# Numerical Simulation Approach for a Dynamically Operated Sorption-Enhanced Water-Gas Shift Reactor

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}-free synthesis gas for e-fuel production from pure CO. The nonlinear model equations describing simultaneous adsorption and reaction are solved with three numerical approaches in MATLAB: a built-in solver for partial differential equations, a semi-discretization method in combination with an ordinary differential equation solver, and an advanced graphic implementation of the latter method in Simulink. The novel implementation in Simulink offers various advantages for dynamic simulations and is expanded to a process model with six reaction chambers. The continuous conditions in the reaction chambers and the discrete states of the valves, which enable switching between reactive adsorption and regeneration, lead to a hybrid system. Controlling the discrete states in a finite-state machine in Stateflow enables automated switching between reactive adsorption and regeneration depending on predefined conditions, such as a time span or a concentration threshold in the product gas. The established chemical reactor simulation approach features unique possibilities in terms of simulation-driven development of operating procedures for intensified reactor operation. In a base case simulation, the sorbent usage for serial operation with adjusted switching times is increased by almost 15%.

## 1. Introduction

_{2}, H

_{2}O, and renewable energy are converted to the target product kerosene [4]. One unit operation within the process chain comprises a compact sorption-enhanced water–gas shift (SEWGS) reactor. Here, purified CO from a plasma reactor is partly converted with steam to produce hydrogen. Two main targets are pursued in this process step:

- 1.
- Providing synthesis gas (syngas) with a H
_{2}/CO ratio of approximately two for the subsequent Fischer–Tropsch synthesis. - 2.
- Removing the by-product CO
_{2}and refeeding it to the plasma reactor.

_{2}on a potassium-impregnated hydrotalcite sorbent (K-HTC). Many experimental studies have proven the feasibility of this technology [6,7,8]; a comprehensive overview is given in [9]. According to Le Chatelier’s principle, the selective removal of the product component CO

_{2}(Equation (2)) shifts the reaction equilibrium towards the product side. Hence, higher yields of the desired product component H

_{2}can be obtained, and the undesired product component CO

_{2}can be recycled. Depending on prevailing operating conditions, H

_{2}O is also adsorbed on K-HTC (Equation (3)).

- pressurization (feed: reactants);
- reactive adsorption (feed: reactants);
- depressurization (no feed, often purged);
- regeneration (feed: purge gas).

`ode15s`solver for pre-defined time spans of reactive adsorption/regeneration steps.

_{2}content in the reaction chamber outlet.

## 2. SEWGS Model

#### 2.1. Model Development

_{2}O

_{3}catalyst particles. Due to the rather low adsorption capacity of K-HTC, the SEWGS reactor in the Kerogreen pilot plant employs a PSA concept with fast switching times for continuous H

_{2}production. Contrary to concepts with large, thick vessels as reactor compartments, where severe back-mixing can occur, the slits are assumed to behave like ideal plug flow reactors.

_{2}with steam) through the tempering channels prior to entering the chambers in desorption mode leads to a negligible temperature difference between adsorption and desorption slits. The model is based on the following assumptions and considerations:

- Homogeneous distribution of uniformly sized catalyst and sorbent particles according to their weight fraction assumed as one phase
- Uniform gas distribution in the reaction chambers
- Isothermal conditions in the slits
- Negligible pressure drop in the slits
- Constant superficial velocity (which is only the case for sufficient dilution, [27])
- No gradients rectangular to flow direction
- Axial dispersion considered with axial dispersion coefficient
- External mass transfer limitations neglected (Maers criterion)
- Internal mass transfer limitations considered (Weisz–Prater criterion) and implemented with linear driving force (LDF) model (Glueckauf criterion)
- Equilibrium-based desorption of H
_{2}O and CO_{2}.

#### 2.2. Model Equations

_{2}O, CO

_{2}, H

_{2}, N

_{2}in the bulk gas phase are given in Equation (4). A heterogeneous LDF approximation was used for intraparticle mass transfer limitations to account for the porous character of the particles.

