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The effect of inaccurate volume calibrations on hydrogen uptake measured by the Sieverts method
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 2 1 6 8 e2 1 7 4
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The effect of inaccurate volume calibrations on hydrogen uptake measured by the Sieverts method C.J. Webb*, E.MacA. Gray Queensland Micro- and Nanotechnology Centre, Griffith University, Nathan, Brisbane 4111, Australia
article info
abstract
Article history:
The Sieverts technique is a popular method of measuring the uptake of gas on a sample
Received 22 October 2013
and is an important tool for the characterisation of hydrogen uptake by potential hydrogen
Received in revised form
storage materials. An analysis of the consequences of errors in the calibration of volumes
23 November 2013
in this technique has been made using hydrogen absorption on an AB2 alloy as an example.
Accepted 28 November 2013
Trends in the error in the uptake as a function of errors in the volumes have been
Available online 31 December 2013
explained in terms of the equations for uptake. Significant advantages are found for calibration techniques for which the ratio of volumes is an intrinsic measurement.
Keywords:
Crown Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All
Volume calibrations
rights reserved.
Hydrogen storage Sieverts
1.
Introduction
The manometric Sieverts technique for measuring the uptake of hydrogen and other gases by materials is popular because it is cheap and easy to set up, physically robust and reasonably reliable. As found empirically [1] and theoretically [2], it has vulnerabilities to uncertain or inaccurate knowledge of relevant parameters, especially the volume occupied by the sample itself when this is large for the amount of gas absorbed or adsorbed, i.e. for samples of low-density, generally speaking. Two round-Robin exercises comparing results for an activated carbon [3] and catalysed magnesium [4] have revealed an alarming spread in the reported hydrogen capacities of these low-density materials. The need for authoritative measurements of hydrogen capacity for new storage materials makes an investigation of this irreproducibility between laboratories urgently necessary. Measurements of pressure and temperature can usually be made with sufficient accuracy and fed into a sufficiently
sophisticated model of the apparatus and sample to produce a notionally accurate result for the hydrogen capacity of the sample. The propagation of uncertainties in measured parameters is analysed in detail by Webb and Gray [2] and supports this view. The difficulty in determining the volume of the sample, particularly for low density samples or samples that change volume during absorption has been treated elsewhere [1] and the need to use the non-ideal gas equation incorporating the compressibility of the gas has also been discussed [5e7]. This leaves calibration of the apparatus as the only apparent culprit in the matter of obtaining accurate results. Methodological and technical approaches to calibration of a Sieverts apparatus have been described and analysed in a number of papers [6e9]. The challenge is to first model the system, in which the various zones may be at different temperatures, then to apply a calibration procedure to measuring the volume in each temperature zone and in particular obtain values for the reference volume and the void (unoccupied) volume of the sample cell.
* Corresponding author. Tel.: þ61 7 3735 5023; fax: þ61 7 3735 7656. E-mail address: [email protected] (C.J. Webb). 0360-3199/$ e see front matter Crown Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.11.121
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The most fundamental calibration is that of the system with no sample, since no amount of care taken to account for the volume occupied by the sample itself will recover an accurate measurement of the hydrogen capacity if the reference and empty-cell volumes are wrong. Therefore, in this paper we analyse the consequences of wrong reference and emptycell volumes and find that the calibration methodology has a great influence on the sensitivity of the calculated hydrogen capacity to errors in these volumes. On that basis we make recommendations about the calibration methodology and propose a routine check to be applied to the basic instrument to confirm that there is no serious error in its calibration.
2.
Methods
The calculation of gas uptake by a sample using the Sieverts technique has been detailed previously [7] and is only summarised briefly here. The technique proceeds in a stepwise fashion. At the start of step k, a quantity of gas (nkref ) is prepared in a reference or dosing volume (Vref). After reaching thermal equilibrium, this gas is released into the cell volume containing the sample (Vcell) which initially contains the amount of gas from the end of the previous step. The gas occupies that volume of the cell not occupied by the sample e the void volume, Vvoid ¼ Vcell Vxk
(1)
where Vxk is the volume of the sample. Since the sample may swell or expand, adsorb or chemically change, the sample volume is a function of the uptake and hence step number. The amount of gas initially is
nki ¼ nkref þ nk1 void ¼
Pkref Vref Zkref ;ref RTkref
þ
k1 Pk1 sys Vcell Vx k1 Zk1 sys;cell RTcell
(2)
where Pkref is the pressure prepared in the reference volume, Pk1 sys is the system pressure after equilibration of the previous step and hence the initial pressure in the void volume, Tkref and Tk1 cell are the temperature of the reference volume at this step and the temperature of the cell volume at the previous step k1 k1 k1 ¼ ZðPsys ; Tcell Þ respectively and Zkref ;ref ¼ ZðPkref ; Tkref Þ and Zsys;cell are the respective compressibilities of the gas which are functions of pressure and temperature. Similarly, the amount of gas in the system after the two volumes are connected and have reached equilibrium is
nkf
¼
nkref ;sys
þ
nkvoid
¼
Pksys Vref Zksys;sys RTksys
þ
Pksys Vcell Vxk Zksys;cell RTkcell
Dnkabs
¼
Pkref Zkref ;ref RTkref þ þ
Zksys;sys RTksys
Pk1 sys k1 Zk1 sys;cell RTcell
Pksys Vxk k Zsys;cell RTkcell
!
Pksys
Vref
Pksys Zksys;cell RTkcell
! Vcell
k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
The total absorption by the sample at the end of step k is the sum of absorption increments of this step and all the previous steps. To calculate the correct or true uptake, the values of the reference and the cell volumes must be known accurately e as must the volume occupied by the sample itself at each step. To show the effect of inaccuracies in the volumes of the reference and/or cell, data from a Sieverts experiment using an AB2 alloy (Ti0.60Zr0.4Cr0.85Mn0.25Fe0.7Ni0.20Cu0.03) have been used as a model for varying the volumes and re-calculating the uptake. The original data are shown in Fig. 1, showing uptake measured in wt%, where wt% is the percentage ratio of mass of gas uptake to the mass of sample plus gas, wt% ¼
mg nabs Mg 100 100 ¼ mx þ mg mx þ nabs Mg
¼nki
¼
Pkref Vref
(3)
k1 Pk1 sys Vcell Vx
þ k1 Zkref ;ref RTkref Zk1 sys;cell RTcell Pksys Vcell Vxk Pksys Vref k k Zsys;sys RTsys Zksys;cell RTkcell nkf
Rearranging with respect to volumes gives
(6)
where mg is the mass of absorbed gas, Mg is the molecular mass of the gas and mx the initial mass of the sample. Fig. 1 shows the uptake of hydrogen at 23 C as a function of applied pressure in bar up to 300 bar with the uptake conventionally on the horizontal axis. It can be seen that the alloy absorbs with a plateau characteristic of metal hydrides to a maximum of 1.4 wt% at a pressure of approximately 150 bar. Between the last two points close to 175 bar and 300 bar there is effectively no further absorption. The volumes of the reference and cell are 2.180 0.005 cc and 6.944 0.015 cc respectively as determined by a gas expansion volume calibration using helium. A calibrated volume was used to experimentally determine the ratios of the reference and (empty) cell volume to the calibrated volume.
