# Optimization of Methanol Synthesis under Forced Periodic Operation

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## Abstract

**:**

_{2}O

_{3}catalysts under steady state conditions. In this paper, the potential of alternative forced periodic operation modes is studied using numerical optimization. The focus is a well-mixed isothermal reactor with two periodic inputs, namely, CO concentration in the feed and total feed flow rate. Exploiting a detailed kinetic model which also describes the dynamics of the catalyst, a sequential NLP optimization approach is applied to compare optimal steady state solutions with optimal periodic regimes. Periodic solutions are calculated using dynamic optimization with a periodicity constraint. The NLP optimization is embedded in a multi-objective optimization framework to optimize the process with respect to two objective functions and generate the corresponding Pareto fronts. The first objective is the methanol outlet flow rate. The second objective is the methanol yield based on the total carbon in the feed. Additional constraints arising from the complex methanol reaction and the practical limitations are introduced step by step. The results show that significant improvements for both objective functions are possible through periodic forcing of the two inputs considered here.

## 1. Introduction

_{2}O

_{3}under steady state conditions. More recently, there has also been a growing interest in dynamic methanol reactor operation in the context of energy storage and power to methanol processes [2]. In these types of processes, green hydrogen is produced from excess wind or solar energy via electrolysis. It reacts with CO/CO

_{2}from biogas or waste streams to methanol [3,4,5]. This will result in a more flexible use of electrical energy from renewable resources and simultaneously in a reduced emission of greenhouse gases to the environment. However, due to the fluctuating availability of the reactants, the reactor will face dynamic varying feeds. The focus of this paper is on improving reactor performance compared to conventional steady state operation by forced periodic operation, which is a specific type of dynamic reactor operation. The feed can be provided by renewable resources, conventional sources, or a mixture of both. Forced periodic operation can be beneficial if the time averaged reactor performance under forced periodic operation is higher than the corresponding steady state value, which is only possible for nonlinear systems (see, e.g., [6]). In general, the idea is not new and has been discussed since the 1960s (see, e.g., [7,8,9,10]). For a recent overview, we refer to [6,11]. For a rigorous evaluation of new forced periodic operation modes, a comparison with optimal steady state conditions is essential. First, work on forced periodic operation of methanol synthesis conducted on some Cu/ZnO and Cu/ZnO/Al

_{2}O

_{3}catalysts at temperatures between 225 and 270 °C and relatively moderate pressures between 1.97 and 2.93 MPa was done by [12,13]. The results indicate a potential of the methanol synthesis for improvement through periodic operation. However, a rigorous optimization of forced periodic operation and corresponding steady states was not performed due to the experimental focus of this work. Hence, a rigorous evaluation of forced periodic operation of methanol synthesis is lacking. This gap is closed in the present paper. Numerical NLP optimization is applied to compare optimal steady state solutions with optimal periodic regimes. Multi-objective optimization is applied to trace out the Pareto fronts to simultaneously maximize the methanol flow rate and the methanol yield with respect to the total carbon feed. Additional constraints arising in practice are integrated. In the first step, a well-mixed isothermal reactor with multiple periodic inputs is considered, namely, CO partial pressure in the feed and total feed flow rate. The underlying mathematical model is based on the lumped kinetic model presented in [14,15]. It accounts for dynamic changes of the catalyst and shows good agreement with steady state and dynamic experimental data from Vollbrecht [16].

## 2. Kinetic Model

_{2}, H

_{2}over a commercial Cu/ZnO/Al

_{2}O

_{3}catalyst considered in this paper comprises the following three reactions

_{2}. The third reaction is the reverse water–gas shift reaction. For the computations in this paper, we use the simplified reaction kinetics presented in [14,15]. It assumes a Langmuir–Hinshelwood mechanism with three different active surface centers. The resulting expressions for the three reaction rates are:

- i
- : ⊙ for oxidized surface centers, also assumed as active center for CO hydrogenation;
- ii
- : * for reduced surface centers, also assumed as active center for CO
_{2}hydrogenation; - iii
- : ⊗ as active surface centers for heterolytic decomposition of hydrogen.

## 3. Reactor Model

_{3}OH, CO

_{2}, CO, H

_{2}, H

_{2}O, N

_{2}, total number ${N}_{k}=6$) and the subscript j describes the reaction (total number ${N}_{r}=3$). The superscript G denotes the gas phase and S the solid phase. Under the abovementioned assumptions, the model equations follow from the overall material balances of each component i according to

## 4. Methods

#### 4.1. Steady State Optimization

#### 4.2. Optimization of Forced Periodic Operation

#### 4.3. Multi-Objective Optimization

## 5. Results

#### 5.1. Steady State Multi-Objective Optimization

_{cat}with a yield of 61%. At the Pareto optimal steady state operating point OP4, the methanol flow rate is 413 mmol/min/kg

_{cat}with a yield of 52%. The corresponding optimal steady state feed concentrations are shown in the first three diagrams of Figure 2. It can be seen from Figure 2 that the optimal feed contains more CO than CO

_{2}. This is a well-known fact, which follows from the equilibrium limitations of the reaction network and in particular the water inhibition for methanol production (see, e.g., Vollbrecht [16] and references therein). The optimal CO and CO

_{2}concentrations in the feed are continuously decreasing from the left to the right along the Pareto front.

