1. Introduction
A big proportion of the greenhouse gas emissions in the atmosphere are caused by the combustion of hydrocarbons [
1]. Hydrogen has been considered as a zero-carbon fuel replacement, in particular in the transport sector for conventional vehicles, since it can be used by internal combustion engines or fuel cells [
2]. Due to the low volumetric energy density of hydrogen, which is its major drawback, 5 MJ·kg
−1 at 700 bar and ambient temperature, compared to 32 MJ·kg
−1 for gasoline [
3], its storage is a challenge. This requires a capacity between 5 and 13 kg of hydrogen for onboard hydrogen storage to meet the driving range for the full range of light-duty vehicle platforms [
3]. To store 5 kg of hydrogen at 700 bar, a type IV compressed hydrogen storage tank requires a volume of 203 L [
4]. Storing hydrogen by compression at such an elevated pressure poses various issues: the high cost of the equipment, difficult maintenance operation, hydrogen contamination with lubricating oil, and embrittlement of metal components that may cause the container to fracture [
5]. The alternative conventional method, which is liquefaction, offers a very high density of liquid hydrogen, namely 70 kg·m
−3 at 20 K and 1 bar, which is far higher than the density of compressed hydrogen gas at 700 bar, 42.6 kg·m
−3 [
6]. However, the cryogenic process is technically complex and energy intensive. Moreover, it induces more potential hazards resulting from boil-off during dormancy, ice formation, and air condensation [
7].
It has been recognized that storing hydrogen in a solid state by adsorption into porous materials can be a viable solution for stationary and on-board applications [
8]. Hydrogen confinement in nanometer-sized pores of high-surface area materials, such as nanoporous carbons (activated chars, nanotubes, fullerenes, expanded graphite), metal organic frameworks (MOFs), or oxides (zeolites) results in a higher volumetric storage density than the bulk gas under the same pressure and temperature conditions [
8]. The higher the difference between the densities of the adsorbed fluid and the bulk gas, the more efficient the hydrogen storage process is. However, due to hydrogen’s low critical temperature (T
c = 32 K), reaching liquid-like densities in the adsorbed phase at ambient temperature (298 K) is particularly difficult when operating at moderate pressures, so that storage capacities do not dramatically exceed those by simple compression at the same pressure. Typically, among the best MOF materials suitable for hydrogen storage, the MOF NU-1103, synthesized by dissolving a zirconium source and organic linkers in dimethilformamide with the addition of benzoic acid, features a bulk density of 345 kg·m
−3 and stores 8.0 gH
2·L
−1 (≈2.3 wt%) at 100 bar and 295 K, while the density of compressed hydrogen under the same conditions is 7.7 gH
2·L
−1 [
9,
10].
Consequently, in order to enhance volumetric and gravimetric adsorption capacities of hydrogen in porous materials, operation at cryogenic temperature (77 K or above) is commonly adopted. At the temperature of liquid nitrogen (77 K) and at a pressure of 100 bar, the same MOF material offers volumetric and gravimetric adsorption capacities of 44.9 gH
2·L
−1 and 13.0 wt%, respectively. Although carbonaceous porous materials such as activated carbons exhibit lower gravimetric adsorption capacities, reaching at best 7–10.4 wt% at 60 bar and 77 K [
11], they are competitive with MOFs because they can demonstrate larger volumetric storage capacities thanks to their higher bulk density, so that a larger mass of adsorbent can be packed in the vessel. Moreover, the activated carbons are inexpensive in comparison to MOFs and commonly available on the market [
5]. Furthermore, several adsorbent materials have been engineered at the simulation scale, like a titanium decorated carbon [
12], exhibiting gravimetric hydrogen storage capacities up to 6.67 wt% under ambient conditions.