_{2}O, CO

_{2}, H

_{2}, N

_{2}in the particle void phase, where reaction and adsorption or desorption are assumed to take place on the solid surface, are given in Equation (7) for an axial dependent averaged particle concentration. The stoichiometric reaction coefficient ${\nu}_{i}$ is $-1$ for the reactants (CO and H

_{2}O), 1 for the products (CO

_{2}, H

_{2}), and 0 for inert N

_{2}. The adsorption rates of H

_{2}O and CO

_{2}are given in Equation (8) and Equation (9), respectively. Initial conditions are defined analogous to the bulk phase.

_{2}O and CO

_{2}on adsorption site A (H

_{2}O only), B (CO

_{2}only), and C (H

_{2}O and CO

_{2}) of K-HTC sorbent are taken from Coenen et al. and are specified in Equation (10) (identical approach for ${q}_{\mathrm{A},{\mathrm{H}}_{2}\mathrm{O}}$) and Equation (11) (identical approach for ${q}_{\mathrm{C},{\mathrm{H}}_{2}\mathrm{O}}$), respectively [29]. The initial conditions are chosen according to the prevailing conditions in the reaction chambers before the switch.

_{2}by H

_{2}O takes place (Equation (11)). The heterogeneity of the surface is considered by a modified Elovich approach, shown for site B in Equation (13) (identical approach for site A and C) [29].

_{2}O

_{3}catalyst is taken from Choi et al. and presented in Equation (14) [28]. The reaction equilibrium constant results from Equation (15) [30].

#### 2.3. Model Parameters

## 3. Numerical Simulation Approaches

#### 3.1. Built-In Solver

`pdepe`for parabolic and elliptic PDEs in one dimension. This solver solves initial-boundary value problems in the form of Equation (18) with two independent variables: one spatial variable and time. The coefficients of Equation (18) have to be provided in the function handle

`pdefun`, whereas initial and boundary conditions are specified in

`icfun`and

`bcfun`. The boundary conditions must be coded according to Equation (19) [33]. At least one PDE must be parabolic. Therefore, the axial dispersion term in Equation (4) is required for the applicability of

`pdepe`and the proper definition of boundary conditions.

`pdepe`solver transforms the system of PDEs into a system of ODEs by means of spatial discretization with a piecewise Petrov–Galerkin method on a set of user-defined nodes (

`xmesh`) [34]. The resulting ODE system is integrated in time with the built-in solver

`ode15s`for stiff differential algebraic equations (DAEs). DAEs arise from elliptic equations in the PDE system. The variable time step integration of

`ode15s`delivers the numerical solution at specified points of time, defined in

`tspan`. With this procedure, both time step and computing formula are adapted dynamically to obtain high accuracy in short computational time [35]. The

`pdepe`solver provides a user-friendly implementation and is capable of solving nonlinear and coupled equations reliably for suitably defined meshes [36].

#### 3.2. Method-of-Lines (MoL)

`ode15s`for best comparability with the

`pdepe`results. This variable-step, variable-order solver is based on numerical differentiation formulas and is recommended for stiff problems and DAEs. A convergence analysis with up to 250 cells is performed to examine the adequacy of the spatial discretization.

#### 3.3. Simulink

#### 3.3.1. Model Implementation and Data Structure

_{2}concentration leaving the 25th cell is addressed as

`N20.N05.CCO2`. This structure simplifies data handling enormously, compared to the methods presented in Section 3.1 and Section 3.2, where the overall time- and space dependent solution is contained in one voluminous matrix.

#### 3.3.2. Solver Selection

`ode23t`, the computation time was reduced by almost half compared to the commonly used solver

`ode15s`with sufficient precision (relative error tolerance of ${10}^{-3}$). This solver is used to speed-up all Simulink simulations presented in this paper.

#### 3.3.3. Cyclic Process Design

_{2}production. The automated switching logic from reactive adsorption to regeneration mode is realized with a finite-state-machine implemented in the Simulink tool Stateflow. Stateflow enables mode logic, fault management, and task scheduling in discrete or hybrid systems. A Stateflow machine contains Stateflow charts with objects such as states, events, transitions, etc. [40].

_{2}content in the reaction chamber outlet is investigated for a partly serial reaction chamber configuration.

