Simulation of Hydrogen Drying via Adsorption in Offshore Hydrogen Production

Abstract

According to the international standard ISO 14687:2019 for hydrogen fuel quality, the maximum allowable concentration of water in hydrogen for use in refueling stations and storage systems must not exceed 5 µmol/mol. Therefore, an adsorption purification process following the electrolyzer is necessary. This study numerically investigates the adsorption of water and the corresponding water loading on zeolite 13X BFK, based on the mass flows entering the adsorption column from three 5 MW electrolyzers coupled to a 15 MW offshore wind turbine. As the mass flow is influenced by wind speed, a direct comparison between realistic wind speeds and adsorption loading is presented. The presented numerical discretization of the model also accounts for perturbations in wind speed and, consequently, mass flows. In addition, adsorption isobars were measured for water on zeolite 13X BFK within the required pressure and temperature range. The measured data was utilized to fit parameters to the Langmuir–Freundlich isotherm.

1. Introduction

One possible solution for the transition from fossil fuels towards green energy in Germany is to build offshore hydrogen production plants, leveraging the abundant resources of seawater and wind energy required for generating significant quantities of hydrogen through electrolysis. There are various methods to produce hydrogen, including water electrolysis, hydrogenase route, and extraction from biomass like methanol and formic acid [1]. However, water electrolysis represents the most effective zero-emission technology for hydrogen production when coupled with renewable energy [2,3]. Powered by renewable energy, electrolysis can be used to split water molecules into hydrogen and oxygen. The produced hydrogen can then be stored for later use, decoupling energy production from consumption and providing a means to store surplus energy during periods of high generation. The H2Mare project aims to address this by coupling three 5 MW electrolyzers with a 15 MW wind turbine on multiple offshore platforms located in the German North Sea [4,5]. Seawater from the North Sea will be desalinated and purified before being fed into the electrolyzers, where it will be separated into hydrogen and oxygen. For the subsequent use of hydrogen, certain purity standards must be achieved. According to the international standard ISO 14687:2019 for hydrogen fuel quality, the maximum concentration of water in hydrogen for use in refueling stations and storage systems is 5 µmol/mol [6]. This is not a requirement for the fuel cell specifically, as it requires the hydrogen to be moistened beforehand anyway, but rather for the operating components. If, for example, the ambient temperature is subzero, the water inside the hydrogen tanks might form ice crystals around the valve openings, hindering the operation of the fueling station [7]. A condensation process cannot extract water to a level of 5 µmol/mol, and further drying through adsorption is required.

The hydrogen to be purified first flows through an adsorber column filled with zeolite adsorbent from bottom to top. While water is adsorbed by the adsorbent, hydrogen exits at the top at a higher purity level. At the same time, another column is being regenerated. Regeneration can be achieved via Temperature Swing Adsorption (TSA) or Pressure Swing Adsorption (PSA). Regeneration is favored at low pressures, using a PSA, and high temperatures, using a TSA. In a TSA, the adsorbent is dried at high temperatures, typically ranging from 150 °C to 200 °C and higher [8]. This can be accomplished either through direct or indirect heating. In a direct heating process, regeneration gas (e.g., hydrogen) is heated, which subsequently transfers the heat to the adsorbent [9]. Indirect heating involves the use of heating elements installed within the fixed-bed reactor. In a PSA, pressure is decreased for regeneration. An extension of PSA is the Vacuum PSA, where the pressure is reduced to a level below atmospheric pressure, which leads to better regeneration of the adsorbent. A PSA typically has shorter operating times compared to a TSA, since regeneration with pressure changes can be achieved more rapidly than heating the adsorbent and subsequently cooling it. Both regeneration methods are already employed for hydrogen drying. For example, Silica Verfahrenstechnik distributes a direct-heating TSA purification system [9], while the research center EPFL demonstrated the feasibility of a VPSA [7].

For a constant mass flow entering the adsorber, simplified scaling methods, such as the Rosen approximation, are available [8]. This approximation allows engineers to calculate the relation between mass flows, column height, and loading time, without the need to solve balance equations. Most adsorption processes, such as carbon capture or natural gas purification, are foresighted processes where mass flows are typically steady [10,11,12,13]. The challenge with offshore hydrogen production powered by wind and its subsequent purification lies in the need to adapt the design and resilience of the adsorption plant to variations in wind speeds and, consequently, mass flows. Since mass flows are variable due to changing wind conditions, the loading time cannot be fixed.

The numerical model presented in this research captures variation in wind speeds and, consequently, fluctuations in mass flow. The balance equations within the model are discretized explicitly, allowing for fast runtime results. The mass flows entering the adsorption column are taken from a multi-physics system model for offshore hydrogen production [5]. In Section 2.1, a possible offshore hydrogen production process and the subsequent purification process are presented. In Section 2.2, the adsorption isobars from experimental water adsorption measurements on zeolite 13X BFK are presented, along with a fit to the Langmuir–Freundlich isotherm and a comparison to fitting parameters from another research study. Additionally, the influence of hydrogen co-adsorption is analyzed using the Ideal Adsorption Solution Theory (IAST). Section 3 discusses the balance equations and their numerical discretization. The model is used in Section 4 to examine how different wind speeds, having means of 6, 10, and 16 m/s, affect mass loading and temperature behavior within an adsorption column. The flowchart in Figure 1 gives an overview of the steps performed in this paper.

2. Materials and Methods

2.1. Purification Process

The scheme for the planned offshore hydrogen production process is displayed in Figure 2 [5]. Hydrogen is stored in three buffer tanks at a pressure of 30 bar, which we assume to be fixed throughout the process. In the deoxygenation (DEOXO) process, the residual oxygen contained in the incoming hydrogen reacts catalytically to water in a reactor filled with a platinum or palladium catalyst. The moist hydrogen is then cooled to 20 °C in a condensation process, where the condensed water is separated. As seawater temperature typically ranges from 5 to 20 °C, lowering the temperature of the hydrogen stream to 20 °C using seawater in a heat exchanger is achievable. A condensation process cannot extract water to a level of 5 µmol/mol. Therefore, further drying through adsorption is required.