#### 5.2. Multi-Objective Optimization of Forced Periodic Operation

## 6. Conclusions

_{2}O

_{3}catalyst can be improved significantly by periodic forcing of the CO feed concentration and the phase-shifted feed flow rate compared to optimal steady state operation. Improvements were measured in terms of methanol flow rate and methanol yield relative to the total carbon in the feed. The focus was on harmonic forcing functions, and a well-mixed isothermal CSTR. Due to the complexity of the methanol synthesis using a Cu/ZnO/Al

_{2}O

_{3}catalyst, the isothermal CSTR should be considered as a first reasonable step to provide some fundamental insight into the effect of forced periodic operation on methanol synthesis.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${A}_{x}$ | amplitude of input CO, N_{2} or volumetric flow rate |

$\Delta G$ [J/mol] | Gibbs free energy |

J | objective function |

${k}_{j}$ [mol/s/kg_{cat}/bar^{3}] | reaction rate constant for $j=1,2$ |

${k}_{3}$ [mol/s/kg_{cat}/bar] | reaction rate constant for $j=3$ |

${k}_{1}^{+},{k}_{2}^{+}$ [1/s] | reaction rate constant for oxidation and reduction of the catalyst |

${K}_{i}$ | adsorption constant |

${K}_{Pj}$ [1/bar^{2}] | equilibrium constant for $j=1,2$ |

${K}_{P3}$$[-]$ | equilibrium constant for $j=3$ |

${K}_{1},{K}_{2}$ | equilibrium constant for oxidation and reduction of the catalyst |

$\dot{n}$ [mol/s] | molar flow rate |

${\dot{n}}_{i}$ [mol/s] | molar flow rate of component i |

p [bar] | pressure |

${p}_{i}$ [bar] | partial pressure of compenent i |

${q}_{sat}$ [mol/kg_{cat}] | specific number of surface centers |

R [J/K/mol] | gas constant |

${r}_{j}$ [mol/s/kg_{cat}] | rate of reaction j |

T [K] | temperature |

u | general input |

${V}_{G}$ [m^{3}] | volume of the gas phase in the reactor |

$\dot{V}$ [m^{3}/s] | volumetric flow rate |

x | optimization variables |

y | general output variables |

${y}_{i}$ | mole fraction of component i |

${Y}_{\mathrm{Carbon}}$ | yield of methanol based on total carbon in the feed |

Greek letters | |

$\mathsf{\Theta}$ | relative number of free surface centers |

$\tau $ [s] | period time |

${\varphi}_{\mathrm{max}}$ | maximum fraction of reduced centers on the catalyst surface |

$\varphi $ | fraction of reduced centers on the catalyst surface |

$\Delta \varphi $ | phase shift |

$\omega $ [1/s] | frequency |

Subscripts | |

periodic | forced periodic feed stream |

$SS$ | steady state |

0 | feed stream |

i | component ($i=1$ CH_{3}OH, $i=2$ CO_{2}, $i=3$ CO, $i=4$ H_{2}, |

$i=5$ H_{2}O, $i=6$ N${}_{2}$) | |

j | reaction ($j=1$ CO hydrogenation, $j=2$ CO_{2} hydrogenation, |

$j=3$ RWGS) | |

Superscripts | |

G | gas phase |

S | solid phase |

* | reduced surface center |

⊙ | oxidized surface center |

⊗ | reduced surface center |

Abbreviations | |

OP | Operating point |

## Appendix A. Partial Derivatives of the Isotherms of the Adsorbed Species

- (i)
- ⊙ oxidized surface centers: CH
_{3}OH, CO_{2}, CO - (ii)
- * reduced surface centers: CH
_{3}OH, CO_{2}, H_{2}, H_{2}O - (iii)
- ⊗ surface centers for heterolytic decomposition of hydrogen: H
_{2}

_{2}from the gas phase, is adsorbed and simultaneously decomposed into elementary hydrogen, so that in the solid phase only elementary hydrogen is present. This explains the square root in Equations (A3) and (A4).

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**Figure 1.**Pareto fronts for methanol synthesis: x: optimal steady state operation, □: optimal forced periodic operation, ◯: chosen operating points.

**Figure 2.**Optimized parameter values along the Pareto fronts in Figure 1: x: optimal steady state operation, □: optimal forced periodic operation. Subplots (

**a**–

**c**) show the optimized feed concentration of CO

_{2}, CO and H

_{2}. Subplots (

**d**–

**h**) show the optimized forcing parameters A${}_{\mathrm{CO}}$, A${}_{{\mathrm{N}}_{2}}$, A${}_{\mathrm{F}}$, period time $\tau $ and phase shift $\Delta \varphi $.