The difference between the amount of gas in the system before and after the volumes are connected is the additional amount of gas absorbed by the sample in this step
Dnkabs
(5)
(4) Fig. 1 e H2 uptake on interstitial alloy at 23 C e uptake calculated using the correct reference and cell volumes.
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The effect of inaccurate reference volume
From Eqn 5, it can be seen that if the cell and sample volumes are accurate, but the reference volume varies from the true reference volume by an amount 3 (Vref), then it is only the first term of Eqn 5 which contributes to an error in the uptake. In this term, the reference volume is multiplied by Pkref Pksys . k k k k Z RT Z RT ref ;ref
ref
sys;sys
sys
Since Pksys is always less than Pkref (for absorption steps), this expression must always be positive and so the calculated uptake will be greater than the true uptake if the assumed value of the reference volume is larger than the true value (and vice versa). This is shown in Fig. 2. The larger value of Vref in Fig. 2, curve (c) causes all the values of uptake to be greater, expanding the isotherm along the uptake axis. This includes the uptake at the two highest pressures (for which the real uptake is zero) causing the graph to show steadily increasing uptake, i.e. a positive slope instead of vertical. Similarly in Fig. 2, curve (a), for a value of the reference volume smaller than the true value, all uptake values are less than the true uptake and where the true uptake is zero, the uptake is negative, causing the graph to slope backwards. The significance of any error in the reference volume only is shown in Fig. 3. Here, the percentage error, i.e. the percentage difference between the calculated maximum uptake and the true maximum uptake has been plotted as a function of the error in the reference volume, 3 (Vref) (the difference between the value of Vref used in the calculation and the true value). The factor ðPkref =Zkref ;ref RTkref Pksys =Zksys;sys RTksys Þ, which multiplies the error Vref is a measure of how large the step is in terms of pressure in the reference volume compared to the final system pressure. In addition, this error in each step of the Sieverts uptake measurement compounds as the incremental uptake at each step is summed to yield the total uptake for each pressure step. The expressed percentage error in the total uptake is for the last point at w300 bar which will have
Fig. 3 e Absolute error in the calculated uptake for the maximum pressure value as a function of error in the reference volume.
the largest error. The importance of Fig. 3 is showing how a quite small error in the reference volume leads to a significant error in the calculated uptake. For example, an error of 0.05 cc gives an error of nearly 9% (of 1.4 wt%) in the uptake.
2.2.
The effect of inaccurate cell volume
Using Eqn 5, with an accurate value of the reference volume but allowing the cell volume to vary produces a corresponding error in the calculated uptake. In this case, the error in the cell Pk1 Pk volume 3 (Vcell), is multiplied by Zk1 sysRTk1 Zk sysRTk . sys;cell
ref ;ref
Fig. 2 e Recalculated hydrogen uptake isotherms with ±10% error in the reference volume, (a) Vref [ 1.962 cc, (b) Vref [ 2.180 cc, true volume and (c) Vref [ 2.398 cc.
cell
sys;cell
cell
For absorption, the term involving the equilibrium system pressure from the previous step (Pk1 sys ) will always be less than the equilibrium system pressure for the current step (Pksys ), so this expression will always be negative. A larger cell volume than the true cell volume will reduce the calculated uptake and vice versa. This is shown in Fig. 4. The sensitivity of the uptake error to the cell volume error is shown in Fig. 5, which resembles Fig. 3 for the reference volume error, except that the magnitude of the uptake error is approximately one-quarter that of corresponding errors for k1 k1 the reference volume. The term ðPk1 sys =Zsys;cell RTcell Pksys =Zksys;cell RTkcell Þ amplifying the cell volume error has a smaller Pk Pk magnitude than Zk refRTk Zk sysRTk amplifying the reference ref
sys;sys
sys
volume error since the difference in system pressures between steps is typically smaller than the difference in reference and equilibrium pressure for a step. We emphasise that the assumed error here is in the volume of the empty cell, excluding any imperfect knowledge of the volume occupied by the sample. In the case of desorption, the reference pressure is always smaller than the resultant system pressure and the system pressure decreases with each step, so the terms multiplying the errors in the reference and cell volume change sign. This means that the desorption isotherm will undergo the same
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along the pressure axis) will always be achieved provided the ratio of volumes is correct. Performing an empty-cell isotherm to check the integrity of the Sieverts apparatus has been advocated previously [8] and is a technique which can discover leaks, gas adsorption to the cell walls and other problems. However, if the values of both cell and reference volume are incorrect e but the ratio of volumes is correct, the empty-cell isotherm will not indicate the problem. If the calculated uptake at step k is split into the true part plus an additional amount representing the error true þ 3 Dnkabs Dnkabs ¼ Dnkabs
(8)
and the errors in the reference and cell volumes are denoted similarly, true þ 3 Vref Vref ¼ Vref true þ 3 Vref Vcell ¼ Vcell
Fig. 4 e Recalculated hydrogen uptake isotherms with ±10% error in the cell volume, (a) Vcell [ 7.638 cc, (b) Vcell [ 6.944 cc, true volume and (c) Vcell [ 6.250 cc.
(9)
Eqn 5 becomes
expansion or contraction along the uptake axis as the absorption isotherm.
k true k þ 3 Dnabs ¼ Dnabs
Pkref Zkref ;ref RTkref
þ
The ratio of the cell volume to the reference volume is an important constant in the Sieverts technique since it is often measured by the common practice of calibrating Vcell against Vref. If we consider the case of an empty cell (no sample, Vx ¼ 0), such that the true absorption for every step should be zero, then Eqn 5 becomes,
Pkref
Zkref ;ref RTkref Vcell ¼ Vref Pksys Zksys;cell RTkcell
Zk
k sys;sys RTsys
Pk1
! Pksys true Vcell þ 3 ðVcell Þ k1 k k Zk1 RT Z RT sys;cell cell sys;cell cell
k1 k1 Pksys Vxk Psys Vx k1 k k Zsys;cell RTcell Zsys;cell RTk1 cell
Since Eqn 5 must hold for the true part of the uptake k true ¼ Dnabs
Pkref Zkref ;ref RTkref
Zksys;sys RTksys
!
Pksys Zksys;cell RTkcell
!
Pksys true Vcell þ
true Vref þ
Pksys Vxk k Zsys;cell RTkcell
Pk1 sys k1 Zsys;cell RTk1 cell k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
(7)
Zk1 sysRTk1 sys;cell
true Vref þ 3 Vref
(10)
Pksys
Zksys;sys RTksys
Pk1 sys
þ
2.3. The effect of inaccurate volumes but maintaining correct volume ratio
!