## 4. Results and Discussion

#### 4.1. Reactive Adsorption

_{2}over the reactor length is shown for 10 to 250 cells at $t=25\mathrm{s}$. The curves exhibit steep steps for simulations with few cells, and the smoothness increases significantly for $N>100$. The grid (in)dependence is depicted in Figure 5b. For $N>100$, the deviation from $N=250$ is less than 5%, except for the initial conditions in the differently sized first cell.

`pdepe`simulations have proven to be extremely sensitive to initial conditions. A sufficiently high number of mesh points in time and space needs to be chosen to ensure numerical stability and to avoid oscillations. The computation time for

`pdepe`was significantly lower compared to MoL simulations with a comparable number of mesh points due to

`pdepe`’s run-time optimization (see Appendix A, Table A2). For both methods, MoL and

`pdepe`, convergence could be reached with appropriate mesh settings.

_{2}loading of the sorbent over the reactor length increases with time until it reaches full saturation after approximately 100 $\mathrm{s}$ (Figure 6a). At the same time, the CO

_{2}concentration at the reactor outlet approaches its steady-state value (Figure 6b). The defined breakthrough (CO

_{2}volume fraction in the product > 0.05) of CO

_{2}at the reactor outlet can be noticed after approximately 40 $\mathrm{s}$. Prior to that point, CO is fully converted in excess of H

_{2}O in the feed, and the produced CO

_{2}is adsorbed completely.

`pdepe`and MoL, exhibit only minor deviations from a closed molar balance (< 2%). The deviation is more pronounced for

`pdepe`close to the reactor inlet at $t<$ 50 $\mathrm{s}$. For MoL, the deviation emerges for a longer period of time at the reactor outlet. In both cases, it approaches zero as soon as steady-state conditions are reached. The deviations were attributed to the adsorption process and could—especially for undiluted feeds—possibly be improved by taking into account prevailing velocity changes, as suggested by DiGiuliano et al. [27].

`pdepe`as well as simple uniform semi-discretization with MoL, lead to satisfying results for the SEWGS simulation, the MoL results were compared with the corresponding Simulink results for $N=100$. As expected, hardly any deviation between the curves of the two methods can be distinguished (Figure 7). Therefore, it was concluded that the graphical approach in Simulink performs with sufficient precision and reliability and can be expanded to a more complex hybrid process model.

#### 4.2. Process Design

_{2}O) and produces CO

_{2}-free WGS product, whereas all other chambers are in regeneration mode (M4). The regeneration feed consists of N

_{2}and H

_{2}O to increase CO

_{2}desorption from both sorption sites (B and C). As soon as the sorbent in chamber 1 is no longer capable of adsorbing all produced CO

_{2}and the gas flow leaving chamber 1 reaches a CO

_{2}volume fraction of 5%, a switch to M3 is triggered. In M3, the gas flow leaving chamber 1 is fed into chamber 2, which is then in M1. While the sorbent loading in chamber 1 is still increasing, CO

_{2}-free WGS product is produced in chamber 2. Chamber 1 switches from M3 to M4 when a threshold of 20% CO

_{2}volume fraction is reached. The cycle then continues in chamber 2 with M2.

_{2}-free product flow leaves the reactor. Figure 9b shows the H

_{2}volume fraction in the product flow of the total reactor (six chambers) and of chamber 1. The H

_{2}volume fraction approaches a constant value of about $0.62$ after the first cycle; the rest of the product gas flow consists of inert N

_{2}and unreacted H

_{2}O.

_{2}volume fraction in the product flow below 5%.

## 5. Conclusions

_{2}concentration in the product stream. For that reason, switching times do not need to be pre-defined and can be optimized dynamically.

`pdepe`solver. These results were compared with those obtained via MoL (spatial discretization with FD and ODE system solution with MATLAB’s

`ode15s`solver) to determine the required number of spatial discretization cells and to justify the discretization scheme.

