HYDROGEN AND METHANE STORAGE IN ADSORBENT MATERIALS FOR AUTOMOTIVE
APPLICATIONS
Matthew Beckner1 and Anne Dailly2
2General
1Optimal CAE, Inc. Plymouth, MI, USA
Motors Global Research and Development Center, Warren, MI, USA
E-mail: matthew.beckner@gm.com
Supporting Information
Section S1 – IRMOF-1
IRMOF-1 was synthesized as previously described in the literature [1]. Hydrogen adsorption isotherms were
measured at ambient temperature up to 350 bar using a custom built manometric adsorption instrument
described previously [2,3]. Excess adsorption for IRMOF-1 is found in Fig. S1.
Fig. S1. Graph showing the ambient temperature hydrogen excess adsorption of IRMOF-1.
Section S2 – Derivation of Equations
In this section, as throughout the main paper, lower
case Latin alphabet symbols indicate quantities
normalized to the sample mass (or volume as
appropriate).
S2.1 Equations 1-4
The excess adsorption 𝑚exc is defined to be the total
mass of gas in the system 𝑚st less the amount of gas
that would be present if the solid was non-adsorbing
𝜌gas (𝑝, 𝑇)𝑣pore so that
𝑚exc ≔ 𝑚st − 𝜌gas (𝑝, 𝑇)𝑣pore
(S1)
By algebraically rearranging Eq. (S1), we arrive at Eq.
(1) in the main text
𝑚st = 𝑚exc + 𝜌gas (𝑝, 𝑇)𝑣pore
(1)
The storage per volume of adsorbent, Eq. (2), is
obtained from by the following steps:
𝑀st = 𝑚st 𝑚𝑠
𝑣st =
𝑀st
𝑚st 𝑚𝑠
=
𝑉tank
𝑉tank
(S2)
Since the tank volume is also the bulk volume as
defined in Table 1 in the main text
𝑚𝑠
𝑣st = 𝑚𝑠𝑡
(S4)
𝑉b
and we arrive at Eq. (2)
𝑣st = 𝑚st 𝜌b
Eq. (3) is readily obtained by substitution Eq. (1) into
Eq. (2)
𝑣st = 𝑚exc 𝜌b + 𝜌gas (𝑝, 𝑇)𝑣pore 𝜌b
(3)
From the definitions of the volumes in Table 1, we see
that the pore volume is the bulk volume minus the
skeletal volume or
𝑉pore = 𝑉b − 𝑉s
(S3)
(2)
(S5)
𝑣pore =
𝑉pore 𝑉b − 𝑉s 𝑉b
𝑉s
=
=
−
𝑚s
𝑚s
𝑚s 𝑚s
(S6)
1
1
−
𝜌b 𝜌s
(S7)
It follows that
and
𝑣pore =
Substituting Eq. (S7) into Eq. (3) we have
1
1
𝑣st = 𝑚exc 𝜌b + 𝜌gas (𝑝, 𝑇) ( − ) 𝜌b
𝜌b 𝜌s
𝜌b
)
𝜌s
(4)
𝑣st = 𝑚exc 𝜌b + 𝜌gas (𝑝, 𝑇)𝑣pore 𝜌b
(3)
𝜌gas (𝑝, 𝑇)(𝑉b − 𝑉𝑠 )
𝑣st = 𝑚exc 𝜌b +
𝜌b
𝑚s
(S9)
𝜌gas (𝑝, 𝑇)
(𝑉b − 0)𝜌b
=0+
𝑚s
(S10)
𝜌gas (𝑝, 𝑇)𝑉b
𝜌b
𝜌b = 𝜌gas (𝑝, 𝑇)
𝑚s
𝜌b
(S11)
(empty)
𝑣st
=
(empty)
𝑣st
The volumetric storage of the tank with an adsorbent is
greater than the storage of an empty tank if 𝑣st >
(empty)
𝑣st
so that
𝜌b
𝑚exc 𝜌b + 𝜌gas (𝑝, 𝑇) (1 − ) > 𝜌gas (𝑝, 𝑇)
(S13)
𝜌s
𝑚exc >
𝜌b
)
𝜌s
(S15)
𝜌gas (𝑝, 𝑇)
𝜌gas (𝑝, 𝑇) 𝜌gas (𝑝, 𝑇)
−(
−
)
𝜌b
𝜌b
𝜌s
(S16)
𝜌gas (𝑝, 𝑇)
𝜌s
(S20)
It follows that
𝑉pore + 𝑉s
1
=
𝜌𝑎
𝑚s
(S21)
1
1
= 𝑣pore +
𝜌𝑎
𝜌s
(S22)
𝜌s 𝑣pore 1
1
=
+
𝜌𝑎
𝜌s
𝜌s
(S23)
𝜌s 𝑣pore + 1
1
=
𝜌𝑎
𝜌s
(S24)
and
𝜌a =
𝜌s
𝜌s 𝑣pore + 1
(9)
S2.5 Equations 10-12
We define the packing factor 𝑓 to be the ratio of the
apparent volume to the volume of the tank
𝑉a
𝑓≔
(S25)
𝑉tank
It follows that
𝑓=
𝑉a 𝑚s
𝑉tank 𝑚s
(S26)
𝜌b
𝜌a
(S27)
𝑓=
𝜌b = 𝑓𝜌a
(5)
S2.3 Equations 6-8
Eq. (7) is an algebraic manipulation of the definition Eq.