Pksys
cell
showing that a Sieverts uptake measurement on an empty cell is equivalent to a measurement of the ratio of cell to reference volumes and that a zero absorption isotherm (i.e., straight line
(11) Substituting Eqn 11 into Eqn 10 gives ! k Pkref Pksys Dnabs ¼ k Zref ;ref RTkref Zksys;sys RTksys 3 Vref
3
Pk1 sys
þ
k1 Zk1 sys;cell RTcell
Pksys
!
Zksys;cell RTkcell
3 ðVcell Þ 3
Vref
(12)
or
k 3 Dnabs
"
Pkref
¼
Zkref ;ref RTkref
Pk1 sys
þ
k1 Zk1 sys;cell RTcell
Pksys
!
Zksys;sys RTksys
Pksys Zksys;cell RTkcell
!
true Vref true Vcell
#
3 Vref
(13)
where we have used the fact that the ratio of the errors in the volumes is the same as the ratio of the true volumes true þ3 V Vref ð ref Þ true þ3 ðV Vcell cell Þ 3 ðVcell Þ 3
Fig. 5 e Absolute error in the calculated uptake for the maximum pressure value as a function of error in the cell volume.
ðVref Þ
¼
Vtrue
¼ Vref true
true Vcell true Vref
cell
(14)
Note that the error in the uptake considered here is only that due to reference and cell volume inaccuracies, hence Eqn 13 does not contain any term involving the sample volume.
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Eqn 13 demonstrates that the error in the uptake is then proportional to the error in the reference volume and the term true k1 Pkref Vref Pksys Psys Pksys determines þ k k k k k1 k1 k k Vtrue Z RT Z RT Z RT Z RT ref ;ref
ref
sys;sys
sys
sys;cell
cell
sys;cell
cell
k true nabs ¼0¼
whether the uptake is greater or less than the true value. The first Pk Pk component Zk refRTk Zk sysRTk is always positive and the secsys;sys sys ref ;ref ref k1 P Pk ond component Zk1 sysRTk1 Zk sysRTk is always negative, so the sys;cell
cell
sys;cell
þ
For the limiting case when the true absorption is zero (from Eqn 5)
Fig. 6 e Recalculated hydrogen uptake isotherms with Vcell/ Vref [ true volume ratio and ±25% error in the reference volume, (a) Vref [ 1.635 cc (L25%), (b) Vref [ 2.180 cc, (true) and (c) Vref [ 2.725 cc (D25%).
Pk1 sys
!
Pksys Zksys;sys RTksys
k1
Zk1 sys;cell RTcell Pksys Vxk
Zksys;cell RTkcell
!
Pksys Zksys;cell RTkcell k1 Pk1 sys Vx
!
k1 Zk1 sys;cell RTcell
true Vcell true Vref
(15)
1 true Vref
Substituting Eqn 15 into Eqn 13 gives
cell
2.4. Correct volume ratio when absorption has largely finished
Zkref ;ref RTkref þ
cell
sign of the error in the calculated uptake depends on these pressure values and the ratio of volumes. For non-zero real absorption, the system equilibrium pressure is always lower than it would be if no absorption took place, and this favours the positive component at the expense of the negative component leading to a net positive effect, such that the error in the uptake has the same sign as the error in the reference volume. Using the AB2 alloy data, this can be investigated by changing the values of both Vcell and Vref such that the ratio remains constant at the correct value. For Fig. 6, curve (a), the reference volume is smaller than the true reference volume. Compared to the true uptake values shown in Fig. 6, curve (b), the calculated uptake is smaller for non-zero real uptake points, confirming that the uptake error has the same sign as the reference volume error. However, for zero real uptake (as shown between the last two points) the uptake has increased e this is explained in the next section. For the case where the reference volume is larger than the true reference volume (Fig. 6, curve (c)), the converse is true. Here, the uptake is larger than the real uptake for nonzero real absorption values but less than zero for the last two points where the real absorption is zero.
Pkref
! k k1 Pksys Vxk Pk1 Dnabs 1 sys Vx ¼ k k1 true V Zsys;cell RTkcell Zsys;cell RTk1 3 Vref ref cell
3
(16)
Hence, k 3 Dnabs ¼
Pksys Vxk Zksys;cell RTkcell
k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
! 3
Vref true Vref
(17)
Eqn 17, which is only valid in the case of zero real absorption, explains the behaviour of the uptake between the last two points in the isotherms shown in Fig. 6 curve (a) and Fig. 6 curve (c). Since the sample volume does not change from the previous step to the current step (since there is no absorption), then the term in parentheses is always positive as the current system equilibrium pressure, Pksys is always greater k1 . Hence than the previous system equilibrium pressure, Psys the error in the uptake is negative for positive reference volume error and vice versa. When the reference volume used in the calculation of uptake is less than the true reference volume (and the same is true of the cell volume since the ratio of volumes is equal to the true ratio), 3 (Vref) < 0, and the error in uptake will be positive. Since the true value is zero, the calculated value will be greater than zero leading to the slope of the line between the last two data points in Fig. 6 curve (a). Conversely, the backwards slope of Fig. 6 curve (c) is due to a larger reference volume than the true reference volume in Eqn 17 giving a negative uptake cf. the true uptake of zero. Thus the different behaviour of the uptake provides some opportunity to analyse the source of volume calibration errors if a standard sample (for which the isotherm is wellestablished) is measured. An incorrect ratio of reference and empty-cell volumes with Vref > Vcell leads to increased apparent uptake for all uptake values, while an incorrect ratio with Vref < Vcell gives a calculated uptake smaller than the real uptake for all uptake values. For the correct volume ratio, however, while the same trend is observed for non-zero real uptake, 3 (Vref) > 0 has a negative uptake for zero real uptake and for 3 (Vref) < 0 the calculated uptake is positive for zero real uptake. However, the trialled increase/decrease in the reference and cell volumes in Fig. 6 is very high e 25%. It is unlikely that a careful volume calculation would be in error by such an amount. The effect of a more moderate but still large error in the volumes of 10% is shown in Fig. 7. Seen individually, any of these isotherms might not immediately suggest a volume calibration problem. The slope of the line between the last two points, where the real uptake is zero is sufficiently close to vertical to suggest the calibrations are correct e especially for Fig. 7 curve (a) where a slight positive slope might be expected from solid solution of hydrogen in the metal hydride. However, the effect on the
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direct technique, where a non-absorbing gas is used to directly measure the void volume, or by the indirect technique, where the sample volume is simply subtracted from the calibrated cell volume [9,10]. The indirect technique has been assumed above, in part due to concerns about the legitimacy of assumptions in the direct technique [11,12] and in part due to the array of samples that expand or swell during absorption thus rendering the direct technique impractical. However, for a sample that does not change volume (other than by adsorbing a layer of the gas, if appropriate), it is useful to interpret the uptake in terms of the reference and void volumes. From Eqn 4, Dnkabs Fig. 7 e Recalculated hydrogen uptake isotherms with Vcell/ Vref [ true volume ratio and ±10% error in the reference volume, (a) Vref [ 1.962 cc, (b) Vref [ 2.180 cc, (true) and (c) Vref [ 2.398 cc.