_{2}volume fraction in the product flow could still be kept below 5%. Due to automated switching between interconnected reaction chambers, an increase of sorbent usage from 72.6% to 87.1% could be elaborated in the presented case.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

The following abbreviations are used in this manuscript: | ||

BFDM | Backward finite difference method of first order | |

CFDM | Centered finite difference method of second order | |

Ch. | Chamber | |

CSTR | Continuously stirred tank reactor | |

DAE | Differential algebraic equation | |

DASOLV | Implicit backward differentiation formula solver in gPROMS | |

DMES | Dimethyl ether synthesis | |

FD | Finite Difference | |

FEM | Finite elements method | |

K-HTC | K_{2}CO_{3}-promoted hydrotalcite | |

LDF | Linear driving force model | |

M1-M4 | Mode 1-4 | |

MoL | Method-of-lines | |

ODE | Ordinary differential equation | |

ode15s | Solver for stiff ODEs in MATLAB | |

PDE | partial differential equation | |

pdepe | Solver for systems of parabolic and elliptic PDEs in MATLAB | |

PLC | Programmable logic controller | |

PSA | Pressure swing adsorption | |

PTSA | Pressure temperature swing adsorption | |

SEWGS | Sorption-enhanced water–gas shift | |

SMR | Steam-methane reforming | |

TSA | Temperature swing adsorption | |

WGS | Water–gas shift | |

The following symbols are used in this manuscript: | ||

${\beta}_{j,\mathrm{des},\left(i\right)}$ | activation energy change | ($\mathrm{J}{\mathrm{mol}}^{-1}$) |

${\epsilon}_{\mathrm{b}}$ | bed void fraction | (–) |

${\epsilon}_{\mathrm{p}}$ | particle void fraction | (–) |

${\nu}_{i}$ | stoichiometric reaction coefficient | (–) |

$\rho $ | bulk density | ($\mathrm{k}$$\mathrm{g}$${\mathrm{m}}^{-3}$) |

${a}_{\mathrm{ads},i}$ | adsorption rate | ($\mathrm{mol}$${\mathrm{kg}}^{-1}$${\mathrm{s}}^{-1}$) |

${c}^{*}$ | pdepe diagonal matric | (–) |

${c}_{i}$ | bulk phase concentration | ($\mathrm{mol}{\mathrm{m}}^{-3}$) |

${\overline{c}}_{i}$ | particle void phase concentration | ($\mathrm{mol}{\mathrm{m}}^{-3}$) |

${D}_{\mathrm{ax},i}$ | axial dispersion coefficient | (${\mathrm{m}}^{2}$ ${\mathrm{s}}^{-1}$) |

${D}_{\mathrm{eff},i}$ | effective diffusion coefficient | (${\mathrm{m}}^{2}$ ${\mathrm{s}}^{-1}$) |

${D}_{\mathrm{mix},i}$ | gas mixture diffusion coefficient | (${\mathrm{m}}^{2}$ ${\mathrm{s}}^{-1}$) |

${E}_{\mathrm{a}}$ | activation energy | ($\mathrm{J}{\mathrm{mol}}^{-1}$) |

$eq$ | equilibrium | (–) |

${f}^{*}$ | pdepe flux term coefficient | (–) |

${F}_{\mathrm{STP}}$ | volumetric flow rate at STP | ($\mathrm{m}$$\mathrm{l}$${\mathrm{min}}^{-1}$) |

h | slit height | (mm) |

$\Delta {H}_{\mathrm{R}}^{\ominus}$ | standard reaction enthalpy | ($\mathrm{k}$$\mathrm{J}{\mathrm{mol}}^{-1}$) |

i | species CO, H_{2}O, CO_{2}, H_{2}, N_{2} | (–) |

j | sorption site A, B, C | (–) |

${k}_{\mathrm{C},\mathrm{rep}1}$ | exchange rate coefficient 1 | (${\mathrm{bar}}^{-1}{\mathrm{s}}^{-1}$) |

${k}_{\mathrm{C},\mathrm{rep}2}$ | exchange rate coefficient 2 | (${\mathrm{bar}}^{-1}{\mathrm{s}}^{-1}$) |

${K}_{\mathrm{eq}}$ | equilibrium constant | (–) |

${k}_{j}$ | Freundlich adsorption coefficient | ($\mathrm{mol}$${\mathrm{kg}}^{-1}{\mathrm{bar}}^{-1}$) |

