permeation in metal membranes, such as the SLiMMs developed in our lab [10]. Comparisons through Sieverts' law to these experiments can be made by plotting the flux of hydrogen, expressed as either mol=m2 s or m3 =m2 s, versus 1= pffiffiffiffiffi pffiffiffiffiffi TðKÞ or Pf Pp , as shown in Figs. 13 and 14, respectively. In both cases, there are two sets of parameters shown for the Sieverts' law model lines. The red dashed lines are Sieverts' law plotted using parameters estimated from the PBVMMP framework [18], which slightly under-predicts the experimental flux. The dashed blue lines use the parameters determined using the Sieverts' apparatus in this paper, specifically the inner solution diffusivity. These predictions are further from the experimental values. The experimental flux is expected to be higher than the predictions due to two possible contributions. The first is that the models used to predict the parameters only account for atomic hydrogen diffusion, and there is the potential in liquids for molecular hydrogen diffusion as well, which would increase the flux. There is also the possibility of poor surface coverage or pinholes developing in the membrane that contribute to increased flux of molecular hydrogen through the holes, as opposed to the diffusion through the metal membrane.
steady-state values are more in line with expected predictions from PBV-MMP and experimental palladium results. These metals show signs of there being promising alternatives to the traditional palladium-based hydrogen purification membranes, and a Sieverts' apparatus is a useful tool for screening potential metal and alloy candidates.
Acknowledgement We acknowledge financial support from the Department of Energy under award number DE-FE0001050.
Nomenclature Symbols A AM AWM Bi
cS;t
pre-exponential factor in Sieverts' constant surface area of metal, cm2 atomic weight of metal M, g/mol dimensionless parameter akin to Biot number, Eq. (19) the lattice constant, cm interstitial hydrogen concentration, mol H$cm3 saturation concentration of interstitial hydrogen, mol H$cm3 total concentration of the surface metal atoms,
ct
mol M$cm2 total concentration of the metal atoms in bulk,
dM DH
mol M$cm3 atomic diameter of metal atom M, m interstitial diffusion coefficient of the hydrogen
DH;0
atoms, cm2 s1 pre-exponential factor of interstitial diffusion
a cH$M cH$M;s
ED
Conclusions fH2 The standard Sieverts' device is capable of providing several characteristics of hydrogen transport and storage behavior in metals, be they in solid or liquid form. Liquid indium and gallium were found to have similar hydrogen diffusion coefficients and Sieverts' constants to each other. Both liquid metals also had similar solubilities, which were an order of magnitude lower than the solubility of hydrogen measured in palladium. The diffusion coefficient is able to be accurately predicted when compared with the PBV-MMP framework [18] when utilizing the inner solution for short times (the first ~1000 s). The diffusivity was found to be on the order of 103 104 cm2 =s for both metals, and the activation energy of diffusion ranged from 20 25 kJ=mol. The sorption thermodynamic parameters are relatively consistent when using the steady-state equilibrium and the inner solution. The enthalpy change is around 11 16 kJ=mol for each metal, comparable to palladium, but significantly lower than the PBV-MMP predictions. The entropy change from the steady-state solution is around 5 to 10 J=mol$K, while the inner solution estimates the indium entropy change of sorption as 33 J=mol$K. The
fH$M fH$X
coefficient of H, cm2 s1 activation energy of interstitial diffusion of H, J/mol H dimensionless partial pressure of hydrogen, defined by Eq. (17) dimensionless mole fraction of hydrogen in bulk metal, defined by Eq. (17) dimensionless mole fraction of hydrogen in the interfacial layer
f H2 ðsÞ
Laplace transform of fH2 ðtÞ
f H$M ðsÞ h KS
Laplace transform of fH$M ðtÞ Plancks's constant Sieverts' equilibrium constant for the hydrogen solution in metal
kB ! kS
Boltzmann constant, 1:38064 1023 J$K1
)
kS M nH$M nM nH2;i
forward rate constant for Sieverts' step, s1 reverse rate constant for Sieverts' step, s1 dimensionless parameter, Eq. (19) number of moles of hydrogen in metal, mol number of moles of metal, mol number of hydrogen molecules introduced initially into Sieverts' apparatus
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 3 ( 2 0 1 8 ) 1 9 0 7 5 e1 9 0 9 0
nH2 ;∞
total amount of hydrogen gas transferred in Sieverts' apparatus, mol H2
NAv
Avogadro's number, 6:022 1023 mol1
NH
flux of atomic species H, mol$cm2 $s1 1
NH2 NM pH2 pH2 ;i
flux of molecular species H2, mol$cm $s number of metal atom monolayers in the metal pool hydrogen partial pressure, Pa or atm initial partial pressure of hydrogen, Pa
PH2 p
membrane permeance, mol H2 $cm2 $s1 $atm1=2 standard atmosphere, 101,325 Pa, or 1 atm
rS
rate of reaction of Sieverts' step, mol H$cm2 $s1 net rate of the elementary reaction r, ! r
rr
r
)
r r $mol$cm2 $s1
! rr
forward rate of elementary reaction r, mol$cm2 $s1
)
rr s sr T TC Th t VH2 VC Vh xH$M xH$M;i
reverse rate of elementary reaction r, mol$cm2 $s1 Laplace variable elementary step r temperature, K temperature of cold zone, K temperature of hot zone, K time, s volume of hydrogen absorbed by metal, cm3 volume of cold zone, cm3 volume of hot zone, cm3 mole fraction of interstitial hydrogen in bulk metal equilibrium mole fraction of hydrogen in metal
xH$M;0 xH$X y z z
corresponding to the initial hydrogen partial pressure initial mole fraction of hydrogen in metal mole fraction of hydrogen in the interfacial layer depth from the metal pool surface, cm axial distance from the metal pool bottom, cm number of nearest neighbor interstitial positions
Greek Symbols a geometric factor in interstitial diffusion coefficient a ratio of hot to cold reservoir volumes of Sieverts' apparatus b root defined in Eq. (35) g root defined in Eq. (35) d membrane thickness, cm
DGS
DHS
DSS
DSzD h h q q lk
dimensionless axial distance, Eq. (18) frequency of vibration of the solute in an interstitial position, s1 density of the gas, g/m3
x n0 rg
2
standard Gibbs free energy change of hydrogen solution in metal, J/mol H standard enthalpy change of hydrogen solution in metal, J/mol H standard entropy change of hydrogen solution in metal, J=mol$K standard entropy change of activation for diffusion, J=mol$K dimensionless penetration theory variable combining time and position pffiffiffi square root of Laplace variable, h ¼ s ratio of hot to cold temperatures of Sieverts' apparatus rescaled dimensionless time for inner solutuion, q ≡ ðNM Þt non-zero kth root of the transcendental equation, Eq. (31)
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rM sr t cM cH$M cH$M;s ! uS ! u S;i
density of metal M, g/m3 stoichiometric number for step r dimensionless time, Eq. (18) atomic ratio of vacant interstitial sites, cM =ct interstitial atomic ratio of absorbed hydrogen, cH$M =ct the saturation value of cH$M , cH$M;s =ct forward step weight for Sieverts' step, s1 initial value of forward step weight for Sieverts' step, corresponding to initial hydrogen pressure, s1
) uS
ur ! ur ) ur
reverse step weight for Sieverts' step, s1 step weight for reaction sr forward step weight for reaction sr reverse step weight for reaction sr
Sub- and super-scripts C cold zone of Sieverts' apparatus f feed side of membrane h hot zone of Sieverts' apparatus p permeate side of membrane r elementary reaction step S unoccupied surface site M unoccupied interstitial site in bulk metal X unoccupied subsurface site in interfacial layer
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Evaluation of hydrogen sorption and permeation parameters in liquid metal membranes via Sieverts' apparatus Nicholas D. Deveau, Pei-Shan Yen, Ravindra Datta* Fuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, 100 Institute Rd, Worcester, MA 01609, USA
article info
abstract
Article history:
We have recently proposed sandwiched liquid metal membranes (SLiMM) for hydrogen
Received 16 June 2018
separation. To evaluate SLiMM, thermodynamic and kinetic parameters such as the sol-
Received in revised form
ubility, diffusion coefficient, and absorption kinetics of hydrogen in the liquid metal are
14 August 2018
needed. While there are some theoretical approaches to estimate these parameters,
Accepted 17 August 2018
including our own published recently, it is important to obtain experimental corroboration.