¼
Pkref Zkref ;ref RTkref
Fig. 8 e Recalculated hydrogen uptake isotherms with (a) of L2%, 3(Vcell) [ 0, (b) Vcell/Vref [ true volume ratio with 3(Vref) of L2% (c) true values, (d) Vcell/Vref [ true volume ratio with 3(Vref) of D2% and (e) 3(Vref) [ L2%, 3(Vcell) [ 0.
Zksys;sys RTksys !
Pksys Zksys;cell RTkcell
Pkref Zkref ;ref RTkref
3(Vref)
! Vref þ
Pk1 sys k1 Zk1 sys;cell RTcell
(18)
Vvoid
Clearly, for zero real uptake 0¼
calculated uptake at the maximum pressure e typically the quoted value e is significant, approximately 10%. For a still smaller error in the reference volume, Fig. 8 compares a 2% error in Vref for the case of correct and incorrect volume ratios. Fig. 8 demonstrates the increased accuracy consequent on having a correct ratio of volumes. For an error of 2% in the reference volume, the error in the calculated uptake at maximum pressure is 1.5% if the ratio of volumes equals the correct ratio e and 7.6% if the other (cell) volume is actually correct. The analysis above has been couched in terms of the reference volume Vref and the cell volume Vcell. However, the cell volume is not the volume that the gas prepared in the reference volume expands into, that is the void (or dead) volume e the volume of the cell not occupied by the sample. Using a gas expansion, this may be measured by either the
Pksys
Vvoid Vref
Zk
k sys;sys RTsys
k1 Psys
Zk1
¼ sys;cellk
Pksys
RTk1 cell
Pk1 Vref þ Zk1 sysRTk1 Zk
Pk sys Zk RTk sys;cell cell
P Pk sys ref k Zsys;sys RTk Zk RTk sys ref ;ref ref
sys;cell
cell
Pksys
RTkcell sys;cell
Vvoid (19)
In theory, for a pressure range on a sample which is known to have zero absorption, the void volume and, in turn, the sample volume could be determined from Eqn 19. In practice, this may require high pressures and large pressure steps and there would be a limited range of samples which would have zero real absorption at these pressures.
3.
Conclusions
An analysis of the consequences of inaccurate reference and cell volumes for the Sieverts technique for uptake of gas in or on to a sample has been made. It has been shown, using hydrogen absorption data for an AB2 alloy, that relatively small errors in the values for these volumes have a significant impact on the accuracy of the uptake of gas calculated for this technique, demonstrating the fundamental reliance of the method on these volumes. In addition, the inherent accumulation of errors associated with the summation of incremental uptake values in the Sieverts technique exacerbates this problem. The equation of the uptake for each step in the measurement process has been used to explain trends in the calculated uptake for different combinations of errors in the reference and cell volumes. In particular, when the value of the reference volume used is smaller than the true value, or the cell volume used is greater than its true volume, the calculated uptake is smaller than the true uptake. Conversely, when the reference volume used is greater than the true volume, or the cell volume used is less than its true volume, the calculated uptake is greater than the true uptake. For a 10% error in one of these volumes (the other being correct), the uptake calculated was shown to be in error by as much as 50% at 300 bar for the trial data.
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In the special case of incorrect reference and cell volumes, but for a correct ratio of Vcell/Vref, the calculated uptake may include a distinguishing feature. This particular situation is important because of the way in which volume calibrations are often performed. If the reference and (empty) cell volumes are calibrated using a third, known volume through a gas expansion method, the two unknown volumes are always determined as a ratio to the calibrated volume (and hence to each other). So, through careful measurements, it is possible to obtain a calibration that has a very accurate cell volume to reference volume ratio. This applies to both the conventional volume calibration technique, employing a separate vessel as a calibrated volume, as well as the displacement volume technique [8]. However, if the volume used to calibrate is in error, then the values are incorrect even though the ratio is the same as the ratio of the true volume values. For this situation, the error in uptake is comparable to the error in the calibrating volume and significantly smaller than the corresponding uptake error for the same error in the reference volume when the cell volume is correct and vice versa. This demonstrates the importance of a calibration technique which determines volumes in terms of their ratios, compared to a technique where the reference and cell volumes are determined independently. For larger errors in the reference volume, the calculated uptake behaves similarly to incorrect ratios for non-zero real uptake, however, when the real uptake is close to zero the trend reverses, with negative uptake steps for larger values of Vref, leading to a distinctive backward sloping isotherm at high pressures instead of a vertical slope (and conversely for smaller values of Vref than the true value). The practice of performing an uptake measurement on an empty cell is extremely valuable in determining that the ratio of reference to cell volume is correct as well as possibly indicating other errors. However, performing an uptake measurement on a known sample can provide additional confirmation that the absolute values of the reference and cell volumes are correct. This is indicated primarily by the measured maximum uptake value, but also by the slope of the isotherm after absorption is mostly complete (if a sufficiently high pressure can be reached). A suitable known sample would need to be one which has been measured by many different laboratories with a well-established uptake and for which the density is a well-known function of uptake in order
to remove any error contributions from the sample volume. This would depend on the measurement system being used, but for hydrogen uptake measurements, virgin LaNi5 might be suitable for many systems.