${k}_{j,\mathrm{ads}}$ | adsorption coefficient | (${\mathrm{s}}^{-1}$) |

${k}_{j,\mathrm{des},\left(i\right)}$ | desorption coefficient | (${\mathrm{s}}^{-1}$) |

${k}_{j,\mathrm{des},\left(i\right)}^{1}$ | desorption coefficient | (${\mathrm{s}}^{-1}$) |

${k}_{\mathrm{LDF},i}$ | linear driving force coefficient | (${\mathrm{s}}^{-1}$) |

${k}_{\infty}$ | frequency factor | (${\mathrm{molbar}}^{-2}{\mathrm{g}}^{-1}$${\mathrm{h}}^{-1}$) |

l | slit length | (mm) |

$\mathrm{m}$ | tiny value exponent | (–) |

${m}^{*}$ | pdepe symmetry constant | (–) |

n | index variable | (–) |

N | number of cells | (–) |

${n}_{j}$ | Freundlich adsorption intensity | (–) |

O | approximation error | (–) |

${p}^{*}$ | pdepe boundary coefficient | (–) |

${p}_{\left(i\right)}$ | (partial) pressure | (bar) |

${q}^{*}$ | pdepe boundary coefficient | (–) |

${q}_{j,i}$ | sorbent loading of species i on site j | ($\mathrm{mol}$${\mathrm{kg}}^{-1}$) |

${r}_{\mathrm{p}}$ | particle radius | (m) |

${r}_{\mathrm{WGS}}$ | WGS reaction rate | ($\mathrm{mol}$${\mathrm{kg}}^{-1}$ ${\mathrm{s}}^{-1}$) |

R | gas constant | ($\mathrm{J}{\mathrm{mol}}^{-1}$ ${\mathrm{K}}^{-1}$) |

${s}^{*}$ | pdepe source term coefficient | (–) |

t | time | ($\mathrm{s}$) |

T | temperature | ($\mathrm{K}$) |

u | gas velocity | ($\mathrm{m}$${\mathrm{s}}^{-1}$) |

v | dependent variable | (–) |

w | slit width | (mm) |

${w}_{\mathrm{cat}}$ | catalyst weight fraction | (–) |

${y}_{i}$ | volume fraction | (–) |

z | axial coordinate | ($\mathrm{m}$) |

## Appendix A

Adsorption / Desorption | |
---|---|

${k}_{\mathrm{A},\mathrm{ads}}$ | $4.18\times {10}^{-2}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{A}}$ | $1.69$$\mathrm{mol}$${\mathrm{kg}}^{-1}{\mathrm{bar}}^{-1}$ |

${n}_{\mathrm{A}}$ | $0.235$ |

${k}_{\mathrm{A},\mathrm{des}}^{1}$ | $7.84\times {10}^{-9}$${\mathrm{s}}^{-1}$ |

${\beta}_{\mathrm{A},\mathrm{des}}$ | $1.72\times {10}^{5}$$\mathrm{J}{\mathrm{mol}}^{-1}$ |

${k}_{\mathrm{B},\mathrm{ads}}$ | $9.29\times {10}^{-2}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{B}}$ | $0.31$$\mathrm{mol}$${\mathrm{kg}}^{-1}{\mathrm{bar}}^{-1}$ |

${n}_{\mathrm{B}}$ | $0.239$ |

${k}_{\mathrm{B},\mathrm{des}}^{1}$ | $4.41\times {10}^{-5}$${\mathrm{s}}^{-1}$ |

${\beta}_{\mathrm{B},\mathrm{des}}$ | $6.27\times {10}^{4}$$\mathrm{J}{\mathrm{mol}}^{-1}$ |

${k}_{\mathrm{C},\mathrm{ads}}$ | $0.1{\mathrm{bar}}^{-1}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{C},\mathrm{rep}1}$ | $5.0\times {10}^{-3}{\mathrm{bar}}^{-1}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{C},\mathrm{rep}2}$ | $1.4\times {10}^{-2}{\mathrm{bar}}^{-1}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{C},\mathrm{des},{\mathrm{H}}_{2}\mathrm{O}}^{1}$ | $5.23\times {10}^{-11}$${\mathrm{s}}^{-1}$ |