Available online 12 September 2018
This study utilizes the classical Sieverts' apparatus in an effort to estimate these parameters by monitoring the change in pressure with time of hydrogen introduced over a pool of
Keywords:
liquid metal within a container. The solubility was calculated from the change in pressure
Sieverts' apparatus
over the entire duration of the experiment as it attained equilibrium, while typically the
Metal-hydrogen system
diffusion coefficient could be determined from the short-time response. The theory behind
Liquid metal
the Sieverts' apparatus is extended here to provide full-time solution by linearizing Sie-
Hydrogen sorption
verts' law, along with early (very short) time solutions to determine the sorption kinetics in
Hydrogen diffusion
addition to the diffusion coefficient. In this manner, new theoretical and experimental results are obtained for hydrogen sorption and diffusion in liquid gallium and indium at different temperatures. The results corroborate the hydrogen permeability in a gallium SLiMM. © 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Due to the increasing demand for pure hydrogen in applications at different scales ranging from chemicals to fuel cells, there is significant interest in developing dense metal membranes for separation and purification of hydrogen. These metal membranes promise to be more energy efficient and selective when compared to traditional hydrogen purification technologies such as pressure swing adsorption [1]. The state-
of-the-art dense metal membrane for hydrogen uses palladium or its alloys with silver and copper, manufactured as thin foils or as thin but dense membranes deposited on a porous support. While very promising, these membranes have some distinct disadvantages. Palladium, and to a lesser extent silver, is expensive. These membranes also come up short in regard to durability, as exceedingly thin membranes (~10e20 mm) are employed to concomitantly reduce cost and increase permeance. Further, these solid but thin films can develop cracks and micropores that allow other gases to pass
* Corresponding author. E-mail address: [email protected] (R. Datta). https://doi.org/10.1016/j.ijhydene.2018.08.101 0360-3199/© 2018 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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through and hence reduce membrane selectivity and product purity. Such defects cannot easily be repaired, requiring membrane replacement. These limitations are driving research into seeking alternative and cheaper metals with adequate hydrogen permeability [2e9], which could be fashioned into thicker, more robust, solid membranes. In this vein, we have for the first time recently proposed thicker films (~200 mm) of low-melting metals or alloys as sandwiched liquid metal membranes (SLiMMs), as shown schematically in Fig. 1 [10]. Such liquid metal membranes could potentially address some of the issues associated with solid metal membranes, e.g., sintering, hydrogen embrittlement, and thermal mismatch between support and membrane. However, liquid metals provide their own challenges. There is limited literature on liquid metal membranes due to their high reactivity with other metals [11e16], as well as wettability issues [17] with ceramic supports. There is limited data on the formation of hydrides with liquid metals and any molecular species diffusion. Although there is limited available experimental results, they indicate that the solubility and diffusivity of hydrogen of a metal increases upon melting [10]. Here, we utilize Sieverts' apparatus to determine the relevant parameters for two low melting metals of interest for SLiMM, namely, gallium (29.8 C, MP) and indium (156.6 C, MP), and compare these to our recent theoretical predictions [18]. The hydrogen permeation in a metal, as proposed by Wang and Roberts [19], consists of five sequential steps, described schematically in Fig. 2. Thus, hydrogen is first dissociatively adsorbed onto the metal surface on the feed side, and then penetrates to the sub-surface of the bulk metal. Next, the hydrogen atoms interstitially diffuse across the membrane, ultimately emerging on the opposite surface. The final step is the associative desorption of hydrogen from the second surface of the metal membrane on the permeate side. The flux of hydrogen (NH2 ) in traditional palladium-based membranes is described using what is commonly known as Sieverts' law [20], first proposed by Richardson et al. [21], NH2 ¼
ct cH$M;s KS DH 1=2 1=2 1=2 1=2 pffiffiffiffiffio pH2 ;f pH2 ;p ≡PH2 pH2 ;f pH2 ;p 2d p
(1)
which relates the hydrogen flux to the partial pressure of hydrogen (pH2 ) on the feed (f) and permeate (p) sides,
Fig. 1 e Schematic of sandwiched liquid metal membrane (SLiMM).
Fig. 2 e Schematic of the five steps of hydrogen diffusion through metal: (1) dissociative adsorption, (2) solution to the bulk, (3) diffusion, (4) dissolution to the surface, and (5) associative desorption.