In this study, the zeolite “Köstrolith 13X BFK” was investigated for water adsorption. This particular zeolite was chosen because its physical properties, such as tortuosity, macropore diameter, thermal conductivity, and specific heat capacity, have been characterized experimentally, for example by Mette et al. [12]. Furthermore, zeolite is a typical adsorption material used for hydrogen drying [9]. It is supplied as beads that measure 1.6–2.5 mm in diameter and is manufactured by Chemiewerk Bad Köstritz (Bad Köstritz, Germany). BFK is the German abbreviation for binder-free spherical granulate.

In order to calculate the molar fraction of water left after condensation, we utilize the empirical expression

$$p_{H_{2}O}=6.1078·{10}^{-3}exp\left(17.2694·\left({\displaystyle \frac{T-273.15}{T-34.85}}\right)\right)\left[\mathrm{atm}\right]$$

(1)

as used in [15] to calculate the partial water pressure on the cathode side of a PEM electrolyzer. In the condensation process, at a temperature of 20 °C, we obtain a partial pressure of approximately $p_{H_{2}O}\approx 0.023\mathrm{atm}\approx 2330\mathrm{Pa}$ using Equation (1). With the assumption of ideal gas mixtures, we obtain the molar fraction c of water in hydrogen $c={\displaystyle \frac{2330\mathrm{Pa}}{30·{10}^5\mathrm{Pa}}}=777\mathrm{ppm}$.

The specialty in the case of hydrogen drying following electrolysis lies in the fluctuation of mass flows entering the adsorption column, as hydrogen mass flow is coupled to wind velocities. Therefore, different mass flows result in non-uniform loading and regeneration times, making the scaling of a plant with conventional methods more difficult.

2.2. Adsorption Isotherms and IAST

2.2.1. Experimental Water Adsorption Measurements

The adsorption equilibrium of zeolite 13X BFK was investigated using a gravimetric sorption analyzer as described in [16]. Pure water adsorption isobars in the temperature range of 25–145 °C at water vapor pressures of 1223 and 2339 Pa were measured, corresponding to dew points of 10 and 20 °C, respectively. The measured adsorption isobars are displayed in Figure 3. The measured point values are displayed in Appendix A.

Using the modified potential theory of adsorption [17], the characteristic curve data for water adsorption on zeolite 13X BFK were calculated (see Figure 4). These data were then fitted using the methodology also described in [17]. The best fit was achieved using the Dubinin–Astakhov (DA) equation. Finally, the characteristic curve was used to calculate the complete adsorption isotherms at 25 °C, 115 °C, and 145 °C, as shown in Figure 5.

A Langmuir–Freundlich isotherm was fitted to obtain the adsorption isotherm parameters via the curvefit function from the Python library SciPy. The Langmuir–Freundlich isotherm has the form

$$X_{s,eq}(T,p)=X_{sat}{\displaystyle \frac{b\left(T\right)p^n}{1+b\left(T\right)p^n}}$$

(2)

with the temperature-dependent coefficient $b\left(T\right)$

$$b\left(T\right)=B_{0}exp\left({\displaystyle \frac{B_1}{RT}}\right)$$

(3)

and

$$n=n_1+{\displaystyle \frac{n_2}{T}}.$$

(4)

The optimized parameters were $X_{sat}$, $B_0$, $B_1$, $n_1$, and $n_2$. Gaeini et al. used the Langmuir–Freundlich isotherms to estimate the equilibrium water loading in the adsorbed phase on zeolite 13X BFK [18]. The experimental data used in Gaeini’s study, provided by the manufacturer CWK Chemiewerk Bad Köstritz GmbH, covered a pressure range of 0 to 2500 Pa at temperatures of 25, 80, and 95 °C. The fitted parameters from Gaeini are displayed in

Table 1. Fitted Langmuir–Freundlich isotherm parameters from Gaeini et al. [18] (Value-Gaeini) and from experimental data and data points generated by the modified potential theory (Value-Experimental).

Parameter	Unit	Value-Gaeini	Value-Experimental
$X_{sat}$	$\mathrm{kg}{\mathrm{kg}}^{-1}$	$0.32403$	$0.3284$
$q_{sat}$	$mol{\mathrm{kg}}^{-1}$	18	$18.24$
$B_0$	${10}^{-4}{\mathrm{bar}}^{-1}$	$0.000308$	$4.37·{10}^{-6}$
$B_1$	$\mathrm{kJ}\mathrm{mol}$	18,016	27,482.81
$n_1$	-	$-0.3615$	$0.3896$
$n_2$	-	$274.23$	$53.76$

in the column ‘Value-Gaeini’, which were used as starting parameters for our curve fitting. The column ‘Value-Experimental’ shows the isotherm fitting parameters based on our measurement data.

Figure 5 shows a comparison of Gaeini’s fitted isotherm parameters and our fitted isotherm parameters, both using the Langmuir–Freundlich isotherms. For the 25 °C isotherm, the curves show similar results. For the 115 °C and 145 °C isotherms, however, there is only one common intersection point. Using Gaeini’s fitting parameters, we see that the gradient of the isotherm is less steep than for the parameters fitted on our experimental data. Thus, at vapor pressure above approximately 1000 Pa, Gaeini underestimates our experimental data. One possible reason for this is that Gaeini uses considerable adsorption data at low partial pressures, which dominate the curve fit. As a result, even in Gaeini’s work, the isothermal fit at higher partial pressures deviates from the measurement data in a similar way. In our simulation presented in Section 4, we assume conditions of 20 °C and a partial pressure of water of 2330 Pa. Due to the similarity of the isotherms at 25 °C above 1110 Pa (Figure 5a), both Gaeini’s and our fitted isotherm parameters are applicable under these conditions. From the adsorption mechanism perspective, we obtain a heterogeneous adsorption, as indicated by $n<1$ at the lowest investigated temperature at $T=20°C$. Moreover, with increasing temperature, the coefficient n further decreases (see Equation (4)).