Most of research efforts on the application of nanoporous materials for hydrogen adsorption have been driven to meet the 5.5 wt% and 40 gH
2·L
−1 DoE 2025 target for onboard H
2 storage systems, including their reservoir components [
13]. A great part of these studies addresses the design and characterization of porous materials and the assessment of their hydrogen adsorption—desorption capacities. But the evaluation of the energetic efficiency of the process is rarely addressed, especially in comparison to conventional methods of hydrogen storage [
14]. If the overall energy consumption of the storage system can be primarily assessed by analyzing the thermodynamic path of the process between different equilibrium states, a more accurate assessment of the heat and power energy requirements should account for the dynamics of the system, and consider the transient variations in the operating parameters such as pressure, temperature, and amounts of hydrogen accumulated in the tank during the charge and discharge steps.
Simulation of the dynamic adsorption process during hydrogen loading and discharge requires the development of models relying on the formulation of mass and energy balances, combined with an equation of state to describe the gas phase behavior and temperature dependent equilibrium adsorption isotherms. Assuming the hydrogen gas phase as ideal can be considered reasonable provided that the operating adsorption pressure remains moderate [
15,
16,
17,
18,
19,
20,
21,
22]. In the pressure range of 150 bar, P. Sridhar and N. S. Kaisare [
23], demonstrated that deviations computed by using either the ideal gas law or the viral equation of state are actually small, and do not exceed 0.2 bar for pressure and 1 K for temperature. However, in the case of higher working pressures, reaching about 700 bar, and cryogenic temperatures, the non-ideality of the hydrogen gas phase needs to be considered and this requires implementing a real gas equation of state [
5,
23,
24].
In order to describe adsorption equilibrium data of hydrogen onto microporous adsorbents such as activated carbons or MOFs, different isotherm models can be applied. The modified Dubinin–Astakhov model (M-D-A) was, for instance, retained in several studies investigating hydrogen adsorption at high pressure and supercritical temperature [
5,
14,
16,
19,
24,
25,
26,
27]. Alternative models such as the Langmuir model [
15,
28], the Radke–Prausnitz model [
17,
29] and the Unilan model [
18,
22,
30] were also chosen. Sridhar and Kaisare [
23] compared the simulation results when charging a reservoir containing MOF-5 adsorbent using three isotherm models: Unilan, M-D-A and Toth, in spite of the good fit between theoretical and experimental isotherms determined in the temperature range between 77 and 300 K, these authors showed that the predicted hydrogen up-takes in the adsorbent bed were considerably impacted by the choice of the isotherm model. The right selection of the temperature-dependent isotherm equation is therefore crucial to obtaining the good predictive ability of the process simulation model.
A comparison of models proposed in the literature in recent decades for the simulation of cryogenic hydrogen storage reservoirs operating by pressure–temperature swing adsorption is given in
Table 1. Moreover, considering or not the non-ideal behavior of the hydrogen gas and describing adsorption equilibrium data according to different forms of isotherm equations, the spatial heterogeneities of the system could or not be assumed. In the case of zero-dimensional (0-D) models, also denominated as lumped models, the variables of the system are supposed to be uniform throughout the entire adsorbent vessel volume, so that overall pressure, temperature, and hydrogen amounts are computed at each time step for the whole system. In such systems, the mass and energy balance equations are derived according to Equations (1) and (2), respectively, resulting in a set of ordinary differential equations (ODEs) which are relatively easy to be solved by applying a first-order numerical method, such as Euler or Runge Kutta. One-dimensional simulation models allow computation of the system variables in the axial direction of the reservoir. The corresponding mass and energy balances (Equations (3) and (4), respectively) are then established in a cylindrical coordinate system, accounting for the shape of the tank. Taking into account the spatial variations in the radial direction of the tank, the mass and energy balance equations take the form of Equations (5) and (6), representative of a 2-D model. The great advantage of the multidimensional models is their accuracy, thanks to the local adjustment of parameters related to the mass and heat transfer kinetics, such as the intra-particle hydrogen diffusivity or heat transfer coefficients. Nevertheless, their numerical solution is much more complicated and most often relies on the spatial discretization of the set of partial differential equations (PDEs), using a finite difference method performed either by in-house codes [
29] or by commercial solvers [
5,
16,
19,
23,
25].