(6). From Table 1, it is clear that
𝑉s < 𝑉c ≈ 𝑉a < 𝑉b
From the definitions in Table 1
𝑚s
𝜌𝑎 ≔
𝑉pore + 𝑉s
and
and finally we arrive at Eq. (5)
𝑚exc >
S2.4 Equation 9
(S14)
𝜌gas (𝑝, 𝑇) 𝜌gas (𝑝, 𝑇)
𝜌b
−
(1 − )
𝜌b
𝜌b
𝜌s
𝑚exc >
(8)
(S12)
= 𝜌gas (𝑝, 𝑇)
𝑚exc 𝜌b > 𝜌gas (𝑝, 𝑇) − 𝜌gas (𝑝, 𝑇) (1 −
(S19)
𝜌s > 𝜌c ≈ 𝜌a > 𝜌b
The volumetric storage of an empty tank is obviously
the density of the gas, but this can be obtained from
Eq. (3) to demonstrate the consistency of these
equations. In an empty tank, 𝑚exc = 0 and 𝑉s = 0 so
that, starting from Eq. (3),
(empty)
𝑚s 𝑚s 𝑚s 𝑚 s
>
≈
>
𝑉s
𝑉c
𝑉a
𝑉b
(S8)
S2.2 Equation 5
𝑣st
(S18)
and finally we arrive at Eq. (8)
and finally arrive at Eq. (4)
𝑣st = 𝑚exc 𝜌b + 𝜌gas (𝑝, 𝑇) (1 −
1
1
1
1
> ≈ >
𝑉s 𝑉c 𝑉a 𝑉b
(S17)
(10)
We arrive at Eq. (11) and (12) by replacing the bulk
density in Eqs. (1) and (4) with Eq. (10).
S2.6 Equations 13 and 14
In this paper, we consider a tank without walls. The
mass includes the mass of the gas and mass of
material. For an 8 GGE tank,
𝑚tank ≔ 𝑚𝑠 + 𝑚8GGE
𝑚tank =
𝑚8GGE
+ 𝑚8GGE
𝑚st
(S28)
𝑉tank = 𝑉tank
(13)
𝑉tank =
The quantity 𝑚8GGE /𝑚st is the total mass of the sample,
keeping in mind that 𝑚st is normalized to the sample
mass. This ratio is the total mass required to store
8GGE of gas. Since we are neglecting the walls of the
container, the volume of the tank is simply the volume
that the material occupies.
𝑉tank =
𝑚s
𝑚s
(S29)
𝑚s
𝜌b
(S30)
𝑚8GGE
𝑚st 𝜌b
(S31)
And finally using Eq. (10)
𝑉tank =
𝑚8GGE
𝑚st 𝑓𝜌a
(14)
Section S3 – Table 1 for Reference
Table 1. Definitions of different derived volumes. Corresponding densities are found by dividing the sample mass by
the appropriate volume.
Inter-particle Void
Volume
Solid Volume
Closed Pore Volume
Open Pore Volume
Volume
Skeletal
X
X
Crystal*
X
X
X
Apparent
X
X
X
Bulk
X
X
X
X
*Only includes closed pores that exist because of the theoretical structure or the structure surmised from x-ray
scattering.
References
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Chae HK, Siberio-Pérez DY, Kim J, Go Y, Eddaoudi M, Matzger AJ, et al. A route to high surface area,
porosity and inclusion of large molecules in crystals. Nature 2004;427:523–7.
[2]
Voskuilen T, Zheng Y, Pourpoint T. Development of a Sievert apparatus for characterization of high pressure
hydrogen sorption materials. Int J Hydrogen Energy 2010;35:10387–95.
[3]
Beckner M, Dailly A. Adsorption Enthalpy Calculations of Hydrogen Adsorption at Ambient Temperature and
Pressures Exceeding 300 bar. Am J Anal Chem 2013;04:8–16.