references
[1] Blach TP, Gray EMacA. Sieverts apparatus and methodology for accurate determination of hydrogen uptake by lightatom hosts. J Alloys Compd 2007;446e447(0):692e7. [2] Webb CJ, Gray EMacA. Analysis of uncertainties in gas uptake measurements using the Sieverts method. Int J Hydrogen Energy 2014;39(1):366e75. [3] Zlotea C, Moretto P, Steriotis T. A round Robin characterisation of the hydrogen sorption properties of a carbon based material. Int J Hydrogen Energy 2009;34(7):3044e57. [4] Moretto P, Zlotea C, Dolci F, Amieiro A, Bobet JL, Borgschulte A, et al. A round Robin test exercise on hydrogen absorption/desorption properties of a magnesium hydride based material. Int J Hydrogen Energy 2013;38(16):6704e17. [5] McLennan KG, Gray EMacA. An equation of state for deuterium gas to 1000 bar. Meas Sci Technol 2004;15(1):211e5. [6] Broom DP. The accuracy of hydrogen sorption measurements on potential storage materials. Int J Hydrogen Energy 2007;32(18):4871e88. [7] Gray EMacA. Reliable measurement of hydrogen uptake. In: Walker G, editor. Solid-state hydrogen storage. Cambridge: Woodhead Publishing Ltd; 2008. pp. 174e204. [8] Gross KJ, Carrington KR, Barcelo S, Karkamkar A, Purewal J, Ma S, et al. Recommended best practices for the characterization of storage properties of hydrogen storage materials. pp. 1e579. Available from, http://www1.eere. energy.gov/hydrogenandfuelcells/pdfs/best_practices_ hydrogen_storage.pdf; 2010. [9] Broom DP. Hydrogen storage materials. London: Springer; 2011. [10] Rouquerol F, Rouquerol J, Sing K. Adsorption by powders and porous solids. London: Academic Press; 1999. [11] Neimark AV, Ravikovitch PI. Calibration of pore volume in adsorption experiments and theoretical models. Langmuir 1997;13(19):5148e60. [12] Silvestre-Albero J, Silvestre-Albero AM, Llewellyn PL, Rodrı´guez-Reinoso F. High-resolution N2 adsorption isotherms at 77.4 K: critical effect of the He used during calibration. J Phys Chem C 2013;117(33):16885e9.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
The effect of inaccurate volume calibrations on hydrogen uptake measured by the Sieverts method C.J. Webb*, E.MacA. Gray Queensland Micro- and Nanotechnology Centre, Griffith University, Nathan, Brisbane 4111, Australia
article info
abstract
Article history:
The Sieverts technique is a popular method of measuring the uptake of gas on a sample
Received 22 October 2013
and is an important tool for the characterisation of hydrogen uptake by potential hydrogen
Received in revised form
storage materials. An analysis of the consequences of errors in the calibration of volumes
23 November 2013
in this technique has been made using hydrogen absorption on an AB2 alloy as an example.
Accepted 28 November 2013
Trends in the error in the uptake as a function of errors in the volumes have been
Available online 31 December 2013
explained in terms of the equations for uptake. Significant advantages are found for calibration techniques for which the ratio of volumes is an intrinsic measurement.
Keywords:
Crown Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All
Volume calibrations
rights reserved.
Hydrogen storage Sieverts
1.
Introduction
The manometric Sieverts technique for measuring the uptake of hydrogen and other gases by materials is popular because it is cheap and easy to set up, physically robust and reasonably reliable. As found empirically [1] and theoretically [2], it has vulnerabilities to uncertain or inaccurate knowledge of relevant parameters, especially the volume occupied by the sample itself when this is large for the amount of gas absorbed or adsorbed, i.e. for samples of low-density, generally speaking. Two round-Robin exercises comparing results for an activated carbon [3] and catalysed magnesium [4] have revealed an alarming spread in the reported hydrogen capacities of these low-density materials. The need for authoritative measurements of hydrogen capacity for new storage materials makes an investigation of this irreproducibility between laboratories urgently necessary. Measurements of pressure and temperature can usually be made with sufficient accuracy and fed into a sufficiently
sophisticated model of the apparatus and sample to produce a notionally accurate result for the hydrogen capacity of the sample. The propagation of uncertainties in measured parameters is analysed in detail by Webb and Gray [2] and supports this view. The difficulty in determining the volume of the sample, particularly for low density samples or samples that change volume during absorption has been treated elsewhere [1] and the need to use the non-ideal gas equation incorporating the compressibility of the gas has also been discussed [5e7]. This leaves calibration of the apparatus as the only apparent culprit in the matter of obtaining accurate results. Methodological and technical approaches to calibration of a Sieverts apparatus have been described and analysed in a number of papers [6e9]. The challenge is to first model the system, in which the various zones may be at different temperatures, then to apply a calibration procedure to measuring the volume in each temperature zone and in particular obtain values for the reference volume and the void (unoccupied) volume of the sample cell.
* Corresponding author. Tel.: þ61 7 3735 5023; fax: þ61 7 3735 7656. E-mail address: [email protected] (C.J. Webb). 0360-3199/$ e see front matter Crown Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.11.121
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The most fundamental calibration is that of the system with no sample, since no amount of care taken to account for the volume occupied by the sample itself will recover an accurate measurement of the hydrogen capacity if the reference and empty-cell volumes are wrong. Therefore, in this paper we analyse the consequences of wrong reference and emptycell volumes and find that the calibration methodology has a great influence on the sensitivity of the calculated hydrogen capacity to errors in these volumes. On that basis we make recommendations about the calibration methodology and propose a routine check to be applied to the basic instrument to confirm that there is no serious error in its calibration.
2.
Methods
The calculation of gas uptake by a sample using the Sieverts technique has been detailed previously [7] and is only summarised briefly here. The technique proceeds in a stepwise fashion. At the start of step k, a quantity of gas (nkref ) is prepared in a reference or dosing volume (Vref). After reaching thermal equilibrium, this gas is released into the cell volume containing the sample (Vcell) which initially contains the amount of gas from the end of the previous step. The gas occupies that volume of the cell not occupied by the sample e the void volume, Vvoid ¼ Vcell Vxk
(1)
where Vxk is the volume of the sample. Since the sample may swell or expand, adsorb or chemically change, the sample volume is a function of the uptake and hence step number. The amount of gas initially is
nki ¼ nkref þ nk1 void ¼
Pkref Vref Zkref ;ref RTkref
þ
k1 Pk1 sys Vcell Vx k1 Zk1 sys;cell RTcell
(2)
where Pkref is the pressure prepared in the reference volume, Pk1 sys is the system pressure after equilibration of the previous step and hence the initial pressure in the void volume, Tkref and Tk1 cell are the temperature of the reference volume at this step and the temperature of the cell volume at the previous step k1 k1 k1 ¼ ZðPsys ; Tcell Þ respectively and Zkref ;ref ¼ ZðPkref ; Tkref Þ and Zsys;cell are the respective compressibilities of the gas which are functions of pressure and temperature. Similarly, the amount of gas in the system after the two volumes are connected and have reached equilibrium is
nkf
¼
nkref ;sys
þ
nkvoid
¼
Pksys Vref Zksys;sys RTksys
þ
Pksys Vcell Vxk Zksys;cell RTkcell
Dnkabs
¼
Pkref Zkref ;ref RTkref þ þ
Zksys;sys RTksys
Pk1 sys k1 Zk1 sys;cell RTcell
Pksys Vxk k Zsys;cell RTkcell
!