membrane characteristics such as metal atom concentration (ct) and maximum hydrogen solubility (cH$M;s ), membrane thickness (d), Sieverts' equilibrium constant (KS ), and diffusion coefficient (DH ). This simple relation is derived assuming that the diffusion step is the rate-determining step (RDS) in the sequence of steps, and that the hydrogen solubility is low, so that the interstitial mole fraction can be approximated via Sieverts' law, following a square-root dependence on pffiffiffiffiffiffiffiffiffiffiffiffiffiffi hydrogen partial pressure, xH$M zKS pH2 =p , where p is standard (atmospheric) pressure. These assumptions may not necessarily remain true under all conditions for the liquid metals of interest, and indeed have even been shown [22] to be invalid under certain conditions for palladium membranes, when the adsorption/desorption steps can be the RDS. While a framework has been developed to ascertain this [22], however, the solubility and diffusion parameters needed for the model for many metals, especially liquid metals, are unknown. Finally, although we are specifically interested here in the potential application of liquid metal-membranes, there are other applications of liquid metals as well where these parameters are of interest, e.g., as coolants in fast neutron reactors, in metal casting, and perhaps even in hydrogen storage [18]. One convenient way to determine some of these kinetic and thermodynamic parameters is by monitoring hydrogen uptake, or evolution, over a liquid metal pool in what is known as the Sieverts' apparatus [23,24]. This apparatus can be set up to operate in one of two different modes: 1) a constant-volume mode, known as a pressure Sieverts' apparatus [23], or 2) a constant-pressure mode, known as a volume Sieverts' apparatus [24]. Both modes of operation follow similar set-ups and theory, having the metal of interest of known surface area and volume within a closed system under hydrogen atmosphere that monitors either the system pressure at a constant volume (pressure Sieverts' apparatus), or the total system volume at a constant pressure (volume Sieverts' apparatus), so as to determine the amount of hydrogen permeating into the metal
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as a function of time. In conjunction with theory, the results of such experiments [25e28] can provide the solubility of a gas and its diffusion coefficient. The diffusion coefficient is typically estimated from experimental data for short times only, in order to utilize the simplified mathematical analysis based on penetration theory, applicable rigorously for a metal pool of infinite depth. Further, the Sieverts' experiments can be performed for gas absorption into a metal or for metal degassing. The absorption experiments begin with the metal sample being initially devoid of hydrogen. The gas chamber is then charged with pure hydrogen gas, and the pressure, for instance, is monitored as a function of time. As hydrogen is absorbed into the metal sample, the pressure in the chamber declines until the sample becomes saturated corresponding to the final equilibrium hydrogen pressure. For degassing, or off-gassing, experiments, on the other hand, the metal sample is initially saturated with dissolved hydrogen before being exposed to an atmosphere devoid of hydrogen. As the hydrogen desorbs from the metal, the pressure increases until an equilibrium between absorbed and gaseous hydrogen is achieved corresponding to the final hydrogen pressure. For complete degassing of a sample, however, multiple such cycles are needed. The available theoretical treatment in the literature regarding analysis of Sieverts' experiments is limited. Not only is it generally limited to short times, rather than to the full duration of the experiment, but it does not account for the sorption steps (Fig. 2). Thus, Sacris and Parlee [27] studied hydrogen diffusion into liquid nickel, copper, silver, and tin, by monitoring the volume of hydrogen VH2 absorbed as a function of time, t. Following a very similar procedure, Depuydt and Parlee [26] studied hydrogen diffusion in liquid iron alloys. In both studies, the hydrogen diffusion constant was estimated from the experimental data using the relation based on penetration theory solution, applicable to short times VH2 ¼
pffiffiffiffiffiffiffiffiffiffiffi d2 rM CS pDH t 200rg
(2)
where VH2 is the volume of hydrogen absorbed, d is the metal pool diameter, rM is the density of the metal, DH is the diffusion coefficient of hydrogen atoms in the metal, and rg is the density of the gas. A goal of this study is to extend such theoretical modeling of Sieverts' experimental apparatus in order to extend its applicability to long times, and to also potentially allow evaluation of sorption kinetic parameters for hydrogen from experimental data at very short, or early, times. This goal is accomplished in two parts. First, a solution is developed that is valid over the full time of an experiment, rather than being limited to short times. Second, absorption step into the bulk material is considered and solution obtained for early times. The model is finally used to analyze experimental data for liquid Ga and In in order to estimate hydrogen sorption and transport parameters that are compared to our recent theoretical predictions [18]. If one is interested in modeling a complete set of such an experiment until the system has reached an equilibrium,
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there are two alternate approaches. In the first, Solar and Guthrie [29] extend the above penetration theory solution, which holds only for a small change in gas pressure, into an infinite series of small pressure changes. The second approach involves solution of the coupled differential equations for the mixed gas phase and the stagnant liquid metal phase. The latter approach has not been developed so far for the Sieverts' experiment, because of the non-linearity (square root dependence on pressure) of the Sieverts' solubility that precludes an analytical solution. Here, we will use the Laplace transform method to obtain such a long-term solution for the analysis of a complete data set from Sieverts' experiments [30e33], by first linearizing the Sieverts' solution step.
Experimental apparatus and methods The experimental apparatus used in this study resembles the constant volume set-up of Small and Pehlke [28]. As shown schematically in Fig. 3, the total gas volume consists of a hightemperature “hot zone” and a room temperature “cold zone”. The volume of the hot and cold zones were estimated using helium gas and the ideal gas law relation P1 V1 ¼ P2 V2 . The cold zone was found to have a volume of 84.8 cm3 while the hot zone, accounting for the graphite crucible, had a volume of 30.69 cm3 . The cold zone (Fig. 3) comprises of tubing and fittings for controlling the gas inlet, in addition to two pressure transducers (Transducers Direct, 0e10 V DC output, 0e500 psi) at room temperature. One transducer is set up to measure the absolute pressure of the system, and is connected to a National Instruments data box (NI USB 6008) for computer acquisition of the pressure versus time data. The other transducer measures the gauge pressure of the system. The cold zone is connected to the gas supply cylinders via a buffer tank and a pressure regulator. The hot zone includes a section of tubing connecting the cold zone tubing to a chamber containing the sample. This zone is enclosed within a cylindrical ceramic heater (Watlow 120 V, 900 W) controlled by a temperature controller (Eurotherm 2116), which also monitors the temperature with a thermocouple inserted through the top of the furnace and situated just above the chamber in the middle of the furnace. Indium or gallium nuggets were placed in a graphite crucible machined from a 2.539 cm diameter, 1.126 cm long graphite rod. The cylindrical hollow of the crucible was machined to a diameter of 1.879 cm and 0.619 cm deep. As an example a sample of pre-melted indium inside the graphite crucible is shown in Fig. 4 prior to the experiments. The indium sample was 5.2550 g in mass, and the gallium sample was 5.8808 g. The copper gasket ring shown in the picture, placed on the visible knife-edge in the depression just inside the ring of bolt holes, acted as a compressive seal when the top half of the chamber was fastened to the bottom half using eight bolts. The metal (Ga or In) samples were, of course, in the molten state under experimental conditions when the temperature was raised above their respective melting points. In these experiments, hydrogen was the solute and helium was the inert gas used during the off-gassing runs.
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Fig. 3 e Schematic of Sieverts' apparatus: Gas inlet on the left, the metal sample (gray) is in a graphite crucible (black), and there are two pressure transducers for pressure monitoring. Hot zone is demarked with dashed red, cold zone with dashed blue. (For interpretation of the references to color/ in this figure legend, the reader is referred to the Web version of this article.)
Once the sample was sealed in the chamber and enclosed in the furnace, the system was purged of air and filled with the inert gas prior to heating. For each of the experiments with liquid metals, five different temperatures were studied in the range of 350e550 C of interest for SLiMM in 50 C increments. The system was then evacuated prior to the start of the absorption run, and was quickly charged with about 4 psig of hydrogen before the system inlet was again closed. The
gallium runs had initial pressures of 960e990 torr, while the indium runs had initial pressures of 880e920 torr. The pressure was monitored and recorded every few minutes until equilibrium was reached. These equilibrium pressures varied with temperature and the metal sample. Gallium runs ended between 450 torr (at 500 C) and 530 torr (at 350 C). The indium runs ended between 530 torr (at 500 C) and 580 torr (at 350 C). Between absorption experiments, the system was purged with the inert gas to completely remove any hydrogen from the metal sample. The system was initially purged of hydrogen by briefly venting to atmosphere and flushing with the inert gas. The system was then pressurized with the inert gas to around 10 psig. The hot zone valve was closed, and the cold zone was released to ambient pressure before opening the hot zone valve to reduce the pressure. Pressure reduction cycle was performed in total two to three times to reduce the pressure to values able to be monitored by the transducer. The pressure again was monitored until a new equilibrium was attained. The absorption runs were performed twice for each metal sample at each temperature.
Theory
Fig. 4 e Pre-melted (shown solidified) indium sample inside graphite crucible. The copper gasket used to seal the Sieverts' apparatus chamber is shown in place.