Experimental validation of these models has been evaluated both at ambient temperature [
17,
29,
31] and at cryogenic temperature [
5,
14,
25]. Experimental tests at cryogenic temperature, which is the process that most concerns us, generally use a tank filled with the adsorbent material, submerged in a liquid nitrogen Dewar. Under these conditions, charging and discharging processes are operated, including steps of cooling, compression, storage, heating, pressure release, and dormancy of the system. Regardless of the use of sophisticated 1-D or 2-D models implemented in Multiphysics software, the simplified approach of the 0-D models can efficiently be employed to simulate reservoirs operating adsorbent masses at g–kg scales [
14,
24,
26]. In most works, simulation models aim to reproduce temperature and pressure profiles in the system in order to determine optimal operating conditions and to upscale the process. Parameter sensitivity studies are also performed in order to evaluate experimental data that are difficult to directly measure, such as heat transfer coefficients and specific heat of the adsorbed phase. Moreover, as introduced earlier, the process simulation can enable accurate assessment of the energy efficiency of the adsorbed hydrogen storage system, which, to the best of our knowledge, has only been carried out in a limited number of works [
14].
Considering the good compromise obtained from the numerical simplification of zero-dimensional mathematical models, implying a limited number of adjusted lumped parameters together with fast computation times to obtain rather good predictive ability of time-dependent profiles of pressure, temperature, and hydrogen storage capacities, this work focuses on the development of a 0-D model applicable to the pressure–temperature swing adsorption for hydrogen storage. This model is based on earlier works [
24,
32], but has been substantially improved: 1. by considering an isotherm model derived from the Dubinin–Astakhov (DA) isotherm equation, that we have modified in order to adapt it for supercritical temperatures and high pressures; 2. by introducing the compressibility factor of hydrogen to account for deviation from ideal gas behavior, that was computed after the equations of state implemented within the NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP-version 8) [
33]; and 3. by taking into account variation in the isosteric heat of adsorption with the amount of hydrogen adsorbed.
The validity of the model so developed was assessed for a variety of adsorption storage systems for which experimental data were reported in the literature. The results obtained so far show the capability of the new developed 0-D model to properly describe temperature and pressure profiles with time, for a variety of adsorbent materials, reservoir configurations, and operating conditions. Furthermore, the process simulation model was completed with the computation of the energy consumptions associated with the pressure–temperature swing adsorption storage system, so that a comparative analysis could be carried out with other conventional technologies, such as H2 compression and liquefaction.
2. Description of Hydrogen Adsorption Equilibria
When simulating hydrogen adsorption systems, the isotherm model, which describes the amount of hydrogen adsorbed at equilibrium under given (
P,
T) conditions, is crucial for proper evaluation of the storage capacities of the system. The Dubinin equation is based on the potential theory developed by Polányi in 1932 [
34], which introduced the notion of a characteristic curve. The characteristic curve describes the relationship between the state of compression of the adsorbed fluid and the forces prevailing at the surface of the adsorbent, represented by the adsorption potential.
For a couple of given adsorbate–adsorbents, a single curve independent of temperature so describes the volume of the adsorbate in the adsorbed phase
(m
3·kg
−1) as a function of the adsorption potential
A. Dubinin and co-workers proposed to describe the fraction of micropore volume filling occupied by the adsorbed phase according to the functional form of the Weibull distribution [
34]:
where
(m
3·kg
−1) is the maximum volume that the adsorbate can occupy, usually estimated as the total volume of the micropores,
ԑ (J·mol
−1) is a characteristic energy representative of the adsorbent–adsorbate system, and
n is a constant which characterizes the pore heterogeneities.