2.2.2. Ideal Adsorption Solution Theory (IAST)

In this section, we estimate the influence of hydrogen co-adsorption. The presence of oxygen and nitrogen were neglected. Oxygen is removed during the DEOXO process and is therefore assumed to be negligible. Nitrogen is only present during the setup of the apparatus and is displaced by the process gas after a certain period of operation. A well-known approach for predicting the adsorption of gas mixtures is the Ideal Adsorption Solution Theory (IAST), which was chosen in this analysis. There exist different other multi-component adsorption models, such as Real Adsorption Solution Theory (RAST) [19], Predictive Real Adsorption Solution Theory (PRAST) [20], and Vacancy Solution Theory (VAST) [21]. For most of these methods experimental data on multi-component adsorption equilibria or grand-canonical Monte-Carlo simulations are needed. Direct measurement of mixture adsorption equilibria as well as molecular simulations remain complicated and time consuming. Due to the aforementioned limitations, IAST was chosen. IAST is a predictive model that uses only pure isotherms of the individual components to calculate a multi-component isotherm. IAST assumes that the mixture of adsorbed gases behaves like an ideal solution and that the gas mixture is in equilibrium with the adsorbed phase. The fitting parameters of the Sips isotherm for $H_2$ adsorption were taken from [14] and are displayed in

Table 2. Fitted Sips isotherm parameters for $H_2$ adsorption from Streb et al. [14].

Parameter	Unit	Value
$q_{sat}$	$mol\mathrm{kg}$	$5.013$
$B_0$	${10}^{-4}{\mathrm{bar}}^{-1}$	$1.034$
$B_1$	$\mathrm{kJ}\mathrm{mol}$	$9.453$
s	-	$1.006$

. The Sips isotherm has the form

$$q\left(p\right)=q_{sat}{\displaystyle \frac{{\left(b\left(T\right)p\right)}^s}{1+{\left(b\left(T\right)p\right)}^s}}$$

(5)

with temperature-dependent coefficient

$$b\left(T\right)=B_{0}exp\left({\displaystyle \frac{B_1}{RT}}\right).$$

(6)

A central quantity in IAST is the reduced grand potential $\psi \left(p_i^{*}\right)$ of sorption pressure $p_i^{*}$, which is

$$\psi \left(p_i^{*}\right)={\displaystyle \frac{\pi A}{RT}}={\int }_0^{p_i^{*}}{\displaystyle \frac{q_i^0\left(p_i\right)}{p_i}}d{p}_i,$$

(7)

where $\pi$ is the spreading pressure, A is the specific surface area of adsorbent in $m^2$, and $q_i^0\left(p_i\right)$ is the equilibrium adsorbed amount of single-component i. The reduced grand potential can be calculated for each specific isotherm model by solving the integral in Equation (7). The resulting equation can then be rearranged to solve for sorption pressure $p_i^{*}$. The resolved integral for various isotherm models to calculate the reduced grand potential and sorption pressures can, for example, be found in [22]. In IAST, the spreading pressure (and also the reduced grand potential) has to be equal for all components for a fixed temperature. The basic equations for IAST are then

$$\begin{array}{cc}\hfill y_{i}p=p_i& =x_i{p}_i^{*}\left(\psi \right),i=1,2,\dots ,N\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \sum _{i=1}^N{x}_i& =1,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \psi ={\psi }_1& ={\psi }_2=\dots ={\psi }_N,\hfill \end{array}$$

(10)

where N denotes the number of single components present in the gas mixture, p is the total pressure, $x_i$ is the mole fraction in the adsorbed phase, and $y_i$ is the mole fraction in the gas phase for component i. Equation (8) is analogous to Raoult’s law. The total amount adsorbed $q_T$ can be obtained from the Gibbs adsorption isotherm and is given by

$${\displaystyle \frac{1}{q_T}}=\sum _{i=1}^N{\displaystyle \frac{x_i}{q\left(p_i^{*}\right)}},$$

(11)

where $q\left(p_i^{*}\right)$ is the adsorbed amount of pure component i at the sorption pressure $p_i^{*}$. For more details on IAST and a detailed derivation of the formulas, the reader is referred to [22,23,24].

The sorption pressures for the Langmuir–Freundlich and the Sips isotherms, respectively, are given as in [22]:

$$p_{LF}^{*}\left(\psi \right)={\left({\displaystyle \frac{exp(n\psi /q_{sat})-1}{b\left(T\right)}}\right)}^{1/n}$$

(12)

$$p_{Sips}^{*}\left(\psi \right)={\displaystyle \frac{1}{b\left(T\right)}}{\left(exp\left({\displaystyle \frac{\psi }{(1/s)q_{sat}}}\right)-1\right)}^{1/s}.$$

(13)

Using Equations (12) and (13) for sorption pressures, parameters from Table 1 (column ‘Value-Experimental’) and Table 2, and considering Equations (8)–(10), $\psi$ is calculated as the root of Equation (14) with

$${\displaystyle \frac{y_{1}p}{p_{LF}^{*}\left(\psi \right)}}+{\displaystyle \frac{y_{2}p}{p_{Sips}^{*}\left(\psi \right)}}-1=0.$$

(14)

For Equation (14), the following variables, according to the gas drying scheme in Figure 2, are used: $p=30\mathrm{bar}$, $T=20°C$, and the molar fractions for water $y_{H_{2}O}=777\mathrm{ppm}$ and for hydrogen $y_{H_2}=1-y_{H_{2}O}$.