Our goal is to develop a theoretical model that relates the observed change in gas pressure pH2 ðtÞ in the Sieverts' device to hydrogen permeation in the liquid metal pool, based on material balance in gas and metal phases, coupled with Fick's law of diffusion and Sieverts' law of solubility in the metal phase. A schematic representation of the Sieverts' apparatus
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is shown in Fig. 5. In fact, it depicts three distinct phases in the apparatus: 1) the gas phase, 2) the metal phase, and 3) the interfacial layer. The interfacial layer is assumed to be of essentially monoatomic thickness that contains the liquid metal surface on the one side and bulk metal on the other. From the total concentration of atoms in the bulk metal, ct ¼ rM =AWM , thus, the concentration of atoms on the surface can be found using the atomic diameter of the metal as cS;t ¼ ct dM . It is further assumed that the gas phase is well-mixed, resulting in a uniform pressure and composition across the hot and cold zones, while the liquid metal pool is stagnant. The gas volumes (Vh ; VC ) and temperatures (Th ; TC ) of the hot and cold zones are assumed to be constant. Starting at time t ¼ 0 when it is introduced into the gas chamber, hydrogen dissociates on the liquid metal surface (step s1 ) and is then atomically dissolved (step s2 ) in the interfacial layer at the rate rS uniformly across the whole surface area of the metal (AM ). Consequently, the mole fraction of atomic hydrogen in the metal (xH$M ), starting at some initial value (xH$M;0 ), varies with time and depth (y) across the total metal pool of thickness (d). The coordinate z ¼ d y. The dissolved hydrogen leaves the interfacial layer via diffusion into the bulk metal at the rate of NH jd (in the schematic, the flux is shown in the positive z direction). At the very start of the experiment, for a very short time, or during the early time period, the difference between these two rates accounts for accumulation of atomic hydrogen within the interfacial layer. Thereafter, it essentially attains, for the long term, quasisteady-state (QSS) within the layer, when rS zNH jd , and any accumulation in the interfacial layer is negligible.
Formulation of the theoretical model Unsteady-state mass balance in the interfacial layer As discussed above and shown schematically in Fig. 2, there are a number of sequential steps involved in hydrogen
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permeation into a metal, although diffusion is often the rate determining step. Under certain conditions, however, it has been shown [22] that the adsorption (s1) and desorption (s5) steps can also have a significant impact on the hydrogen flux. On the other hand, steps s2 (dissolution from the surface into the subsurface) and s4 (evolution to the surface) are almost invariably rapid. Assuming, thus, that step s2 is rapid in comparison with adsorption step s1, the first two steps can be combined into an overall Sieverts' dissolution step (sS ) sr sr s1 : H2 þ 2S#2H$S þ1=2 s2 : H$S þ X#H$X þ S þ1 sS : ½H2 þ X#H$X
(3)
here, S represents a surface adsorption site, X represents the interfacial layer site, and sr is the stoichiometric number of step sr , such that an overall reaction (OR) is given by the linear P combination of steps, OR ¼ sr sr . As s2 is assumed to be rapid r
[6], the hence combined dissolution step (sS ) in equation (3) may be treated as an elementary step, with mass-action kinetics, i.e., u S cS;t ð1 xH$X Þ u S cS;t xH$X rS ¼ ! )
(4)
where xH$X is the mole fraction of hydrogen in the interfacial layer. For convenience, further, the kinetic parameters are lumped in the so-called step weights ! ! uS ¼ k S
rffiffiffiffiffiffiffiffi ) pH2 ) ; uS ¼ k S p
! ! where k S ¼ k 2
(5)
qffiffiffiffiffiffi qffiffiffiffiffiffiffi ) ) ) ! k 1 and k S ¼ k 2 k 1 are the dissolution step
rate constants, while those with subscripts 1 and 2 correspond to rate constants for steps 1 and 2. Further, these are related to the Sieverts' equilibrium constant via the microscopic ! ) reversibility relation KS ¼ k S = k S .
Fig. 5 e Simplified representation of the Sieverts apparatus. The hot zone, which includes the metal sample (represented as the shaded blue region), is on the left in this representation, while the cold zone is on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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The diffusive flux of atomic hydrogen in the metal phase is assumed to be given by Fick's Law of the form vxH$M NH ¼ ct DH vz
(6)
where the mole fraction of hydrogen in the bulk metal phase, xH$M , can be simplified under the assumption that the number of moles of hydrogen are much less than those of the metal, nH$M ≪nM , so that, xH$M ≡nH$M =ðnH$M þ nM ÞznH$M =nM ¼ cH$M =ct , where ct is the concentration of the metal atoms in the bulk. Using Fick's law along with the dissolution kinetics, Eq. (4), in the unsteady-state mass balance in the interfacial layer (Fig. 5), cS;t ðvxH$X =vtÞ ¼ rS þ NH jz¼d , thus yields cS;t
h i vxH$X vxH$M ) u S ð1 xH$X Þ u S xH$X ct DH ¼ cS;t ! vt vz z¼d
dnH2 ¼ dt
VC Vh þ RTC RTh
dpH2 dt
(12)
Combining this with the gas phase mass balance (dnH2 =dt ¼ AM rS =2) results in VC a dpH2 1 vxH$M jz¼d 1 vxH$M 1þ ¼ AM cS;t AM ct DH q 2 2 RTC dt vt vz z¼d
where the hot/cold volume ratio is a ¼ Vh =VC and the temperature ratio is q ¼ Th =TC . This is subject to the initial condition IC : at t ¼ 0; pH2 ¼ pH2 ;i
(14)
Unsteady-state mass balance in the metal phase (7)
where the left-hand side represents any accumulation in the interfacial layer. Further, if we assume that the solution of hydrogen in liquid metal is dilute, xH$X ≪1, and that xH$X zxH$M jz¼d , the mass balance takes the form
The mass balance of the metal phase is given by Fick's second law, vxH$M v2 xH$M ¼ DH vt vz2
(15)
subject to
8 IC : at t 0; xH$M ¼ xH$M;0 ¼ 0 > > > > > vxH$M < ¼ 0 BC1 : at z ¼ 0; vz > > > > vxH$M jz¼d vxH$M ) > : BC2 : at z ¼ d; cS;t u S u S xH$M jz¼d ¼ ct DH þ cS;t ! vt vz z¼d
vxH$M jz¼d vxH$M ) u S u S xH$M jz¼d ¼ ct DH þ cS;t ! cS;t vt vz z¼d
(8)
The forward step weight ! u S in this relation depends on the square root of hydrogen partial pressure, which is a function of time. In order to utilize the Laplace transform method for solution, consequently, we must linearize this relation. We do so about the dimensionless initial partial pressure pH2 ;i =p , and retain only the linear term, resulting in ! u S;i ! u Sz 2
pH 1þ 2 pH2 ;i
(9)
rffiffiffiffiffiffiffiffiffi pH2 ;i p
(10)
Further, we define the equilibrium bulk mole fraction corresponding to the initial hydrogen partial pressure xH$M;i ≡
! u S;i ) uS
¼ KS
rffiffiffiffiffiffiffiffiffi pH2 ;i p
Dimensionless form We recast the equations in terms of the following dimensionless variables and parameters. Thus, the dependent variables, the partial pressure and mole fraction of hydrogen, are defined as fH2
! 1 pH2 xH$M ≡ 1þ ; fH$M ≡ 2 pH2 ;i xH$M;i
The ideal gas law can be used to relate the change in the total number of moles of hydrogen in the gas phase (nH2 ) to the pressure
(17)
while the independent variables, time and position, are defined as DH z t≡ 2 t; x ≡ d d
(18)
A dimensionless parameter, Bi, similar to the Biot number, is defined along with the number of metal atom monolayers NM in the metal pool, and an experimental parameter, M
(11)
Unsteady-state mass balance in the gas phase
(16)
where the first boundary condition represents no flux at the bottom of the liquid metal pool, while the second represents mass balance at the gas-metal interface.