As initially proposed by Dubinin and Radushkevich (1947) [
35],
n equals 2 for carbonaceous solids with low degree of burn-off (Dubinin Radushkevich model, DR). According to later works by Dubinin and Astakhov (1971), it ranges between 1.2 and 1.8 for carbons with high burn-off (DA model). For solids having narrow micropore size distributions, such as carbon molecular sieves or zeolites, the parameter
n may be found to lie between 3 and 6. The predictive ability of the DR and DA equations was illustrated in the original works of Dubinin and his co-workers for a variety of fluids below their critical temperature [
34]. In comparison with other classical isotherm models applicable to microporous materials, such as Langmuir, Langmuir–Freundlich, Toth, etc., these models have the advantage of being directly derived from physically meaningful data, both representative of the adsorbent microporosity and of adsorbate properties.
In Equation (7), the differential molar work of adsorption can be expressed as [
34]:
where
Ps (Pa) is the saturated vapor pressure of the adsorbate,
f and
fs are the adsorbate fugacities at temperature
T. Furthermore, assuming that for any adsorbate, at a same fraction of the micropore filling volume the ratio
A over
ԑ is constant, a similarity coefficient is defined according to a reference adsorbate [
36]:
where
are, respectively the characteristic energy and adsorption potential of the reference adsorbate. For activated carbons, the reference adsorbate chosen is benzene. A variety of correlations were proposed to estimate
β, the affinity coefficient. According to Dubinin [
36], this coefficient can be expressed as the ratio of the parachors of the adsorbate and benzene molecules:
(cm
3·g
1/4·s
−1/2·mol
−1):
For hydrogen
β equals 0.165.
is a reverse function of the micropore half-width
x (nm) and can be determined according to Equation (11) [
37], obtained empirically for carbonaceous microporous solids.
The characteristic free energy can then be calculated as follows:
Knowing the fraction of micropore volume
occupied by the adsorbed phase, the molar adsorption capacity
(mol·kg
−1) is given as a function of the density of the adsorbed phase
(kg·m
−3).
where
M (kg·mol
−1) is the molar mass of the adsorbate.
At boiling temperature (
Tb) or below,
can be assumed to be equal to the density of the bulk liquid. For the range of temperatures from the boiling point to the critical temperature
Tcr (K), the density of the adsorbed phase is determined as a function of the thermal coefficient of limiting adsorption
α (K
−1) and can be calculated according to the Dubinin–Nikolaev equation [
34]:
where
(mol·kg
−1) is the density of the liquid at boiling temperature
(K). The thermal coefficient
α being a constant, it can be derived from the ratio of the density of the bulk liquid at boiling temperature
to the density of the fluid at critical temperature
(m
3·kg
−1), according to Equation (15) [
34]:
Assuming that the density of the adsorbate at the critical temperature corresponds to maximal compression,
ρcr is derived from the constant
b in the van der Waals equation of state and is expressed as [
34]:
The constant
b (L·mol
−1) is then calculated by the familiar formula:
and for hydrogen it equals 0.026 L·mol
−1. Above the critical temperature, the density of the adsorbed phase may be considered as not dependent upon temperature and equals
(77.3 kg·m
−3) as given by Equation (16).
In order to compute the adsorption potential in Equation (8), it is necessary to derive the fugacities
f and
fs of the adsorbate. For the adsorbate in the gaseous phase, its fugacity at temperature
T and pressure
P is simply given by:
where
z is the compressibility factor of the gaseous adsorbate.
The computation of the adsorbate fugacity at saturation
fs differs whether the temperature
T is below or above the critical temperature. Under the critical temperature, the fugacity at saturation
fs is derived from Equation (18), replacing
P with
Ps (T). The saturated vapor pressure
Ps is then estimated from Antoine equation:
where coefficients
K and
N are determined according to the critical pressure and temperature of the adsorbate, and from its normal boiling point at 1 atm. As illustrated by numerous works [
5,
14,
16,
19,
24,
25,
26,
27], the classical DA equation can be satisfactorily employed to describe the adsorption process of gases onto microporous adsorbents under supercritical conditions, provided that parametric adjustment is performed for proper fitting of the isotherm curves.