The reduced grand potential $\psi$ is inserted into Equations (12) and (13) to find sorption pressures $p_1^{*}=p_{LF}^{*}\left(\psi \right)$ and $p_2^{*}=p_{Sips}^{*}\left(\psi \right)$. Then, with the adsorbed mole fractions $x_{H_{2}O}={\displaystyle \frac{y_{H_{2}O}p}{p_1^{*}}}$ and $x_{H_2}={\displaystyle \frac{y_{H_2}p}{p_2^{*}}}$ (Equation (8)), the total adsorbed amount in $\mathrm{mol}{\mathrm{kg}}^{-1}$ is calculated using Equation (11):

$$q_T={\displaystyle \frac{1}{\frac{x_{H_{2}O}}{q^{LF}\left(p_1^{*}\right)}+\frac{x_{H_2}}{q^{Sips}\left(p_2^{*}\right)}}}.$$

(15)

From the total loading, the partial adsorbed amount can be calculated with $q_{H_{2}O}=x_{H_{2}O}{q}_T$ and $q_{H_2}=x_{H_2}{q}_T$. We can generate the IAST predictions for each component $q_{H_{2}O}$ and $q_{H_2}$ as a function of pressure, which are displayed alongside the pure component’s adsorption isotherm in Figure 6. Figure 6a shows that, for water adsorption, both graphs overlap for all pressures. The maximum deviation between the pure water isotherm and the water isotherm calculated using IAST was calculated to be on the order of ${10}^{-6}$$\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1}$. At the same time, Figure 6b shows that hydrogen uptake is suppressed, as the blue line (IAST hydrogen component) remains zero throughout the considered pressure range of 0 to 30 bar. As hydrogen co-adsorption predicted by IAST is considered negligible, it is omitted in the simulations presented in Section 4.

3. Model

3.1. Balance Equations

For the mathematical model, the following assumptions were made for the heat, mass, and momentum balance:

• The size, shape, and porosity of particles are assumed to be uniform;
• Axial dispersed plug flow;
• The mixture (hydrogen and water) is considered to be an ideal gas mixture;
• Gradients of temperature, mass, and velocity in the radial direction are neglected;
• The solid and gas phases are in thermal equilibrium;
• The Linear Driving Force (LDF) model represents the mass transfer from the gas phase to the adsorbent;
• Heat transfer to the ambient is neglected.

3.1.1. Mass Balance of the Solid Phase

The adsorption rate can be described by the Linear Driving Force (LDF) model [12]:

$${\displaystyle \frac{\partial X_s}{\partial t}}=k_{LDF}(X_{s,eq}(T,p_{H_{2}O})-X_s)$$

(16)

with the Glueckauf factor $k_{LDF}$

$$k_{LDF}={\displaystyle \frac{15{\delta }_{eff}}{{\left(0.5d\right)}^2}}.$$

(17)

In Equation (16), the gradient of the adsorbed amount of water ${\displaystyle \frac{\partial X_s}{\partial t}}$ is described by the difference between the actual loading $X_s$ in $\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1}$ and $X_{s,eq}(T,p_{H_{2}O})$, which is the equilibrium loading at temperature T and partial pressure $p_{H_{2}O}$ of water, multiplied by the mass transfer resistance $k_{LDF}$. The constant d is the average diameter of a particle. The effective diffusion coefficient ${\delta }_{eff}$ is calculated using Equations (18) and (19) [25]:

$${\delta }_{eff}={\displaystyle \frac{D_{ges}/\mu }{1+\alpha \left(X_s\right)}}$$

(18)

and the expression

$$\alpha \left(X_s\right)={\displaystyle \frac{RT{\rho }_s}{{ϵ}_p{M}_{H_{2}O}}}{\displaystyle \frac{\partial X_{s,eq}}{\partial p_{H_{2}O}}}.$$

(19)

Here, ${ϵ}_p$ is the inner porosity of zeolite particles, $\mu$ is the tortuosity factor, ${\rho }_s$ is the density of dry adsorbent, and $M_{H_{2}O}$ is the molar mass of water. ${\displaystyle \frac{\partial X_{s,eq}}{\partial p_{H_{2}O}}}$ is the gradient with respect to the partial water pressure $p_{H_{2}O}$ of the Langmuir–Freundlich isotherm, which is

$${\displaystyle \frac{\partial X_{s,eq}}{\partial p_{H_{2}O}}}=X_{sat}{\displaystyle \frac{b\left(T\right)n{p}_{H_{2}O}^{n-1}}{{(1+b\left(T\right)p_{H_{2}O}^n)}^2}}.$$

(20)

The total diffusion coefficient $D_{ges}$ is calculated as the sum of the Knudsen diffusion $D_{Kn}$ and the free gas diffusion $D_{H_2,H_{2}O}$ [26]:

$$D_{ges}={\left[{\displaystyle \frac{1}{D_{Kn}}}+{\displaystyle \frac{1}{D_{H_2,H_{2}O}}}\right]}^{-1}$$

(21)

with

$$D_{Kn}={\displaystyle \frac{4}{3}}{d}_{p,makro}\sqrt{{\displaystyle \frac{RT}{2\pi M_{H_{2}O}}}}$$

(22)

and the binary diffusion coefficient [27]

$$D_{H_2,H_{2}O}={\displaystyle \frac{{10}^{-4}·0.00143T^{1.75}{\left[\frac{1}{M_{H_2}}+\frac{1}{M_{H_{2}O}}\right]}^{1/2}}{\sqrt{2}p{\left[{\left({\nu }_{H_2}\right)}^{1/3}+{\left({\nu }_{H_{2}O}\right)}^{1/3}\right]}^2}},in[c{m}^2/s]$$

(23)

where p denotes the total pressure in bar and $M_{H_{2}O}$, $M_{H_2}$ molar masses in $\mathrm{g}{\mathrm{mol}}^{-1}$. $d_{p,makro}$ denotes the pore diameter of the particle. The dimensionless diffusion volumes of the corresponding gases are given as follows: ${\nu }_{H_2}=6.12$ and ${\nu }_{H_{2}O}=13.1$ [27].

The particle porosity ${ϵ}_p=0.6$, bed porosity $ϵ=0.4$, density of dry adsorbent ${\rho }_s=1150\mathrm{k}\mathrm{g}/{\mathrm{m}}^{-3}$, macro-pore diameter $d_{p,makro}=300\mathrm{n}\mathrm{m}$, and tortuosity factor $\mu =4$ are taken from experimental investigations on water vapor adsorption of zeolite 13X BFK [12]. For the particle diameter, we set $d=0.002\mathrm{m}$, which is the average of the bead size 1.6–$2.5\mathrm{m}\mathrm{m}$.