!
where the initial forward step weight, which is a constant, is defined as ! ! u S;i ¼ k S
(13)
Bi ≡
) 1 d2 u S d 1 nM ! u S;i ; NM ≡ ; M≡ NM DH dM 2 2nH2;i ) uS
! (19)
Interfacial Phase: The interfacial phase mass balance in dimensionless form is
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vfH$M 1 vfH$M x¼1 þ Bi fH2 fH$M x¼1 ¼ NM vx x¼1 vt
(20)
Clearly, the number of atomic monolayers in the liquid metal pool, NM is a very large number (~107), which would justify neglect of the accumulation term in the interfacial layer and hence support the QSS assumption made in the long-term analysis. In such a case, the interfacial phase mass balance simplifies to vfH$M ¼ Bi fH2 fH$M x¼1 vx x¼1
(21)
Gas Phase: The mass balance in the gas phase in dimen-
d2 f H$M sf H$M ¼ 0 dx2 8 df H$M > > BC1 : at x ¼ 0; ¼0 > > > dx < > > df H$M > > > BC2 : at x ¼ 1; ¼ Bi f H2 f H$M : x¼1 dx x¼1
Eq. (27) can be solved using the characteristic equation for a second order differential equation, in this case having roots of pffiffiffi ± s. This yields a result of C1 ex þ C2 ex . However, substituting in ex ¼ coshðxÞ þ sinhðxÞ and ex ¼ coshðxÞ sinhðxÞ and defining new constants of A1 ¼ C1 þ C2 and A2 ¼ C1 C2 gives pffiffiffi pffiffiffi f H$M ¼ A1 cosh s x þ A2 sinh s x
sionless form is dfH2 vfH$M ¼ M dt vx x¼1
(22)
IC : at t ¼ 0; fH2 ¼ 1 Metal Phase: The mass balance in the metal phase in
(27)
(28)
When applying the boundary condition BC 1, the constant A2 ¼ 0. Applying BC 2 gives the remaining constant as A1 ¼
dimensionless form is
f 2 pffiffiffi H pffiffiffi pffiffiffi cosh s þ Bi1 ssinh s
(29)
Substituting the resulting expression for the metal phase vfH$M v2 fH$M ¼ vt vx2 8 IC : at t 0; fH$M ¼ 0 > > > > > > > > < BC1 : at x ¼ 0; vfH$M ¼ 0 vx > > > > > > vfH$M > > ¼ Bi fH2 fH$M x¼1 : BC2 : at x ¼ 1; vx
into Eq. (26) for the gas phase and solving for the variable f H2 , yields (23)
lk tanðlk Þ ¼
Full solution To solve this system of the coupled differential equations, thus, we take the Laplace transform of the dependent variables, i.e., L½fH$M ðtÞ ¼ f H$M ðsÞ and L½fH2 ðtÞ ¼ f H2 ðsÞ, so that the 2nd order partial differential equation (PDE) representing mass balance in the liquid metal pool reduces to a 2nd order ordinary differential equation (ODE), while the 1st order ODE representing mass balance in the gas phase reduces to an algebraic equation. These can be solved simultaneously in the Laplace domain, and eventually inverted back into the time domain. Applying the Laplace transform to the gas phase, Eq. (22), results in df ¼ 1 M H$M dx
(30)
fraction has roots at s ¼ 0 and sk ¼ l2k , where lk represents a non-zero root of the denominator expressed as
Model solution
l2k M Bi1 l2k
(31)
From these roots, the dimensionless pressure in the time domain can be found using Cauchy's residue theorem [30,31]. fH2 ¼ Resð0Þ þ
∞ X
Resðsk Þ
(24)
(32)
k¼1
The solution is, in fact, provided by Carslaw and Jeager [31], which in rearranged form is pH2 ðtÞ ¼
1M pH2 ;i 1þM þ 4pH2 ;i MBi2
∞ X k¼1
x¼1
2
elk t MBi2 ð1 þ MÞ þ Bið1 þ Bi 2MÞl2k þ l4k (33)
Short-Time Approximation of Laplace Solution: The full
At the interface, Eq. (21) becomes df H$M dx
pffiffiffi pffiffiffi 1 þ 1 stanh s Bi 1 pffiffiffi pffiffiffi s þ M þ Bi s stanh s
To invert this expression back into the time domain, we use the Heavyside expansion theory. The denominator of the
x¼1
sf H2
f H2 ¼
¼ Bi f H2 f H$M
x¼1
(25)
x¼1
and combining this with Eq. (24) yields sf H2 ¼ 1 BiM f H2 f H$M
x¼1
(26)
For the metal phase, the Laplace transform of Eq. (23) yields
solution, Eq. (33) at short times can require an inordinately large number of terms in the summation to converge approximately to the initial condition. Therefore, it is of not a very suitable form for computation at short time. On the other hand, at larger times, the number of terms needed is quite small, and the solution provided is convenient. It is, therefore, desirable to obtain alternate closed forms solution for short times. In fact, this can be readily obtained from the Laplace domain solution as follows. Starting from Eq. (30) by noting
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pffiffiffi pffiffiffi s 3, which results in tanhð s Þ/1. As a pffiffiffi result, and using a change of variable h ¼ s, the Laplace domain solution can be reduced to
that when t/∞,
h þ Bi h þ Bi ¼ f H2 z 3 h þ ðBiÞh2 þ ðMBiÞh hðh þ bÞðh þ gÞ
(34)
b¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bi2 4MBi Bi þ Bi2 4MBi ;g ¼ 2 2
(35)
Eq. (34) can be rewritten using partial fraction expansion as f H2
at q' 0 fH$X ¼ 0
(39)
The system can then be modeled with a single partial differential equation for the metal phase, with a change in the coordinate based on y ¼ z d (or x' ¼
y d
¼ x 1). Further, the
mass balance in the bulk metal takes the form
where Bi
IC :
1 A2 B2 C2 þ þ ¼ bgðg bÞ h hþb hþg
(36)
where the constants A2 ¼ Biðg bÞ, B2 ¼ ðBi bÞð gÞ, and C2 ¼ ðBi gÞðbÞ. The terms in the brackets can each be inverted using pffiffiffi Laplace Transform Inversion Tables using h ¼ s , followed by combination and simplification finally provides pH2 ¼pH2 ;i
pffiffiffi
pffiffiffi 2 bexp g2 t erfc g t gexp b2 t erfc b t 1 bg (37)
subject to the conditions 8 < IC : BC1 : : BC2 :
at q' 0 fH$M ¼ 0 at x' ¼ 0 fH$M ¼ fH$X ðq'Þ at x'/∞ fH$M ¼ 0
for 0 x' ∞ for all q' > 0 for all q' > 0
(41)
Interfacial Phase: The mass balance for the interfacial phase can be solved independently of the metal phase. The result of this is fH$X ¼