When T does not significantly differ from Tcr, the linearity of adsorption isosthers during the transition from sub-critical to super-critical conditions suggests that this model is still valid outside the super-critical temperature domain. But for temperature largely above the critical point, it is not only the assessment of fs that becomes difficult, but also the thermal invariance of both the characteristic energy ԑ and the heterogenetity parameter n, as originally postulated by Dubinin, can no longer be considered.
M.A. Richard et al. [
38], therefore, proposed an empirical modification of the DA equation to describe the adsorption of supercritical gases in large temperature intervals. Assuming a linear temperature dependence of the characteristic energy of adsorption
ԑ expressed as the sum of two contributions, the enthalpic factor
a (J·mol
−1) and the entropic factor
B (J·mol
−1·K
−1), the equation proposed takes the form:
where
(mol·kg
−1) is the absolute molar adsorption capacity in equilibrium with the gas phase, and
(mol·kg
−1) is the maximal molar amount adsorbed at saturation of the micropore volume. According to Equation (20), the density of the adsorbed phase is then assumed constant along the micropore volume filling until saturation.
With these assumptions, this model was employed in several works [
24,
25,
26,
39,
40] to describe adsorption–desorption capacities in hydrogen storage reservoirs with quasi-perfect fitting of hydrogen adsorption isotherms between 77 and 298. K. Ramirez-Vidal et al. [
41] derived experimental correlations between the isotherm parameters and textural properties of activated carbons for hydrogen adsorption: the accessible micropore volume
and the limiting adsorption capacity
where thus found to be linearly correlated with the BET surface area, while “
a” and “
B” coefficients were related to the average micropore size and total pore volume, respectively. The saturation pressure
Ps, considered as a fitting constant parameter to compute the adsorption potential
A was found to be lower for activated carbons with smaller micropores [
42].
In our study, we still consider the linear temperature dependence of the characteristic energy as proposed by Richard et al. [
38], but we further propose to account for the saturation fugacity
fs instead of the saturation pressure as derived from the NIST standard reference database [
33]. Moreover, in accordance with the fundamental of the potential theory, we also propose to account for the variations of the adsorbed phase volume as a function of the number of moles adsorbed and the density of the adsorbate:
may be computed from Equation (14) assuming the state of the adsorbed phase is close to a liquid, or from Equation (16), considering it closer to the critical state. By combining Equations (7), (8), and (13) and expanding the term
A, the revisited Dubinin–Astakhov (rev-D-A) equation is obtained, which takes into account the temperature dependence of the molar adsorption capacity:
The characteristic free energy was estimated from the micropore width
x (nm) according to Equation (12) whilst the term
B (J·mol
−1·K
−1), was considered as a best fit parameter:
2.1. Hydrogen Density in the Adsorbed Phase
In order to determine which assumption to retain to describe the hydrogen density in the adsorbed phase, we preliminary reviewed some data from the literature.
Numerous works attempted to determine the density of adsorbed hydrogen at saturation under different (
P,
T) conditions. The density data were derived either from molecular simulation studies or from in situ measurements by small-angle neutron scattering [
43,
44,
45,
46].
Figure 1 compares hydrogen adsorbed densities determined by various authors under saturation conditions at different temperatures onto different adsorbents, including activated carbons and MOFs, with the density of the liquid computed by Equation 14 and Equation 15 for different values of α. From data from the literature, significant variations of the adsorbed phase density are so reported, spreading in the range from 8 up to 72 kg·m
−3, below the critical density. These variations do not appear solely explained by temperature effect, so that the determination of the density of the adsorbed phase at saturation conditions appears uncertain whatever the temperature domain considered.
Given such deviations, we tested one or the other assumption in the simulation of an experimental process [
14], considering either the adsorbed phase density as a liquid, dependent on temperature and on α in the range 5 × 10
−4 up to 5 × 10
−3, or as a constant equal to the critical density.
Figure 2 shows that pressure and temperature profiles do not significantly differ by varying the adsorbed phase density between the lower and upper limits. Note that reference [
19], in line with our observation, also pointed out the low sensitivity of the hydrogen adsorbed phase density parameter on simulation data derived from a 2-D model. A better fit was obtained assuming the critical state of the adsorbed phase density, so that this assumption was retained in the coming parts of the study.