3.1.2. Mass Balance of the Gas Phase

The change in time of the water content $X_f$ in the gas phase is described by [12]:

$${\displaystyle \frac{\partial X_f}{\partial t}}=-{\displaystyle \frac{1}{ϵ}}{\displaystyle \frac{\partial \left(u_0{X}_f\right)}{\partial z}}+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{D_{ax}}{ϵ}}{\displaystyle \frac{\partial X_f}{\partial z}}\right)-{\displaystyle \frac{{\rho }_s}{{\rho }_f}}{\displaystyle \frac{(1-ϵ)}{ϵ}}{\displaystyle \frac{\partial X_s}{\partial t}}.$$

(24)

Here, $u_0$ denotes the bed velocity with $u_0=uϵ$, where u is the interstitial velocity (velocity of the fluid between the particles). The fluid density ${\rho }_f$ is assumed to be constant, as we consider a fixed molar fraction of water in hydrogen at the inlet. We calculate the axial dispersion coefficient $D_{ax}$ according to Tsotsas [28]:

$$D_{ax}={\delta }_{bed}+{\displaystyle \frac{P{e}_d}{2}}{D}_{H_2,H_{2}O}$$

(25)

with the Peclet number $P{e}_d$ and the diffusion coefficient ${\delta }_{bed}$ calculated as in [29]:

$$P{e}_d={\displaystyle \frac{u_{0}d}{D_{H_2,H_{2}O}}}$$

(26)

$${\delta }_{bed}=D_{H_2,H_{2}O}(1-\sqrt{1-ϵ}).$$

(27)

As the mass flow—and consequently the bed velocity $u_0$—varies in our use case, ${\displaystyle \frac{\partial }{\partial z}}{D}_{ax}$ is not constant, and the gradient must be calculated for each axial section.

3.1.3. Energy Balance for Gas and Solid Phases

Considering the assumption that the temperature is the same for the gas and solid phases, the energy balance is

$$\begin{array}{cc}\hfill & [ϵ{\rho }_f{c}_{p,f}+(1-ϵ){\rho }_s(c_{p,s}+X_s{c}_{p,ads}\left(T\right))]{\displaystyle \frac{\partial T}{\partial t}}\hfill \\ \hfill & =-{\rho }_f{c}_{p,f}{\displaystyle \frac{\partial \left(u_{0}T\right)}{\partial z}}-{\rho }_f{c}_{p,w}{\displaystyle \frac{\partial \left(u_{0}T{X}_f\right)}{\partial z}}+{\displaystyle \frac{\partial }{\partial z}}\left({\Lambda }_{ax}{\displaystyle \frac{\partial T}{\partial z}}\right)+{\rho }_s(1-ϵ)\underset{={\displaystyle \frac{\partial }{\partial t}}\left(\Delta h_{ads}{X}_s\right)}{\underbrace{{\displaystyle \frac{\partial X_s}{\partial t}}(\Delta h_{ads}+X_s{\displaystyle \frac{\partial \Delta h_{ads}}{\partial X_s}})}},\hfill \end{array}$$

(28)

where $c_{p,w}$, $c_{p,ads}\left(T\right)$, $c_{p,f}$, and $c_{p,s}$ denote the specific heat capacities of the water, the adsorbed water, the fluid, and the solid, respectively. For the specific heat coefficients, we set the following values: $c_{p,f}=14,369\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$, which is the approximate specific heat coefficient for hydrogen at 30 bar and 20 °C [30], and $c_{p,w}=4184\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$, which is the specific heat coefficient for water at 30 bar and 20 °C. From [12], we obtain the specific heat capacity of zeolite 13X BFK as $c_{p,s}=880\mathrm{J}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$. The temperature-dependent heat capacity of adsorbed water $c_{p,ads}\left(T\right)$ was interpolated from Vucelic’s measurements of adsorbed water on zeolite NaA [31].

The enthalpy of vaporization is given by the Watson Equation [27]:

$$\Delta h_v\left(T\right)=A_V{(1-T_r)}^{B_V+C_V{T}_r+D_V{T}_r^2+E_V{T}_r^3}.$$

(29)

The critical temperature of water is $T_c=647.1\mathrm{K}$ and the reduced temperature is calculated with $T_r=T/T_c$. The coefficients for the enthalpy of vaporization can be calculated from the enthalpy of the vaporization curve taken from [27], and we obtain $A_V=2872019$, $B_V=0.28184$, $C_V=-0.109110$, $D_V=0.147096$, $E_V=0.044874$. The effective axial thermal conductivity ${\Lambda }_{ax}$ is calculated as in [28]:

$${\Lambda }_{ax}={\lambda }_{bed}+{\displaystyle \frac{P{e}_{\lambda }}{2}}{\lambda }_f,$$

(30)

with the molecular Peclet number $P{e}_{\lambda }$ [28]:

$$P{e}_{\lambda }={\displaystyle \frac{u_0{\rho }_f{c}_{p,f}d}{{\lambda }_f}}.$$

(31)

The effective thermal conductivity of the bed ${\lambda }_{bed}$ can be obtained by [29]:

$${\lambda }_{bed}={\lambda }_f\left(1-\sqrt{1-ϵ}+\sqrt{1-ϵ}{k}_c\right)$$

(32)

with

$$k_c={\displaystyle \frac{2}{N}}\left({\displaystyle \frac{2}{N}}{\displaystyle \frac{k_p-1}{k_p}}ln{\displaystyle \frac{k_p}{B}}-{\displaystyle \frac{B+1}{2}}-{\displaystyle \frac{B-1}{N}}\right)$$

(33)

and

$$N=1-{\displaystyle \frac{B}{k_p}},k_p={\displaystyle \frac{{\lambda }_p}{{\lambda }_f}},B=1.25{\left({\displaystyle \frac{1-ϵ}{ϵ}}\right)}^{10/9}.$$

(34)

The thermal conductivity of zeolite 13X BFK, ${\lambda }_p=0.4\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, is taken from [12]. The thermal conductivity of the fluid is ${\lambda }_f=0.186\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, which corresponds to the thermal conductivity of hydrogen at 20 °C and 30 bar [30]. The adsorption enthalpy $\Delta h_{ads}$ is defined as the sum of the vaporization enthalpy $\Delta h_v$ and the binding enthalpy [32]:

$$\Delta h_{ads}(X_s,T)=\Delta h_v\left(T\right)+E{\left(ln{\displaystyle \frac{X_{sat}}{X_s}}\right)}^{1/n}+{\displaystyle \frac{E\beta T}{n}}{\left(ln{\displaystyle \frac{X_{sat}}{X_s}}\right)}^{-(n-1)/n}.$$