Pksys
Vref
Pksys Zksys;cell RTkcell
! Vcell
k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
The total absorption by the sample at the end of step k is the sum of absorption increments of this step and all the previous steps. To calculate the correct or true uptake, the values of the reference and the cell volumes must be known accurately e as must the volume occupied by the sample itself at each step. To show the effect of inaccuracies in the volumes of the reference and/or cell, data from a Sieverts experiment using an AB2 alloy (Ti0.60Zr0.4Cr0.85Mn0.25Fe0.7Ni0.20Cu0.03) have been used as a model for varying the volumes and re-calculating the uptake. The original data are shown in Fig. 1, showing uptake measured in wt%, where wt% is the percentage ratio of mass of gas uptake to the mass of sample plus gas, wt% ¼
mg nabs Mg 100 100 ¼ mx þ mg mx þ nabs Mg
¼nki
¼
Pkref Vref
(3)
k1 Pk1 sys Vcell Vx
þ k1 Zkref ;ref RTkref Zk1 sys;cell RTcell Pksys Vcell Vxk Pksys Vref k k Zsys;sys RTsys Zksys;cell RTkcell nkf
Rearranging with respect to volumes gives
(6)
where mg is the mass of absorbed gas, Mg is the molecular mass of the gas and mx the initial mass of the sample. Fig. 1 shows the uptake of hydrogen at 23 C as a function of applied pressure in bar up to 300 bar with the uptake conventionally on the horizontal axis. It can be seen that the alloy absorbs with a plateau characteristic of metal hydrides to a maximum of 1.4 wt% at a pressure of approximately 150 bar. Between the last two points close to 175 bar and 300 bar there is effectively no further absorption. The volumes of the reference and cell are 2.180 0.005 cc and 6.944 0.015 cc respectively as determined by a gas expansion volume calibration using helium. A calibrated volume was used to experimentally determine the ratios of the reference and (empty) cell volume to the calibrated volume.
The difference between the amount of gas in the system before and after the volumes are connected is the additional amount of gas absorbed by the sample in this step
Dnkabs
(5)
(4) Fig. 1 e H2 uptake on interstitial alloy at 23 C e uptake calculated using the correct reference and cell volumes.
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The effect of inaccurate reference volume
From Eqn 5, it can be seen that if the cell and sample volumes are accurate, but the reference volume varies from the true reference volume by an amount 3 (Vref), then it is only the first term of Eqn 5 which contributes to an error in the uptake. In this term, the reference volume is multiplied by Pkref Pksys . k k k k Z RT Z RT ref ;ref
ref
sys;sys
sys
Since Pksys is always less than Pkref (for absorption steps), this expression must always be positive and so the calculated uptake will be greater than the true uptake if the assumed value of the reference volume is larger than the true value (and vice versa). This is shown in Fig. 2. The larger value of Vref in Fig. 2, curve (c) causes all the values of uptake to be greater, expanding the isotherm along the uptake axis. This includes the uptake at the two highest pressures (for which the real uptake is zero) causing the graph to show steadily increasing uptake, i.e. a positive slope instead of vertical. Similarly in Fig. 2, curve (a), for a value of the reference volume smaller than the true value, all uptake values are less than the true uptake and where the true uptake is zero, the uptake is negative, causing the graph to slope backwards. The significance of any error in the reference volume only is shown in Fig. 3. Here, the percentage error, i.e. the percentage difference between the calculated maximum uptake and the true maximum uptake has been plotted as a function of the error in the reference volume, 3 (Vref) (the difference between the value of Vref used in the calculation and the true value). The factor ðPkref =Zkref ;ref RTkref Pksys =Zksys;sys RTksys Þ, which multiplies the error Vref is a measure of how large the step is in terms of pressure in the reference volume compared to the final system pressure. In addition, this error in each step of the Sieverts uptake measurement compounds as the incremental uptake at each step is summed to yield the total uptake for each pressure step. The expressed percentage error in the total uptake is for the last point at w300 bar which will have
Fig. 3 e Absolute error in the calculated uptake for the maximum pressure value as a function of error in the reference volume.
the largest error. The importance of Fig. 3 is showing how a quite small error in the reference volume leads to a significant error in the calculated uptake. For example, an error of 0.05 cc gives an error of nearly 9% (of 1.4 wt%) in the uptake.
2.2.
The effect of inaccurate cell volume
Using Eqn 5, with an accurate value of the reference volume but allowing the cell volume to vary produces a corresponding error in the calculated uptake. In this case, the error in the cell Pk1 Pk volume 3 (Vcell), is multiplied by Zk1 sysRTk1 Zk sysRTk . sys;cell
ref ;ref
Fig. 2 e Recalculated hydrogen uptake isotherms with ±10% error in the reference volume, (a) Vref [ 1.962 cc, (b) Vref [ 2.180 cc, true volume and (c) Vref [ 2.398 cc.
cell
sys;cell
cell
For absorption, the term involving the equilibrium system pressure from the previous step (Pk1 sys ) will always be less than the equilibrium system pressure for the current step (Pksys ), so this expression will always be negative. A larger cell volume than the true cell volume will reduce the calculated uptake and vice versa. This is shown in Fig. 4. The sensitivity of the uptake error to the cell volume error is shown in Fig. 5, which resembles Fig. 3 for the reference volume error, except that the magnitude of the uptake error is approximately one-quarter that of corresponding errors for k1 k1 the reference volume. The term ðPk1 sys =Zsys;cell RTcell Pksys =Zksys;cell RTkcell Þ amplifying the cell volume error has a smaller Pk Pk magnitude than Zk refRTk Zk sysRTk amplifying the reference ref
sys;sys
sys
volume error since the difference in system pressures between steps is typically smaller than the difference in reference and equilibrium pressure for a step. We emphasise that the assumed error here is in the volume of the empty cell, excluding any imperfect knowledge of the volume occupied by the sample. In the case of desorption, the reference pressure is always smaller than the resultant system pressure and the system pressure decreases with each step, so the terms multiplying the errors in the reference and cell volume change sign. This means that the desorption isotherm will undergo the same
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along the pressure axis) will always be achieved provided the ratio of volumes is correct. Performing an empty-cell isotherm to check the integrity of the Sieverts apparatus has been advocated previously [8] and is a technique which can discover leaks, gas adsorption to the cell walls and other problems. However, if the values of both cell and reference volume are incorrect e but the ratio of volumes is correct, the empty-cell isotherm will not indicate the problem. If the calculated uptake at step k is split into the true part plus an additional amount representing the error true þ 3 Dnkabs Dnkabs ¼ Dnkabs
(8)
and the errors in the reference and cell volumes are denoted similarly, true þ 3 Vref Vref ¼ Vref true þ 3 Vref Vcell ¼ Vcell
Fig. 4 e Recalculated hydrogen uptake isotherms with ±10% error in the cell volume, (a) Vcell [ 7.638 cc, (b) Vcell [ 6.944 cc, true volume and (c) Vcell [ 6.250 cc.