2.2. Hydrogen in the Gas Phase
Accounting for the non-ideal gas behavior through the computation of its compressibility factor, the molar amount of H
2 in the gas phase (n
g) was derived according to Equation (24):
Figure 3 summarizes the deviations in the compressibility factor
z computed from both the van der Waals (VDW) and from the NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP-version 10.0). At low pressures, deviations in the compressibility factor
z either computed from the two methods are quite small, less than 6% at 10 MPa, but become significant at higher pressure and low temperature, reaching 26% at 20 MPa, 80 K. The NIST-REFPROP database was thus retained for the determination of the compressibility factor
z [
33]. In Equation 24,
Vg is the volume of hydrogen in the bulk phase and is determined as follows:
where
Vtank (m
3) is the internal volume of tank,
Vs (m
3) is the volume occupied by the solid adsorbent, which can be calculated as the ratio between the mass of the adsorbent and the skeletal density of the adsorbent (
ms/ρs).
3. Mass Balance
The mass balances are derived in order to account for the temporal variations in the molar quantities of hydrogen in both the adsorbed phase (na) and in the gas phase (ng), with pressure (P) and temperature (T), when the reservoir is submitted to the different steps of cooling, pressurization, gas charging, discharging, and heating. The mass balance equations are formulated according to the following assumptions:
Pressure, temperature, and phase composition inside the tank are assumed to be uniform in the entire volume of the system (0-D model);
At each time, equilibrium between the adsorbed and gas phases is assumed to be established (no mass transfer resistance is considered);
The hydrogen adsorption equilibrium is described according to the temperature dependent rev-D-A model (Equation (22));
The normal hydrogen thermodynamic data (compressibility factor z, fugacity at saturation fs, enthalpiy h) were derived from the NIST REFPROP database;
The isosteric adsorption enthalpies Qa were estimated using the Clausius–Clapeyron equation applied to the rev D-A isotherms.
The accumulation of hydrogen in the storage tank in both the gas and adsorbed phases results from the net hydrogen flow rate at the boundaries of the tank. The mass conservation equation is expressed as:
where
ntot (mol) is the total mass of hydrogen in the tank and
(mol·s
−1) and
(mol·s
−1) are the inlet and outlet molar flow rate, respectively. The amount of hydrogen in the gas phase refers to the hydrogen molecules that are not adsorbed, existing instead in the gaseous or supercritical states.
Equation (22) is derived with respect to time to obtain the differential equation that expresses the variations in the amount of adsorbed H
2 in the system:
where
mads (kg) is the amount of adsorbent contained in the tank. The partial derivatives of
na with respect to pressure, temperature, and fugacity are obtained by differentiating Equation (22) and are, respectively:
Finally, all the partial derivatives are replaced in Equation (27):
By differentiating Equation (24) with respect to time and solving for
, the differential equation for the H
2 evolution in the gas phase is obtained:
The variation with time of the H
2 gas phase is obtained by differentiating Equation (25) with respect to time:
By replacing the differential equation of the amount of hydrogen both in the gas and adsorbed phase, Equation (31) and Equation (32), respectively, in Equation (26) and rearranging for
, the differemtial equation for the pressure in the system is obtained:
The values of
,
are approximated by numerical differentiation over every step size:
The NIST REFPROP database makes it possible to compute at each step the gas compressibility factor zi and the fugacity at saturation fsi as functions of temperature Ti and pressure Pi.
4. Energy Balance
The total internal energy of the system
U (J), which includes the storage tank, the adsorbent material, the amount of hydrogen contained in both the adsorbed and gas phases, is expressed at temperature
T as:
where
mads (kg) and
mw (kg) are, respectively, the mass of the adsorbent and of the tank, and their corresponding specific heat capacities are
Cs (J·kg
−1K
−1) and
Cw (J·kg
−1·K
−1),
ma (kg) and
mg (kg) are, respectively the mass of hydrogen in the adsorbed and gas phases, their specific heats are
Cv,a (J·kg
−1·K
−1) and
Cv,g (J·kg
−1·K
−1), obtained with the reference fluid thermodynamic and transport properties database (REFPROP) software version 10.0 [
33]. For the calculation of
Cv,g the conditions of pressure and temperature of the system were applied.