(35)

The Dubinin isotherm parameters $E=1192.3\mathrm{k}\mathrm{J}{\mathrm{kg}}^{-1}$ and $n=1.55$ are adopted from [12]. The partial derivative of the adsorption enthalpy is

$${\displaystyle \frac{\partial \Delta h_{ads}}{\partial X_s}}=-E{\displaystyle \frac{1}{n{X}_s}}{\left(ln{\displaystyle \frac{X_{sat}}{X_s}}\right)}^{1/n-1}+{\displaystyle \frac{(n-1)E\beta T}{n}}{\displaystyle \frac{1}{n{X}_s}}{\left(ln{\displaystyle \frac{X_{sat}}{X_s}}\right)}^{1/n-2}.$$

(36)

$\beta$ is the thermal expansion coefficient of water, which is $\beta =0.207·{10}^{-3}{\mathrm{K}}^{-1}$ at 20 °C. The energy balance in Equation (28) neglects ambient heat transfer. The transfer of heat to the environment can be considered by adding the term ${\displaystyle \frac{4k}{d_R}}(T-T_{amb})$, where $d_R$ denotes the radius of the reactor and k is a heat transfer coefficient that depends on the wall material used.

3.1.4. Momentum Equation

Due to fluctuations in wind speed, the mass flow and therefore the velocity profile also vary when entering the adsorption column. The momentum balance is described by the Ergun equation [28]:

$$-{\displaystyle \frac{\partial p}{\partial z}}=150{\displaystyle \frac{{(1-ϵ)}^2}{{ϵ}^3}}{\displaystyle \frac{{\eta }_{eff}}{d^2}}{u}_0+1.75{\displaystyle \frac{(1-ϵ)}{{ϵ}^3}}{\displaystyle \frac{{\rho }_f}{d}}{u}_0^2.$$

(37)

The effective dynamic viscosity ${\eta }_{eff}$ is described with

$${\displaystyle \frac{{\eta }_{eff}}{{\eta }_f}}=2.0exp\left(3.5·{10}^{-3}R{e}_0\right),R{e}_0={\displaystyle \frac{u_{0}d{\rho }_f}{{\eta }_f}}.$$

(38)

The dynamic viscosity of hydrogen at 30 bar and 20 °C ${\eta }_f=8.82·{10}^{-6}\mathrm{Pa}\mathrm{s}$ [30].

3.2. Discretization

The equations for mass balance for the gas phase and the energy balance for the gas and solid phases were discretized explicitly using the Finite-Difference Method and calculated in Python 3.11.11 Explicit methods are often computationally cheaper compared to implicit methods because they do not require solving a system of equations at each time step. The gradients that depend on the bed velocity $u_0$, e.g., the gradients ${\displaystyle \frac{\partial }{\partial z}}{D}_{ax}$ and ${\displaystyle \frac{\partial }{\partial z}}{\Lambda }_{ax}$, were considered fluctuating. The column is divided into $N-1$ sections containing N spatial discretization points. The mass balance of the solid phase, Equation (16), is implicitly discretized since this equation only depends on the unknown variable $X_s$:

$$X_s^{j,t+1}={\displaystyle \frac{X_s^{j,t}+\Delta t{k}_{LDF}{X}_{s,eq}(T^{j,t},p_{H_{2}O}^{j,t})}{1+\Delta t{k}_{LDF}}}.$$

(39)

Here, the index j denotes the axial section and t the temporal discretization step. The mass balance of the gas phase (Equation (24)) is discretized explicitly:

$$\begin{array}{cc}\hfill X_f^{j,t+1}& =X_f^{j,t}+{\displaystyle \frac{\Delta t}{ϵ}}(-{\displaystyle \frac{(u_0^{j,t}{X}_f^{j,t}-u_0^{j-1,t}{X}_f^{j-1,t})}{\Delta z}}+{\displaystyle \frac{D_{ax}^{j,t}-D_{ax}^{j-1,t}}{\Delta z}}{\displaystyle \frac{X_f^{j,t}-X_f^{j-1,t}}{\Delta z}}\hfill \\ & +D_{ax}^{j,t}{\displaystyle \frac{X_f^{j+1,t}-2X_f^{j,t}+X_f^{j-1,t}}{\Delta z^2}}-(1-ϵ){\displaystyle \frac{{\rho }_s}{{\rho }_f}}{k}_{LDF}(X_{s,eq}(T^{j,t},p_{H_{2}O}^{j,t})-X_s^{j,t+1}))\hfill \end{array}$$

(40)

with the Dirichlet boundary condition at the beginning of the column (spatial discretization point at 0 cm)—

$$X_f^{0,t+1}=777·{10}^{-6},\forall t=0,\dots ,t_{max}-1$$

(41)

—and $t_{max}$ as the maximum number of time steps. The Neumann boundary condition is given at the last spatial discretization point N at the end of the column:

$$X_f^{N,t+1}={\displaystyle \frac{2}{3}}\left(-{\displaystyle \frac{1}{2}}{X}_f^{N-1,t}+2X_f^{N,t}\right),\forall t=0,\dots ,t_{max}-1.$$

(42)

Defining $C_1$ and $C_2$—

$$C_1^j=ϵ{\rho }_f{c}_{p,f}+(1-ϵ){\rho }_s(c_{p,s}+X_s^{j,t+1}{c}_{p,ads}\left(T^{j,t}\right))$$

(43)

$$C_2^j={\rho }_s(1-ϵ)k_{LDF}(X_{s,eq}(T^{j,t},p_{H_{2}O}^{j,t})-X_s^{j,t+1})\left(\Delta h_{ads}(X_s^{j,t+1},T^{j,t})+X_s^{j,t+1}{\displaystyle \frac{\partial \Delta h_{ads}(X_s^{j,t+1},T^{j,t})}{\partial X_s}}\right)$$

(44)