${k}_{\mathrm{C},\mathrm{des},\mathrm{C}{\mathrm{O}}_{2}}^{1}$ | $2.06\times {10}^{-10}$${\mathrm{s}}^{-1}$ |

${\beta}_{\mathrm{C},\mathrm{des},{\mathrm{H}}_{2}\mathrm{O}}$ | $5.41\times {10}^{4}$$\mathrm{J}{\mathrm{mol}}^{-1}$ |

${\beta}_{\mathrm{C},\mathrm{des},\mathrm{C}{\mathrm{O}}_{2}}$ | $5.00\times {10}^{4}$$\mathrm{J}{\mathrm{mol}}^{-1}$ |

$\mathrm{m}$ | $1\times {10}^{-16}$ |

WGS Reaction | |

${k}_{\infty}$ | $2.96\times {10}^{5}$$\mathrm{mol}{\mathrm{bar}}^{-1}{\mathrm{g}}^{-1}$${\mathrm{h}}^{-1}$ |

${E}_{\mathrm{a}}$ | $47.400$$\mathrm{J}{\mathrm{mol}}^{-1}$ |

**Table A2.**Computation time for

`pdepe`and MoL simulations for reactive adsorption in one reaction chamber.

MoL | pdepe | ||||||
---|---|---|---|---|---|---|---|

Number of Cells | 10 | 30 | 50 | 100 | 200 | 250 | 3000 |

(-) | |||||||

Computation Time | 0.04 | 0.22 | 0.27 | 4.96 | 46.81 | 90.93 | 12.19 |

(min) |

**Figure A1.**Time and space-dependent deviation from molar balance for reactive adsorption in one reaction chamber: (

**a**)

`pdepe`, and (

**b**) MoL ($N=100$) simulation results.

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**Figure 1.**Pilot plant SEWGS reactor with six individually fed reaction chambers. Every chamber consists of seven slits. Isothermal conditions are ensured with adjacent channels between the slits for tempering with purge gas.

**Figure 2.**Bottom-up Simulink implementation of the model equations for one reaction chamber discretized analogous to MoL. Exemplary ODE implementation of the CO concentration in the bulk phase. The integrator block is marked in yellow. Fourteen ODEs arrange the cell subsystem; N (here: $N=50$) cells form the reaction chamber.

**Figure 3.**Required computation time depending on the number of implemented cells, chosen solver, and relative error tolerance. Filled symbols: relative error tolerance 10

^{$-6$}; open symbols: relative error tolerance 10

^{$-3$}. The best results for $N=100$ are obtained for solver

`ode23t`with a relative error tolerance of ${10}^{-3}$. Computation times below 0 indicate that a simulation did not converge.

**Figure 4.**Simulink model consisting of six reaction chambers (blue), which are operated in either reaction or regeneration mode after an initialization step. Subsequent reaction chambers can optionally be connected. The cyclic process operation is controlled with a Stateflow machine (yellow).

**Figure 5.**MoL grid analysis with $N=10\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}250$ cells for reactive adsorption in one reaction chamber. (

**a**) CO

_{2}loading, and (

**b**) deviation of CO

_{2}loading from $N=250$ as a function of the reactor length at $t=25\mathrm{s}$.

**Figure 6.**Comparison of

`pdepe`and MoL ($N=100$) simulation results for reactive adsorption in one reaction chamber. (

**a**) CO

_{2}loading at various adsorption times as a function of the reactor length, and (

**b**) bulk phase concentrations at reactor outlet as a function of time. Solid lines: MoL; dotted lines:

`pdepe`.

**Figure 7.**Comparison of MoL and Simulink ($N=100$) simulation results for reactive adsorption in one reaction chamber. (

**a**) CO

_{2}loading at various adsorption times as a function of the reactor length, and (

**b**) bulk phase concentrations at reactor outlet as a function of time. Solid lines: MoL; dotted lines: Simulink ($N=100$).

**Figure 8.**Process operation modes. M1: feed from previous chamber, outlet product; M2: reaction feed, outlet product; M3: reaction feed, outlet to subsequent chamber; M4: regeneration feed, outlet exhaust.