Cv,a, was computed at the same temperature accounting for the adsorbed phase density (
ρcr).
With the changes in both kinetic and potential energies ignored, the rate of change in the internal energy of the system is derived from the difference in the enthalpy of the fluid streams at the inlet and the outlet and accounts for the heat fluxes either generated or absorbed by the internal sources during the adsorption and desorption steps and exchanged throughout the tank walls with the surroundings. It can be expressed as:
These terms are calculated with the following equations:
where
(kg·s
−1) and
(kg·s
−1) are the inlet and outlet mass flowrates of hydrogen. The specific enthalpy,
h, (J·kg
−1) is derived from the REFPROP software version 10.0, for
hin, and
hout, respectively, the temperature of inlet H
2 flow and the temperature of the tank are used. Both internal and external heat soures are accounted for throughout the variable
:
(J·s
−1) is the rate of heat transferred between the tank and the environment, which is calculated using a global heat transfer coefficient,
Hc (W·m
−2·K
−1):
S (m
2) is the wall surface area of the tank,
Ta (K) is the temperature of the surroundings and
T (K) is the temperature of the system.
(J·s
−1) is the rate of adsorption heat, which represents the flux of heat released when adsorption takes place or the flux of heat adsorbed when desorption occurs. To determine the isosteric heat of adsorption,
Qa (J·mol
−1), the Clausius–Clapeyron equation (Equation (43)) is applied to the isotherm model (Equation (22)).
Its temporal variation can be expressed as:
The application of the Clausius–Clapeyron equation to the rev-D-A isotherm of adsorption considers the ideal gas behavior. A study conducted by A. F. Kloutse et al. [
47] investigated the hydrogen isosteric heats on five representative metal–organic frameworks using both experimental methods and the model’s predictions with the Clausius–Clapeyron method applied to three different adsorption isotherms (M-D-A, Unilan, and Toth). The results demonstrated good agreement between the model’s predictions and the experimental method.
Accounting for the terms described above, the energy balance as given by Equation (38) takes the form of the following differential equation:
Developing and rearranging Equation (45), the temperature derivative of the system is given as:
The system consists of four ordinary differential equations (ODEs) that describe the behavior of four unknown variables over time: hydrogen adsorbed (na, Equation (27)), hydrogen in the gas phase (ng, Equation (32)), pressure (P, Equation (34)), and temperature (T, Equation (46)). To solve the system, the fourth order Runge–Kutta (RK4) method is employed, setting initial values to each variable. The mathematical model is implemented using an in-house MATLAB code, which incorporates a link to the REFPROP software version 10.0 for computing hydrogen properties. Throughout the simulation, a time step of 0.1 s was selected based on a balance between result accuracy and computational time. Although smaller step sizes are possible, they do not significantly enhance result accuracy and considerably prolong the computation time.
6. Energy Analysis of the H2 Storage System
The process energy requirements highly rely on the amount of heat exchanged between the reservoir and the cryogenic bath. To the best of our knowledge, the heat exchanges during hydrogen loading in the cryogenic process were only determined in the work of Richard et al. [
14]. Two computation methods can be considered to estimate the cooling requirements of the cryogenic process during hydrogen loading: Method 1, by determining the amounts of heat absorbed by the hydrogen feeding flow to be cooled at cryogenic temperature, and that absorbed by the adsorbent bed to compensate the heat released by the exothermal adsorption process, or Method 2, by determining the total amount of heat exchanges throughout the reservoir walls between the adsorbent bed and the cryogenic bath.