—and $\Delta h_{ads}$—

$$\Delta h_{ads}(X_s^{j,t+1},T^{j,t})=\Delta h_v\left(T^{j,t}\right)+E{\left(ln{\displaystyle \frac{X_{sat}}{X_s^{j,t+1}}}\right)}^{1/n}+{\displaystyle \frac{E\beta T^{j,t}}{n}}{\left(ln{\displaystyle \frac{X_{sat}}{X_s^{j,t+1}}}\right)}^{-(n-1)/n}$$

(45)

—we discretize the energy balance in the following way:

$$\begin{array}{cc}\hfill T^{j,t+1}=& T^{j,t}+{\displaystyle \frac{\Delta t}{C_1^j}}(-{\rho }_f{c}_{p,f}\left({\displaystyle \frac{u_0^{j,t}{T}^{j,t}-u_0^{j-1,t}{T}^{j-1,t}}{\Delta z}}\right)\hfill \\ \hfill -& {\rho }_f{c}_{p,w}{\displaystyle \frac{(u_0^{j,t}{T}^{j,t}{X}_f^{j,t}-u_0^{j-1,t}{T}^{j-1,t}{X}_f^{j-1,t})}{\Delta t}}\hfill \\ \hfill +& {\displaystyle \frac{{\Lambda }_{ax}^{j,t}-{\Lambda }_{ax}^{j-1,t}}{\Delta z}}{\displaystyle \frac{T^{j,t}-T^{j-1,t}}{\Delta z}}+{\Lambda }_{ax}^{j,t}{\displaystyle \frac{T^{j+1,t}-2T^{j,t}+T^{j-1,t}}{\Delta z^2}}+C_2^j).\hfill \end{array}$$

(46)

Again, the Dirichlet boundary condition is given at the beginning of the column—

$$T^{0,t+1}=20°C,\forall t=0,\dots ,t_{max}-1$$

(47)

—and the Neumann boundary condition at the end:

$$T^{N,t+1}={\displaystyle \frac{2}{3}}\left(-{\displaystyle \frac{1}{2}}{T}^{N-1,t}+2T^{N,t}\right),\forall t=0,\dots ,t_{max}-1.$$

(48)

The momentum equation was the first one to be solved. A bed velocity matrix ${\mathbf{u}}_{0,\mathrm{matrix}}$ with the dimensions $N\times t_{max}$ was first calculated from the mass flows. The column diameter was set to 30 $\mathrm{c}$$\mathrm{m}$. Then, with the bed velocity matrix, pressure loss was calculated according to the Ergun Equation (37):

$$\Delta \mathbf{p}=\Delta z\left(150{\displaystyle \frac{{(1-ϵ)}^2}{{ϵ}^3}}{\displaystyle \frac{{\eta }_{eff}}{d^2}}{\mathbf{u}}_{0,\mathrm{matrix}}+1.75{\displaystyle \frac{(1-ϵ)}{{ϵ}^3}}{\displaystyle \frac{{\rho }_f}{d}}{\mathbf{u}}_{0,\mathrm{matrix}}^2\right).$$

(49)

We obtain a pressure loss matrix $\Delta \mathit{p}$, which stores the pressure differences and is again a $N\times t_{max}$ matrix. We define a pressure matrix ${\mathit{p}}_{\mathit{M}}$, which tells us the pressure in each section of the column. The pressure at the beginning of the column for each time step is set to $p_M^{0,t}=30\mathrm{bar},\forall t=0,\dots ,t_{max}$. Then, iteratively, for each pressure section in ${\mathit{p}}_{\mathit{M}}$, starting from the beginning of the column, we subtract the corresponding pressure loss section in $\Delta \mathit{p}$:

$$p_M^{j,t}=p_M^{j-1,t}-\Delta p^{j-1,t},\forall j=1,\dots ,N,\forall t=1,\dots ,t_{max}.$$

(50)

This operation is computationally time-expensive. Furthermore, because we consider decent variations in mass flows, the pressure loss is very small compared to the high pressure of 30 bar. Comparing a simulation using the momentum balance and a simulation using fixed pressure of 30 bar throughout the whole column shows that the difference in mass loading $X_s$ deviates maximally in the order of ${10}^{-7}\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1}$, and therefore the Ergun Equation can be neglected for loading simulations.

The initial conditions for $t=0$ are given as follows:

$$X_s^{j,0}={10}^{-8}\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1},X_f^{j,0}=0.0069·{10}^{-6}\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1},T^{j,0}=20°C,\forall j=0,\dots ,N.$$

(51)

$X_s^{j,0}$ was chosen to be very small to simulate adsorption after an almost full regeneration. The mass loading $X_f^{j,0}$ is calculated using the mole fraction $c=777·{10}^{-6}$, resulting in

$$X_f^{j,0}={\displaystyle \frac{c{M}_{H_{2}O}}{c{M}_{H_{2}O}+(1-c)M_{H_2}}}=0.0069\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1}.$$

(52)

The partial pressure $p_{H_{2}O}$ of water is used to calculate the equilibrium loading of the Langmuir–Freundlich isotherm in each section j and time step t, and it is given as follows:

$$p_{H_{2}O}^{j,t}={\displaystyle \frac{p_{total}^{j,t}{X}_f^{j,t}/M_{H_{2}O}}{X_f^{j,t}/M_{H_{2}O}+(1-X_f^{j,t}/M_{H_2})}}.$$

(53)

4. Simulation Results

As mentioned before, we assume that the temperature at the inlet is constant at 20 °C and that the inlet loading of water is constant at 777 ppm. Both criteria can be achieved through a condensation process. We consider the pressure in the hydrogen buffer tank, and therefore at the column inlet, to be fixed at 30 bar, which results in a variable mass flow. The pressure at the cathode of an electrolyzer must be kept constant to stabilize the electrolysis process and maximize efficiency. It is particularly important to keep the pressure balanced on both sides (anode and cathode) to minimize mechanical stress on the membrane and extend the service life of the electrolyzer [33].