1 exp Bi xH$M;i þ 1 q' xH$M;i þ 1
(42)
Metal Phase: The metal phase equations can be solved using the expression for fH$X in Eq. (42) and a composite vari'
2
The results obtained above, including those for short times, are based on the QSS assumption for the interfacial layer, which may alternately be viewed as the “outer solution” in terms of the singular perturbation theory [34], in which the simplified differential equations were first obtained by setting the singular perturbation parameter, 1=NM ¼ 0. It may be of interest to also obtain the “inner solution,” i.e., for the early time, when the QSS assumption for the interfacial layer does not apply, i.e., accumulation term in it is not negligible, from which additional information on the interfacial solution kinetics might be gleaned. For this case, we wish to further consider the full Sieverts' kinetics, i.e., without the dilute solution assumption that was made for the above QSS analysis. The scaling of dimensionless time t ¼ ðDH =d2 Þt used for the outer solution is further inappropriate for the inner solution. Therefore, we rescale time by defining a new dimensionless time as follows [34]: q'≡ðNM Þt. Using this in the dimensionless mass balance equations, and again setting 1=NM ¼ 0, results in dfH2 =dq' ¼ 0 and vfH$M =vq' ¼ 0. In other words, i.e., both gasphase and bulk metal phase concentrations remain virtually unchanged corresponding to their initial values over the very small period of interest for the inner solution, while the mass balance for the interfacial layer remains unchanged. In other words, pH2 /pH2 ;i , or fH2 /1, and xH$M /0, i.e., fH$M /0. Further, H$M since fH$M /0, one might argue that vfvx /0, so that the x¼1
diffusion term in the interfacial mass balance may be dropped at early times, which decouples this from bulk metal balance. Dropping the diffusion term, the mass balance equation for the interfacial layer reduces to
which is subject to the initial condition
(40)
x able h ¼ pffiffiffiffiffiffiffiffiffi , as '
Inner solution: early-time approximation
vfH$X zBi 1 xH$M;i þ 1 fH$X vq'
vfH$M 1 v2 fH$M ¼ NM vx'2 vq'
(38)
fH$M ¼
q =NM
1 exp Bi xH$M;i þ 1 q' erfcðhÞ xH$M;i þ 1
(43)
Thus, the flux of interstitial hydrogen at the metal/gas interphase is vxH$M ct xH$M;0 DH vfH$M vh ¼ vy y¼0 d vh h¼0 vx' x' ¼0 rffiffiffiffiffiffiffi
ct xH$M;0 DH NM 1 exp Bi xH$M;i þ 1 q' ¼ ' xH$M;i þ 1 d pq
NH jy¼0 ¼ ct DH
(44)
Using the relation in gas-phase mass balance, Eq. (12), along with AM NH jy¼0 =2 ¼ dnH2 =dt, results in rffiffiffiffiffiffiffi AM ct RTC DH xH$M;i pH2 ¼ pH2 ;i 2ð1 þ a=qÞ VC p xH$M;i þ 1 8 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = < pffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ) 2 t ) u S xH$M;i þ 1 t erf : ; u S xH$M;i þ 1
(45)
Below we use the theoretical results obtained above to estimate sorption and diffusion parameters for hydrogen diffusion in liquid metals.
Results and discussion Due to the complexity of Eqs. (33) and (37), a simple leastsquares regression of the experimental data was not able to provide the best sorption and diffusion parameters. Therefore, the following approach was adopted to determine the parameters: 1) first, the steady-state final pressure was used with the first term of Eq. (33), i.e. pH2 ðt/∞Þ ¼ 1M 1þM pH2;i , to calculate the
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parameter M, and hence the solubility xH$M;i and the Sieverts' constant KS ; 2) next, the inner solution, Eq. (45), was used with a leastsquares regression over the first ~1000 s of experimental ) data to determine initial estimates of u S and DH ; and 3) finally, the full solution, Eq. (33), is plotted against the ) experimental data and the values of u S and DH and manually adjusted to provide the best fit.
Steady-state data
nH$M nH$M 2nH2 ;∞ z ¼ nH$M þ nM nM nM
(46)
At a pressure of 18.7 psia, thus, the hydrogen solubility in gallium ranges from xH$M ¼ 0.042 at 350 C to 0.063 at 500 C, while solubility in indium is slightly higher, ranging from xH$M ¼ 0.055 at 350 C to 0.077 at 500 C. The calculated solubilities are plotted in Fig. 6. In both cases, there is a general trend of increasing solubility with increasing temperature, as would be expected because of the endothermic heat of solution of hydrogen in these metals as noted below. Linear trendlines are included in Fig. 6 simply as a visual aide. Substituting Eq. (46) into the Sieverts' solubility relation mentioned in the Introduction allows the Sieverts' constant to be calculated as
with the pre-exponential factor is defined as DSS A ≡ exp R
(49)
Gallium was found to have a pre-exponential factor of A ¼ 0:5192 and an enthalpy of solution of DHS ¼ 15:8 kJ/mol. For indium, the pre-exponential factor was found to be A ¼ 0:3267
and the enthalpy of solution DHS ¼ 11:9 kJ/mol. A van't Hoff plot of the Sieverts' constants is provided in Fig. 7. The hence determined thermodynamic parameters of hydrogen solubility in indium and gallium are presented in Table 1 along with those for palladium [22], simply for the purpose of comparison. These values differ somewhat from those predicted from the Pauling Bond Valence e Modified Morse Potential (PBVMMP) model developed in our lab [18]. Thus, the measured enthalpy changes are smaller than those predicted by the PBV-MMP model, while the entropy change is in good agreement. This results in a larger Sieverts' constants than those predicted from the thermodynamic estimates provided by the PBV-MMP framework.