According to [
14], the net amount of heat evacuated at the reservoir walls was experimentally reported to be 139 kJ ± 35 KJ (0.99 ± 0.25 KWh·kg
−1·H
2), which supports the results obtained by this work, according to both methods, Method 1: 157 kJ (1.13 KWh·kg
−1·H
2) and Method 2: 180 kJ (1.29 KWh·kg
−1·H
2), as shown in
Figure 13. According to the data obtained, the heat of adsorption contributes close to one third of the reservoir cooling duty.
To compare the adsorbents for H
2 storage, we applied the simulation conditions of the experience of Richard et al. [
14] evaluated at 20 bar and 50 bar, in terms of amount of H
2 stored and energy consumption. The materials evaluated include activated carbons: AX-21 [
14], MSC-30, and MSP-20X [
42] and metal–organic frameworks: Cu-BTC, MOF-5, and MOF-177 [
44]. The process cooling duty allows to determine the amount of liquid nitrogen (L-N
2) to keep the system at cryogenic conditions, which was derived from its heat of vaporization, that is 5.632 kJ·mol
−1 [
57]. The results are presented in
Table 6 and
Table 7, as well as in
Figure 14. The adsorbent showing the best performance for maximizing H
2 storage with lowest L-N
2 requirement is the AC AX-21.
The energy consumption of the process operating hydrogen loading should include the mechanical energy required to feed the tank (H2 compression) in addition to the heat transferred for cooling the hydrogen inlet gas flow and the heat exchanged to compensate the H2 adsorption exothermicity (internal source). Assuming a cyclic process, the heat required for initial cooling of the tank from ambient to cryogenic temperature was not considered.
Usually, gas compression is carried out in several stages, because this reduces the required energy. However no significant energy reduction is obtained beyond three stages [
58]. Thus, the mechanical energy required to compress the hydrogen flow feeding the process was estimated assuming a 3-stage set of compressors, with an isothermal efficiency of 70% [
14]. The power consumption for the adiabatic compression can be computed from [
58]:
where
is the inlet flow (mol·s
−1),
k is the H
2 heat capacity ratio (1.31), ƞ is the isothermal efficiency of the compressor,
P2 is the final pressure and
P1 is the initial pressure. As presented in
Figure 15, the compression energy can represent around 10% of the total energy consumption.
The specific total energy requirement of the process operating hydrogen storage in the adsorbed state was further compared with the ones estimated for both the compressed and liquefaction processes. The mechanical energy consumed to store hydrogen by compression in a system operating at elevated pressures of 300 and 700 bar was derived using Equation (48). The energy consumption for liquid nitrogen production is based on an industrial Collins-based process, 0.474 KWh·kg
−1 [
59], which is more than twice the ideal consumption, 0.21 KWh·kg
−1 [
60]. When comparing the energy consumption with another compound of similar boiling temperature (77 K for N
2), such as methane, whose boiling temperature is 111 K, the energy consumption is lower: 0.29 KWh·kg
−1 [
61]; thus, the data agree with thermodynamic laws.
Figure 15 presents the results obtained, disclosed by fractions of energy for hydrogen inlet flow compression, and cryogenic cooling from 298 K to 80 K, including compensation of heat losses due to adsorption. The energy required to store hydrogen in an adsorbent-based system is around 13 KWh.kgH
2−1, which is close to a H
2-liquefaction process, that has been reported to be between 7 and 13 KWh.kgLH
2−1 [
62,
63,
64,
65,
66]. It also represents around 32% of the hydrogen high heating value (HHV). The lowest energy duties were computed for storage by compression, with values of 3.52 KWh·kgH
2−1 and 4.2 KWh·kgH
2−1 at 300 bar and 700 bar, respectively, which agree well with data from the literature [
64,
65,
67,
68].
Hydrogen storage by adsorption under cryogenic conditions so appears energetically not better efficient than other modes of storage, by compression at elevated pressures or by liquefaction. But the application potential of that technology should also consider other factors, such as equipment compactness, capital, operation and maintenance costs, and safety concerns, which were not covered in this study.