**Figure 3.**Initial values $\mathbf{y}\left(\mathbf{0}\right)$ for forced periodic operation along the Pareto fronts in Figure 1. Subplots (

**a**–

**f**) show the initial values of CH

_{3}OH, CO

_{2}, CO, H

_{2}, H

_{2}O and N

_{2}. Subplot (

**g**) shows the intial values of the relative amount of reduced surface centers $\varphi $. Subplot (

**h**) shows the initial values of the flow rate at the outlet.

Parameter | Value | Units |
---|---|---|

${A}_{k,CO}$ | 0.00673 | mol/kg_{cat}/s/bar^{3} |

${B}_{CO}$ | 26.4549 | - |

${A}_{k,C{O}_{2}}$ | 0.0430 | mol/kg_{cat}/s/bar^{3} |

${B}_{C{O}_{2}}$ | 1.5308 | - |

${A}_{k,RWGS}$ | 0.0117 | mol/kg_{cat}/s/bar^{3} |

${B}_{RWGS}$ | 15.6154 | - |

$\sqrt{{K}_{{H}_{2}}}$ | 1.1064 | $1/\sqrt{bar}$ |

${K}_{C{H}_{3}OH}^{*}$ | 0 | 1/bar |

${K}_{{H}_{2}O}$ | 0 | 1/bar |

${K}_{O}$ | 0 | - |

${K}_{CO}$ | 0.1497 | 1/bar |

${K}_{C{H}_{3}OH}^{\odot}$ | 0 | 1/bar |

${K}_{C{O}_{2}}^{*}$ | 0.0629 | 1/bar |

${K}_{C{O}_{2}}^{\odot}$ | 0 | 1/bar |

$\Delta {G}_{1}$ | 0.3357 × 10^{3} | J/mol |

$\Delta {G}_{2}$ | 21.8414 ×10^{3} | J/mol |

${k}_{1}^{+}$ | 7.9174 × 10 ^{−3} | 1/s |

${k}_{2}^{+}$ | 0.188 × 10^{−4} | 1/s |

${\varphi}_{max}$ | 0.9 | - |

Parameter | Value | Units |
---|---|---|

p | 60 | bar |

T | 473 | K |

${y}_{{\mathrm{N}}_{2},0}$ | 0.15 | - |

${\dot{V}}_{0}$ | 1.14 × 10^{−7} | m^{3}/s |

${V}^{G}$ | 1.03 × 10^{−4} | m^{3} |

${m}_{\mathrm{kat}}$ | 0.00395 | kg |

${q}_{\mathrm{sat}}$ | 0.98 | mol/kg |

**Table 3.**Operating conditions for presented operating points from Figure 1.

OP | ${\mathit{y}}_{{\mathbf{CO}}_{2},0,\mathbf{SS}}$ | ${\mathit{y}}_{\mathbf{CO},0,\mathbf{SS}}$ | ${\mathit{y}}_{{\mathbf{H}}_{2},0,\mathbf{SS}}$ | A${}_{\mathbf{F}}$ | A${}_{\mathbf{CO}}$ | A${}_{{\mathbf{N}}_{2}}$ | $\mathit{\tau}$ in s | $\Delta \mathit{\varphi}$ |
---|---|---|---|---|---|---|---|---|

OP1 | $0.0296$ | $0.184$ | $0.6453$ | - | - | - | - | - |

OP2 | $0.0266$ | $0.152$ | $0.6714$ | 1 | $0.9866$ | 1 | 34 | $0.1637$ |

OP3 | $0.0176$ | $0.1206$ | $0.7118$ | $0.97$ | 1 | $0.804$ | $18.5$ | $-0.253$ |

OP4 | $0.0314$ | $0.273$ | $0.5433$ | - | - | - | - | - |

OP5 | $0.0345$ | $0.238$ | $0.577$ | $0.9378$ | $0.6291$ | 1 | 18 | $-0.04$ |

OP6 | $0.0266$ | $0.152$ | $0.6714$ | 1 | $0.9866$ | 1 | 34 | $0.1637$ |

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**MDPI and ACS Style**

Seidel, C.; Nikolić, D.; Felischak, M.; Petkovska, M.; Seidel-Morgenstern, A.; Kienle, A.
Optimization of Methanol Synthesis under Forced Periodic Operation. *Processes* **2021**, *9*, 872.
https://doi.org/10.3390/pr9050872

**AMA Style**

Seidel C, Nikolić D, Felischak M, Petkovska M, Seidel-Morgenstern A, Kienle A.
Optimization of Methanol Synthesis under Forced Periodic Operation. *Processes*. 2021; 9(5):872.
https://doi.org/10.3390/pr9050872

**Chicago/Turabian Style**

Seidel, Carsten, Daliborka Nikolić, Matthias Felischak, Menka Petkovska, Andreas Seidel-Morgenstern, and Achim Kienle.
2021. "Optimization of Methanol Synthesis under Forced Periodic Operation" *Processes* 9, no. 5: 872.
https://doi.org/10.3390/pr9050872