Inner solution The steady-state pressure was used to calculate the dimensionless parameter M from the first term of Eq. (33), i.e. pH2 ðt/∞Þ ¼ 1M 1þMpH2;i . With this and the solubility and Sieverts'
sffiffiffiffiffiffiffiffi 2nH2 ;∞ p0 KS ¼ nM pH2
(47)
These were used then to calculate the thermodynamic
(48)
The solubility of hydrogen in the metal sample can be calculated from the total moles of hydrogen absorbed nH2 ;∞ . The solubility, expressed as a mole fraction xH$M;S can be expressed with a simplification, assuming the moles of hydrogen in the metal are much less than the moles of metal present, as xH$M;S ¼
DHS KS ¼ A exp RT
parameters, DHS and DSS , using the van't Hoff relation for temperature dependence
constant, we next utilize the inner solution, Eq. (45), for data )
fitting over the first ~1000 s to estimate the parameters u S and DH . Fig. 8 depicts an example regression over the first 1000 s of an experiment run with indium at 450 C, along with a theoretical prediction using parameters predicted from the PBV-
Fig. 6 e Solubility plot versus inverse temperature (18.7 psia). Gallium data are red circles, indium data are blue squares. Linear trendlines included only as a visual aide. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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Fig. 7 e Arrhenius plot of Sieverts' constants. Gallium data are red circles, indium data are blue squares. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
MMP framework [18]. The theoretical results closely mirror the regression. Both also provide a general mirror of the trend of the experimental data, however the models have a more pronounced initial curvature than the data. The diffusion coefficient is commonly represented via an Arrhenius relation
Table 1 e Comparison of entropy and enthalpy of solution for palladium, indium, and gallium. PBV-MMP estimates for indium and gallium also provided. Pd
DHS (kJ/mol) DSS (J/mol K)
12:1 63
Ga 15:8 5:4
Ga (PBV-MMP)
In
27 5:5
11:9 9:3
In (PBV-MMP) 50.9 9:7
ED DH ¼ DH;0 exp RT
(50)
while the pre-exponential factor is [21]. DH;0 ¼ aa2 zk
z kB T hn0 DSD 2 sinh exp h 2kB T R
(51)
Using the diffusivities derived from the regressions and the Arrhenius relation of Eq. (50), the gallium pre-exponential factor was calculated as DH;0 ¼ 8:63 104 cm2 =s while the activation energy was calculated as ED ¼ 20:0 kJ/mol. For indium, the pre-exponential factor was found to be DH;0 ¼ 1:32 103 cm2 =s while the activation energy was ED ¼ 25:7 kJ/mol. The pre-exponential factors are about three times smaller compared to the predictions from the PBV-MMP framework
Fig. 8 e Inner solution regression for an indium 450 C run, zoomed to the relevant area (~1000 s) of the data. The data points are blue, the experimental regression is red, and the theoretical prediction using PBV-MMP parameters is dashed red. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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Fig. 9 e Arrhenius plot of diffusivities. Gallium data are red circles, indium data are blue squares. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
Table 2 e Comparison of entropy and activation energy of diffusion for palladium, indium, and gallium. PBV-MMP estimates for indium and gallium also provided.
ED (kJ/mol) DSzD (J/mol K) DH;0 (cm2/s)
Pd
Ga
Ga (PBV-MMP)
In
In (PBV-MMP)
27.0 2.3 2:83 102
20.0 29 8:63 104
10 15 2:5 103
25.7 20 1:32 103
8.8 5:1 1:2 103
[18], while the activation energies of diffusion are roughly twice as large as the PBV-MMP predictions. These diffusion coefficient results are slightly lower than hydrogen diffusion in solid palladium, which typically ranges from 104 to 1 cm2/ s. An Arrhenius plot of the diffusion coefficients is provided in Fig. 9. The thermodynamic parameters for indium and gallium are presented in Table 2 along with those for palladium [22]. )
While the reverse sorption weight constant u S was determined from the least-squares regression of the short-time
experimental data, the forward weight constant ! u S was calculated from the reverse constant and the Sieverts' constant. The forward sorption step weights follow an Arrhenius relation, with the results for indium plotted in Fig. 10 below. The forward sorption rate constant is found from the linear regression in Fig. 10, and after factoring out the pressure dependence the van't Hoff form of the equation is ! 29100 k S ¼ 2:64 1015 e RT
Fig. 10 e Plot of adsorption step weight versus 1=T for indium.
(52)
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Fig. 11 e Arrhenius plot of desorption step weight for indium.
The reverse sorption step weight can also be modeled with the Arrhenius relation. A plot of the desorption step weight for indium along with the Arrhenius regression is shown in Fig. 11. The resulting Arrhenius form of the reverse sorption rate constant is )
16900 RT
k S ¼ 1:50 1017 e
(53)
Comparing these relations to the PBV-MMP predictions is more difficult, however taking the ratio of the forward to reverse rate constants gives the Sieverts' constant, which is modeled via the van't Hoff relation to give the change in enthalpy and change in entropy of solution according to Eqs. (48) and (49). For indium, the experimental change in enthalpy is 12.2 kJ/mol, which is significantly lower than the PBV-MMP framework prediction of 50:9 kJ/mol [18]. The entropy change
is 33:6 J/mol K, which is larger than the PBV-MMP prediction of 9:7 J/mol K.
Full solution Due to the complexity of Eq. (33), a least-squares regression was not able to be utilized to determine the sorption and diffusion parameters. Instead, the results from the short time with sorption analysis were used as starting points for a guess-and-check estimation, changing the parameters to generate the best fitting plot. Also, while the solution calls for an infinite summation especially as t/0, a finite number of terms must be used. The model was, thus, split over two timeperiods of the experiment: the initial short-time section (0e25,000 s) and the long-time portion (~75,000 s to the end of the experiment). For the short-time portion, the
Fig. 12 e Long-time solution for indium at 450 C, using the first two terms of the summation only. The data points are blue, the long-time solution is the solid red line, and the short-time approximation is the dashed red line. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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 4 3 ( 2 0 1 8 ) 1 9 0 7 5 e1 9 0 9 0
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Fig. 13 e Plot of hydrogen flux through a SLiMM [10] versus inverse temperature with various Sieverts' law estimates. The Sieverts' apparatus parameters (dashed blue) is a worse fit than the PBV-MMP parameters [18] (dashed red), while both predict lower flux compared to the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
approximation developed as Eq. (37) was used. The long-time portion used two terms in the summation from Eq. (33). The previously calculated Sieverts' constant from the equilibrium pressure was utilized here. Next, the diffusion coefficient was varied until reasonable agreement with the mid-time portion of the experiment was modeled. Due to their relation to the Sieverts' constant, only one of the sorption rate constants needs to be determined. However, it was found that the value of the rate constants does not have an impact on the model predictions. An example, for an indium run at 450 C, is shown
in Fig. 12. The short-time approximation is plotted from 0 to 75,000 s while the long-time solution is plotted from 25,000 s to the end of the run. While the short-time approximation has a good fit from 0 to 25,000 s and the long-time solution fits the data well from ~75,000 s on, there is a region from 25,000e75,000 s, when neither equation provides a reasonable fit to the data. The diffusion coefficients found from the long-time model were found to be two orders of magnitude smaller than the corresponding coefficients from the short-time approximation
Fig. 14 e Plot of hydrogen flux through a SLiMM [10] versus the difference of partial pressure square roots with various Sieverts' law estimates. The Sieverts' apparatus parameters (dashed blue) is worse than the PBV-MMP parameters [18] (dashed red), while both predict lower flux than the experiment. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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above. It is not known why the diffusion coefficients would differ so drastically between the theoretical models. The preexponential factor is around 200 times smaller compared to the predictions from the PBV-MMP framework [18] for each liquid metal, while the activation energy of diffusion is only slightly larger than the PBV-MMP predictions. These diffusion coefficient results are lower than hydrogen diffusion in solid palladium.