**Figure 9.**(

**a**) Operation scheme for six reaction chambers, and (

**b**) H

_{2}volume fraction in the product for all chambers combined ("total") and for chamber 1 for the first 900$\mathrm{s}$. Switching condition M2 to M3: CO

_{2}volume fraction in product > 0.05; M3 to M4: CO

_{2}volume fraction to subsequent chamber > 0.2.

**Figure 10.**Cyclic operation in six reaction chambers; results shown for nine cycles in chamber 1. (

**a**) Time span of the modes depending on the cycle number, and (

**b**) relative sorbent loading at the end of the modes.

**Table 1.**Overview of selected studies employing different numerical methods for solving (reactive) adsorption models.

Reaction | Adsorption | Numerical Solution | Reference |
---|---|---|---|

- | CO_{2} on K-HTC | MATLAB: MoL (N = 500) / ode15s solver | [13] |

SMR | CO_{2} on K-HTC | MATLAB: MoL / ode15s solver | [14] |

WGS | CO_{2} on K-HTC | gPROMS: CFDM (N = 600) / DASOLV solver | [15] |

WGS | CO_{2} | COMSOL Multiphysics: FEM | [16] |

SMR | CO_{2} on CaO-mayenite | MATLAB: pdepe solver | [17] |

DMES | H_{2}O on LTA zeolite | MATLAB: MoL (N = 30) / ode15s solver | [18] |

DMES | H_{2}O on LTA zeolite 3A | gPROMS: BFDM (N = 60) / DASOLV solver | [19,20] |

WGS | CO_{2} on K-HTC | MATLAB: MoL (N = 250) / ode15s solver | [21] |

SMR | CO_{2} on CaO | MATLAB: pdepe solver | [22] |

- | CO_{2} on K-HTC | gPROMS | [23] |

General Parameters | |
---|---|

${\epsilon}_{\mathrm{b}}$ | $0.4$ |

${\epsilon}_{\mathrm{p}}$ | $0.5$ |

$\rho $ | 1096$$$\mathrm{kg}$ ${\mathrm{m}}^{-3}$ |

${r}_{\mathrm{p}}$ | 100 $\mathsf{\mu}$$\mathrm{m}$ |

${w}_{\mathrm{cat}}$ | $0.05$ |

Adsorption Parameters | |

p | 8 bar |

T | 250 ${}^{\circ}\mathrm{C}$ |

${F}_{\mathrm{STP}}$ | 2000 mL /$\mathrm{min}$ |

${y}_{\mathrm{CO},\mathrm{feed}}$ | $0.3$ |

${y}_{{\mathrm{H}}_{2}\mathrm{O},\mathrm{feed}}$ | $0.6$ |

${y}_{{\mathrm{N}}_{2},\mathrm{feed}}$ | $0.1$ |

Desorption Parameters | |

p | 1 bar |

T | 250 ${\text{}}^{\circ}\mathrm{C}$ |

${F}_{\mathrm{STP}}$ | 1000 mL /$\mathrm{min}$ |

${y}_{{\mathrm{H}}_{2}\mathrm{O},\mathrm{feed}}$ | $0.4$ |

${y}_{{\mathrm{N}}_{2},\mathrm{feed}}$ | $0.6$ |

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**MDPI and ACS Style**

Stadler, T.J.; Knoop, J.-H.; Decker, S.; Pfeifer, P.
Numerical Simulation Approach for a Dynamically Operated Sorption-Enhanced Water-Gas Shift Reactor. *Processes* **2022**, *10*, 1160.
https://doi.org/10.3390/pr10061160

**AMA Style**

Stadler TJ, Knoop J-H, Decker S, Pfeifer P.
Numerical Simulation Approach for a Dynamically Operated Sorption-Enhanced Water-Gas Shift Reactor. *Processes*. 2022; 10(6):1160.
https://doi.org/10.3390/pr10061160

**Chicago/Turabian Style**

Stadler, Tabea J., Jan-Hendrik Knoop, Simon Decker, and Peter Pfeifer.
2022. "Numerical Simulation Approach for a Dynamically Operated Sorption-Enhanced Water-Gas Shift Reactor" *Processes* 10, no. 6: 1160.
https://doi.org/10.3390/pr10061160