(9)
Eqn 5 becomes
expansion or contraction along the uptake axis as the absorption isotherm.
k true k þ 3 Dnabs ¼ Dnabs
Pkref Zkref ;ref RTkref
þ
The ratio of the cell volume to the reference volume is an important constant in the Sieverts technique since it is often measured by the common practice of calibrating Vcell against Vref. If we consider the case of an empty cell (no sample, Vx ¼ 0), such that the true absorption for every step should be zero, then Eqn 5 becomes,
Pkref
Zkref ;ref RTkref Vcell ¼ Vref Pksys Zksys;cell RTkcell
Zk
k sys;sys RTsys
Pk1
! Pksys true Vcell þ 3 ðVcell Þ k1 k k Zk1 RT Z RT sys;cell cell sys;cell cell
k1 k1 Pksys Vxk Psys Vx k1 k k Zsys;cell RTcell Zsys;cell RTk1 cell
Since Eqn 5 must hold for the true part of the uptake k true ¼ Dnabs
Pkref Zkref ;ref RTkref
Zksys;sys RTksys
!
Pksys Zksys;cell RTkcell
!
Pksys true Vcell þ
true Vref þ
Pksys Vxk k Zsys;cell RTkcell
Pk1 sys k1 Zsys;cell RTk1 cell k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
(7)
Zk1 sysRTk1 sys;cell
true Vref þ 3 Vref
(10)
Pksys
Zksys;sys RTksys
Pk1 sys
þ
2.3. The effect of inaccurate volumes but maintaining correct volume ratio
!
Pksys
cell
showing that a Sieverts uptake measurement on an empty cell is equivalent to a measurement of the ratio of cell to reference volumes and that a zero absorption isotherm (i.e., straight line
(11) Substituting Eqn 11 into Eqn 10 gives ! k Pkref Pksys Dnabs ¼ k Zref ;ref RTkref Zksys;sys RTksys 3 Vref
3
Pk1 sys
þ
k1 Zk1 sys;cell RTcell
Pksys
!
Zksys;cell RTkcell
3 ðVcell Þ 3
Vref
(12)
or
k 3 Dnabs
"
Pkref
¼
Zkref ;ref RTkref
Pk1 sys
þ
k1 Zk1 sys;cell RTcell
Pksys
!
Zksys;sys RTksys
Pksys Zksys;cell RTkcell
!
true Vref true Vcell
#
3 Vref
(13)
where we have used the fact that the ratio of the errors in the volumes is the same as the ratio of the true volumes true þ3 V Vref ð ref Þ true þ3 ðV Vcell cell Þ 3 ðVcell Þ 3
Fig. 5 e Absolute error in the calculated uptake for the maximum pressure value as a function of error in the cell volume.
ðVref Þ
¼
Vtrue
¼ Vref true
true Vcell true Vref
cell
(14)
Note that the error in the uptake considered here is only that due to reference and cell volume inaccuracies, hence Eqn 13 does not contain any term involving the sample volume.
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Eqn 13 demonstrates that the error in the uptake is then proportional to the error in the reference volume and the term true k1 Pkref Vref Pksys Psys Pksys determines þ k k k k k1 k1 k k Vtrue Z RT Z RT Z RT Z RT ref ;ref
ref
sys;sys
sys
sys;cell
cell
sys;cell
cell
k true nabs ¼0¼
whether the uptake is greater or less than the true value. The first Pk Pk component Zk refRTk Zk sysRTk is always positive and the secsys;sys sys ref ;ref ref k1 P Pk ond component Zk1 sysRTk1 Zk sysRTk is always negative, so the sys;cell
cell
sys;cell
þ
For the limiting case when the true absorption is zero (from Eqn 5)
Fig. 6 e Recalculated hydrogen uptake isotherms with Vcell/ Vref [ true volume ratio and ±25% error in the reference volume, (a) Vref [ 1.635 cc (L25%), (b) Vref [ 2.180 cc, (true) and (c) Vref [ 2.725 cc (D25%).
Pk1 sys
!
Pksys Zksys;sys RTksys
k1
Zk1 sys;cell RTcell Pksys Vxk
Zksys;cell RTkcell
!
Pksys Zksys;cell RTkcell k1 Pk1 sys Vx
!
k1 Zk1 sys;cell RTcell
true Vcell true Vref
(15)
1 true Vref
Substituting Eqn 15 into Eqn 13 gives
cell
2.4. Correct volume ratio when absorption has largely finished
Zkref ;ref RTkref þ
cell
sign of the error in the calculated uptake depends on these pressure values and the ratio of volumes. For non-zero real absorption, the system equilibrium pressure is always lower than it would be if no absorption took place, and this favours the positive component at the expense of the negative component leading to a net positive effect, such that the error in the uptake has the same sign as the error in the reference volume. Using the AB2 alloy data, this can be investigated by changing the values of both Vcell and Vref such that the ratio remains constant at the correct value. For Fig. 6, curve (a), the reference volume is smaller than the true reference volume. Compared to the true uptake values shown in Fig. 6, curve (b), the calculated uptake is smaller for non-zero real uptake points, confirming that the uptake error has the same sign as the reference volume error. However, for zero real uptake (as shown between the last two points) the uptake has increased e this is explained in the next section. For the case where the reference volume is larger than the true reference volume (Fig. 6, curve (c)), the converse is true. Here, the uptake is larger than the real uptake for nonzero real absorption values but less than zero for the last two points where the real absorption is zero.
Pkref
! k k1 Pksys Vxk Pk1 Dnabs 1 sys Vx ¼ k k1 true V Zsys;cell RTkcell Zsys;cell RTk1 3 Vref ref cell
3
(16)
Hence, k 3 Dnabs ¼
Pksys Vxk Zksys;cell RTkcell
k1 Pk1 sys Vx k1 Zsys;cell RTk1 cell
! 3
Vref true Vref
(17)
Eqn 17, which is only valid in the case of zero real absorption, explains the behaviour of the uptake between the last two points in the isotherms shown in Fig. 6 curve (a) and Fig. 6 curve (c). Since the sample volume does not change from the previous step to the current step (since there is no absorption), then the term in parentheses is always positive as the current system equilibrium pressure, Pksys is always greater k1 . Hence than the previous system equilibrium pressure, Psys the error in the uptake is negative for positive reference volume error and vice versa. When the reference volume used in the calculation of uptake is less than the true reference volume (and the same is true of the cell volume since the ratio of volumes is equal to the true ratio), 3 (Vref) < 0, and the error in uptake will be positive. Since the true value is zero, the calculated value will be greater than zero leading to the slope of the line between the last two data points in Fig. 6 curve (a). Conversely, the backwards slope of Fig. 6 curve (c) is due to a larger reference volume than the true reference volume in Eqn 17 giving a negative uptake cf. the true uptake of zero. Thus the different behaviour of the uptake provides some opportunity to analyse the source of volume calibration errors if a standard sample (for which the isotherm is wellestablished) is measured. An incorrect ratio of reference and empty-cell volumes with Vref > Vcell leads to increased apparent uptake for all uptake values, while an incorrect ratio with Vref < Vcell gives a calculated uptake smaller than the real uptake for all uptake values. For the correct volume ratio, however, while the same trend is observed for non-zero real uptake, 3 (Vref) > 0 has a negative uptake for zero real uptake and for 3 (Vref) < 0 the calculated uptake is positive for zero real uptake. However, the trialled increase/decrease in the reference and cell volumes in Fig. 6 is very high e 25%. It is unlikely that a careful volume calculation would be in error by such an amount. The effect of a more moderate but still large error in the volumes of 10% is shown in Fig. 7. Seen individually, any of these isotherms might not immediately suggest a volume calibration problem. The slope of the line between the last two points, where the real uptake is zero is sufficiently close to vertical to suggest the calibrations are correct e especially for Fig. 7 curve (a) where a slight positive slope might be expected from solid solution of hydrogen in the metal hydride. However, the effect on the
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direct technique, where a non-absorbing gas is used to directly measure the void volume, or by the indirect technique, where the sample volume is simply subtracted from the calibrated cell volume [9,10]. The indirect technique has been assumed above, in part due to concerns about the legitimacy of assumptions in the direct technique [11,12] and in part due to the array of samples that expand or swell during absorption thus rendering the direct technique impractical. However, for a sample that does not change volume (other than by adsorbing a layer of the gas, if appropriate), it is useful to interpret the uptake in terms of the reference and void volumes. From Eqn 4, Dnkabs Fig. 7 e Recalculated hydrogen uptake isotherms with Vcell/ Vref [ true volume ratio and ±10% error in the reference volume, (a) Vref [ 1.962 cc, (b) Vref [ 2.180 cc, (true) and (c) Vref [ 2.398 cc.