Application to membrane permeation The diffusion parameters identified, along with the Sieverts' constant, can be used in Eq. (1) to model the hydrogen permeation in metal membranes, such as the SLiMMs developed in our lab [10]. Comparisons through Sieverts' law to these experiments can be made by plotting the flux of hydrogen, expressed as either mol=m2 s or m3 =m2 s, versus 1= pffiffiffiffiffi pffiffiffiffiffi TðKÞ or Pf Pp , as shown in Figs. 13 and 14, respectively. In both cases, there are two sets of parameters shown for the Sieverts' law model lines. The red dashed lines are Sieverts' law plotted using parameters estimated from the PBVMMP framework [18], which slightly under-predicts the experimental flux. The dashed blue lines use the parameters determined using the Sieverts' apparatus in this paper, specifically the inner solution diffusivity. These predictions are further from the experimental values. The experimental flux is expected to be higher than the predictions due to two possible contributions. The first is that the models used to predict the parameters only account for atomic hydrogen diffusion, and there is the potential in liquids for molecular hydrogen diffusion as well, which would increase the flux. There is also the possibility of poor surface coverage or pinholes developing in the membrane that contribute to increased flux of molecular hydrogen through the holes, as opposed to the diffusion through the metal membrane.
steady-state values are more in line with expected predictions from PBV-MMP and experimental palladium results. These metals show signs of there being promising alternatives to the traditional palladium-based hydrogen purification membranes, and a Sieverts' apparatus is a useful tool for screening potential metal and alloy candidates.
Acknowledgement We acknowledge financial support from the Department of Energy under award number DE-FE0001050.
Nomenclature Symbols A AM AWM Bi
cS;t
pre-exponential factor in Sieverts' constant surface area of metal, cm2 atomic weight of metal M, g/mol dimensionless parameter akin to Biot number, Eq. (19) the lattice constant, cm interstitial hydrogen concentration, mol H$cm3 saturation concentration of interstitial hydrogen, mol H$cm3 total concentration of the surface metal atoms,
ct
mol M$cm2 total concentration of the metal atoms in bulk,
dM DH
mol M$cm3 atomic diameter of metal atom M, m interstitial diffusion coefficient of the hydrogen
DH;0
atoms, cm2 s1 pre-exponential factor of interstitial diffusion
a cH$M cH$M;s
ED
Conclusions fH2 The standard Sieverts' device is capable of providing several characteristics of hydrogen transport and storage behavior in metals, be they in solid or liquid form. Liquid indium and gallium were found to have similar hydrogen diffusion coefficients and Sieverts' constants to each other. Both liquid metals also had similar solubilities, which were an order of magnitude lower than the solubility of hydrogen measured in palladium. The diffusion coefficient is able to be accurately predicted when compared with the PBV-MMP framework [18] when utilizing the inner solution for short times (the first ~1000 s). The diffusivity was found to be on the order of 103 104 cm2 =s for both metals, and the activation energy of diffusion ranged from 20 25 kJ=mol. The sorption thermodynamic parameters are relatively consistent when using the steady-state equilibrium and the inner solution. The enthalpy change is around 11 16 kJ=mol for each metal, comparable to palladium, but significantly lower than the PBV-MMP predictions. The entropy change from the steady-state solution is around 5 to 10 J=mol$K, while the inner solution estimates the indium entropy change of sorption as 33 J=mol$K. The
fH$M fH$X
coefficient of H, cm2 s1 activation energy of interstitial diffusion of H, J/mol H dimensionless partial pressure of hydrogen, defined by Eq. (17) dimensionless mole fraction of hydrogen in bulk metal, defined by Eq. (17) dimensionless mole fraction of hydrogen in the interfacial layer
f H2 ðsÞ
Laplace transform of fH2 ðtÞ
f H$M ðsÞ h KS
Laplace transform of fH$M ðtÞ Plancks's constant Sieverts' equilibrium constant for the hydrogen solution in metal
kB ! kS
Boltzmann constant, 1:38064 1023 J$K1
)
kS M nH$M nM nH2;i
forward rate constant for Sieverts' step, s1 reverse rate constant for Sieverts' step, s1 dimensionless parameter, Eq. (19) number of moles of hydrogen in metal, mol number of moles of metal, mol number of hydrogen molecules introduced initially into Sieverts' apparatus
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 4 3 ( 2 0 1 8 ) 1 9 0 7 5 e1 9 0 9 0
nH2 ;∞
total amount of hydrogen gas transferred in Sieverts' apparatus, mol H2
NAv
Avogadro's number, 6:022 1023 mol1
NH
flux of atomic species H, mol$cm2 $s1 1
NH2 NM pH2 pH2 ;i
flux of molecular species H2, mol$cm $s number of metal atom monolayers in the metal pool hydrogen partial pressure, Pa or atm initial partial pressure of hydrogen, Pa
PH2 p
membrane permeance, mol H2 $cm2 $s1 $atm1=2 standard atmosphere, 101,325 Pa, or 1 atm
rS
rate of reaction of Sieverts' step, mol H$cm2 $s1 net rate of the elementary reaction r, ! r
rr
r
)
r r $mol$cm2 $s1
! rr
forward rate of elementary reaction r, mol$cm2 $s1
)
rr s sr T TC Th t VH2 VC Vh xH$M xH$M;i
reverse rate of elementary reaction r, mol$cm2 $s1 Laplace variable elementary step r temperature, K temperature of cold zone, K temperature of hot zone, K time, s volume of hydrogen absorbed by metal, cm3 volume of cold zone, cm3 volume of hot zone, cm3 mole fraction of interstitial hydrogen in bulk metal equilibrium mole fraction of hydrogen in metal
xH$M;0 xH$X y z z
corresponding to the initial hydrogen partial pressure initial mole fraction of hydrogen in metal mole fraction of hydrogen in the interfacial layer depth from the metal pool surface, cm axial distance from the metal pool bottom, cm number of nearest neighbor interstitial positions
Greek Symbols a geometric factor in interstitial diffusion coefficient a ratio of hot to cold reservoir volumes of Sieverts' apparatus b root defined in Eq. (35) g root defined in Eq. (35) d membrane thickness, cm
DGS
DHS
DSS
DSzD h h q q lk
dimensionless axial distance, Eq. (18) frequency of vibration of the solute in an interstitial position, s1 density of the gas, g/m3
x n0 rg
2
standard Gibbs free energy change of hydrogen solution in metal, J/mol H standard enthalpy change of hydrogen solution in metal, J/mol H standard entropy change of hydrogen solution in metal, J=mol$K standard entropy change of activation for diffusion, J=mol$K dimensionless penetration theory variable combining time and position pffiffiffi square root of Laplace variable, h ¼ s ratio of hot to cold temperatures of Sieverts' apparatus rescaled dimensionless time for inner solutuion, q ≡ ðNM Þt non-zero kth root of the transcendental equation, Eq. (31)
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rM sr t cM cH$M cH$M;s ! uS ! u S;i
density of metal M, g/m3 stoichiometric number for step r dimensionless time, Eq. (18) atomic ratio of vacant interstitial sites, cM =ct interstitial atomic ratio of absorbed hydrogen, cH$M =ct the saturation value of cH$M , cH$M;s =ct forward step weight for Sieverts' step, s1 initial value of forward step weight for Sieverts' step, corresponding to initial hydrogen pressure, s1
) uS
ur ! ur ) ur
reverse step weight for Sieverts' step, s1 step weight for reaction sr forward step weight for reaction sr reverse step weight for reaction sr
Sub- and super-scripts C cold zone of Sieverts' apparatus f feed side of membrane h hot zone of Sieverts' apparatus p permeate side of membrane r elementary reaction step S unoccupied surface site M unoccupied interstitial site in bulk metal X unoccupied subsurface site in interfacial layer
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