The mass flow was taken from a multi-physics simulation for offshore hydrogen production carried out in [5]. In their study, a co-simulation of three models was performed: a 15 MW wind turbine, a desalination process through evaporation and three 5 MW water electrolyzers. The mass flow of the hydrogen is influenced by the offshore wind speed. Taking into account data from the Federal Maritime and Hydrographic Agency from the FINO1 Research Platform, wind speeds in the North Sea range from 0 m/s to approximately 30 m/s [34]. However, at high wind speeds above 16 m/s, a trade-off between hydrogen production and the aging of the wind turbine must be considered, and at higher speeds the wind turbine is usually throttled down. Therefore, three wind speeds were simulated: 6 m/s, 10 m/s, and 16 m/s. The mentioned wind speeds are given as means, but due to the volatile character of the wind, the velocities are reproduced with variations around these means. The first 10 min of generated wind velocities are shown in Figure 7a. As three hydrogen electrolyzers were considered, the three mass flows were summed, as shown in Figure 7b.

The dynamics were calculated for an adsorber column having a height of 12 cm and a diameter of 30 cm. The column was divided into 24 sections, i.e., 25 spatial discretization points, with two points designated as boundary points. Therefore, each section has a height of 0.5 cm. The arrangement of the sections is shown in Figure 8. A time step size of $\Delta t=0.001\mathrm{s}$ was found to deliver stable results with the chosen spatial grid. The discretization was tested on the 10 m/s wind profile considering the loading $X_s$, where for a fix $\Delta t$ the discretization points were repeatedly doubled, starting with six sections. The testing was carried out at five spatial discretization points (2 cm, 4 cm, 6 cm, 8 cm, and 10 cm). The subsequent discretization with 2x sections was accepted with an absolute tolerance of $0.02\mathrm{k}\mathrm{g}{\mathrm{kg}}^{-1}$; i.e., if for all five segments, if the absolute difference of the loading values $X_s$ of the simulation with $2x$ sections and the preceding simulation with x sections was smaller than the absolute tolerance, the discretization with x sections was accepted. In order to achieve a smaller tolerance level, the number of sections has to be increased while also decreasing the time step to avoid numerical instability (see CFL condition).

The simulation was executed on a system equipped with a 12th Gen Intel Core i5-1240P (1.7 GHb) and 16 GB of RAM, running a 64-bit operating system. Simulating a 1-hour adsorption loading cycle using the discussed discretization and time step requires approximately 7 to 8 min. The link to the simulation code is provided in the Supplementary Materials section of this paper.

In Figure 9, the loading $X_s$ at the discretization point located at 2 cm of the column is shown in dependence on the wind velocity. For the wind velocity profiles of 10 m/s and 16 m/s, the segment is fully loaded after 18 to 20 min, with the 16 m/s profile reaching full loading slightly faster. For the wind velocity profile of 6 m/s, the segment is still not fully loaded after 60 min. Thus, different wind velocities result in significant variations in water loading profiles.

Figure 10 presents the differences in temperature, T, water loading in the gas, $X_f$, and water loading in the adsorbent, $X_s$, for velocity profiles of 6 m/s and 10 m/s at discretization points located at 2 cm, 4 cm, and 12 cm. As the loading profiles for wind velocities of 10 m/s and 16 m/s in Figure 9 showed similar results, we neglected 16 m/s in the comparison graph. As mentioned before, the loading curve $X_s$ becomes steeper for higher wind speeds (see Figure 10a,b). The concentration of water in the fluid $X_f$ for 10 m/s is also strictly higher than that for 6 m/s, as this corresponds to the loading. In other words, the sooner full adsorption loading is achieved at a section, the sooner the concentration in the fluid $X_f$ at that section returns to its level of 777 ppm. The temperature profile behaves typically for an adsorption cycle. The temperature front rises vertically at the beginning of the adsorption for each segment, then it drops first at the 2 cm segment, and last at the 12 cm segment.The faster decreasing temperature for a wind profile of 10 m/s can be explained by the fact that heat is dissipated more quickly due to the increased gas velocity.

5. Discussion and Conclusions

This study introduced a numerical model of an adsorption process that captures fluctuations in mass flows. The parameters for the Langmuir–Freundlich isotherm were derived from our own water adsorption measurement data. Using IAST, it was estimated that the co-adsorption of hydrogen is hardly present. In addition, differences in water loading in the adsorbent and fluid, as well as temperature variations, were presented and discussed.

In the case of variable mass flows, it is essential to end the loading cycle on time before the unwanted component can break through and to switch to regeneration when the column is full. As shown in the simulations, wind velocity significantly affects the loading profile. To achieve this, the adsorption process must be supplemented with methods that monitor water loading, such as temperature front measurements or moisture detection. Another approach involves establishing communication with the electrolyzer to track the mass flow using flow meters and a control unit that solves the balance equations in Section 3. Another challenge is that there may be periods when no wind energy is present, resulting in a lack of mass flow from the electrolyzer. The industrial manufacturer Silica Verfahrenstechnik, which uses a direct TSA for regeneration, solved this problem by integrating a gas booster to provide regeneration gas and additional battery storage to ensure that the regeneration process finishes during windless times. In future work, the presented model can be used to simulate both the loading and the regeneration process. Furthermore, the model can be used to estimate the design of an adsorption process for both onshore and offshore hydrogen production powered by fluctuating energy sources such as wind or solar power. As the model captures variations in wind speeds, it can simulate worst-case scenarios, including wind reversals or sudden drops in wind velocity, using historical wind data. By considering critical scenarios, the simulation can be especially helpful in scaling components like a purge gas reservoir, battery storage, or an adsorption column.

However, experimental studies, such as breakthrough studies for the determination and verification of the mass transfer parameters $k_{LDF}$, are necessary to validate the model, which should be addressed in future research. Other quantities to be verified experimentally are axial dispersion $D_{ax}$ and axial thermal conductivity ${\Lambda }_{ax}$. In future research, the prediction obtained from IAST in this work should be verified either by multi-component adsorption isotherm measurements or by grand-canonical Monte-Carlo simulations. Further computational research can include the comparison of PSA and TSA in terms of energy efficiency, cycle times, and safety, considering performance indicators such as purity, recovery, and productivity. Additional work could involve comparative studies between different adsorbents, such as other types of zeolite, silica gel, and other materials.