¼
Pkref Zkref ;ref RTkref
Fig. 8 e Recalculated hydrogen uptake isotherms with (a) of L2%, 3(Vcell) [ 0, (b) Vcell/Vref [ true volume ratio with 3(Vref) of L2% (c) true values, (d) Vcell/Vref [ true volume ratio with 3(Vref) of D2% and (e) 3(Vref) [ L2%, 3(Vcell) [ 0.
Zksys;sys RTksys !
Pksys Zksys;cell RTkcell
Pkref Zkref ;ref RTkref
3(Vref)
! Vref þ
Pk1 sys k1 Zk1 sys;cell RTcell
(18)
Vvoid
Clearly, for zero real uptake 0¼
calculated uptake at the maximum pressure e typically the quoted value e is significant, approximately 10%. For a still smaller error in the reference volume, Fig. 8 compares a 2% error in Vref for the case of correct and incorrect volume ratios. Fig. 8 demonstrates the increased accuracy consequent on having a correct ratio of volumes. For an error of 2% in the reference volume, the error in the calculated uptake at maximum pressure is 1.5% if the ratio of volumes equals the correct ratio e and 7.6% if the other (cell) volume is actually correct. The analysis above has been couched in terms of the reference volume Vref and the cell volume Vcell. However, the cell volume is not the volume that the gas prepared in the reference volume expands into, that is the void (or dead) volume e the volume of the cell not occupied by the sample. Using a gas expansion, this may be measured by either the
Pksys
Vvoid Vref
Zk
k sys;sys RTsys
k1 Psys
Zk1
¼ sys;cellk
Pksys
RTk1 cell
Pk1 Vref þ Zk1 sysRTk1 Zk
Pk sys Zk RTk sys;cell cell
P Pk sys ref k Zsys;sys RTk Zk RTk sys ref ;ref ref
sys;cell
cell
Pksys
RTkcell sys;cell
Vvoid (19)
In theory, for a pressure range on a sample which is known to have zero absorption, the void volume and, in turn, the sample volume could be determined from Eqn 19. In practice, this may require high pressures and large pressure steps and there would be a limited range of samples which would have zero real absorption at these pressures.
3.
Conclusions
An analysis of the consequences of inaccurate reference and cell volumes for the Sieverts technique for uptake of gas in or on to a sample has been made. It has been shown, using hydrogen absorption data for an AB2 alloy, that relatively small errors in the values for these volumes have a significant impact on the accuracy of the uptake of gas calculated for this technique, demonstrating the fundamental reliance of the method on these volumes. In addition, the inherent accumulation of errors associated with the summation of incremental uptake values in the Sieverts technique exacerbates this problem. The equation of the uptake for each step in the measurement process has been used to explain trends in the calculated uptake for different combinations of errors in the reference and cell volumes. In particular, when the value of the reference volume used is smaller than the true value, or the cell volume used is greater than its true volume, the calculated uptake is smaller than the true uptake. Conversely, when the reference volume used is greater than the true volume, or the cell volume used is less than its true volume, the calculated uptake is greater than the true uptake. For a 10% error in one of these volumes (the other being correct), the uptake calculated was shown to be in error by as much as 50% at 300 bar for the trial data.
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In the special case of incorrect reference and cell volumes, but for a correct ratio of Vcell/Vref, the calculated uptake may include a distinguishing feature. This particular situation is important because of the way in which volume calibrations are often performed. If the reference and (empty) cell volumes are calibrated using a third, known volume through a gas expansion method, the two unknown volumes are always determined as a ratio to the calibrated volume (and hence to each other). So, through careful measurements, it is possible to obtain a calibration that has a very accurate cell volume to reference volume ratio. This applies to both the conventional volume calibration technique, employing a separate vessel as a calibrated volume, as well as the displacement volume technique [8]. However, if the volume used to calibrate is in error, then the values are incorrect even though the ratio is the same as the ratio of the true volume values. For this situation, the error in uptake is comparable to the error in the calibrating volume and significantly smaller than the corresponding uptake error for the same error in the reference volume when the cell volume is correct and vice versa. This demonstrates the importance of a calibration technique which determines volumes in terms of their ratios, compared to a technique where the reference and cell volumes are determined independently. For larger errors in the reference volume, the calculated uptake behaves similarly to incorrect ratios for non-zero real uptake, however, when the real uptake is close to zero the trend reverses, with negative uptake steps for larger values of Vref, leading to a distinctive backward sloping isotherm at high pressures instead of a vertical slope (and conversely for smaller values of Vref than the true value). The practice of performing an uptake measurement on an empty cell is extremely valuable in determining that the ratio of reference to cell volume is correct as well as possibly indicating other errors. However, performing an uptake measurement on a known sample can provide additional confirmation that the absolute values of the reference and cell volumes are correct. This is indicated primarily by the measured maximum uptake value, but also by the slope of the isotherm after absorption is mostly complete (if a sufficiently high pressure can be reached). A suitable known sample would need to be one which has been measured by many different laboratories with a well-established uptake and for which the density is a well-known function of uptake in order
to remove any error contributions from the sample volume. This would depend on the measurement system being used, but for hydrogen uptake measurements, virgin LaNi5 might be suitable for many systems.
references
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