Forced Periodic Operation of Methanol Synthesis in an Isothermal Gradientless Reactor

2.1 Kinetic Model

The calculations in this paper are based on the lumped kinetic model for the three essential reactions which take place in the heterogeneously catalyzed synthesis of methanol from synthesis gas. Details of this model are presented in ¹⁴, ¹⁵ with readjusted parameters from ¹⁶, ¹⁸ using the comprehensive set of steady-state and dynamic experimental data from ¹¹. These data were obtained over a wide range of operating conditions in a micro Berty reactor using a commercial Cu/ZnO/Al₂O₃ catalyst in the presence of 15 % of nitrogen in the feed gas used to quantify the degree of contraction. The model accounts for hydrogenation of CO and CO₂ and the reverse water-gas shift reaction according to:

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The lumped kinetic model assumes a Langmuir-Hinshelwood mechanism comprising the adsorption of reactants, surface reaction, and desorption of products. Adsorption and desorption are assumed to be in quasistatic equilibrium, which seems reasonable in view of the time constants of the dynamic processes to be considered subsequently, which lie in the range of 10 s with inert nitrogen up to 10 min without inert nitrogen.

The resulting expressions for the reactions rates r are:

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with the reformulated Arrhenius equation

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The corresponding surfaces are:

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Depending on the reducing/oxidizing potential of the reaction gas, oxidized sites are transformed reversibly into reduced sites and vice versa. This is described with an additional dynamic equation in the model ¹⁴, ¹⁵:

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The equilibrium constants can be expressed as function of free energies according to:

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The model describes with the parameters given in Tab. 1 the steady state and experimental data from ¹¹ with good accuracy. A comparison with dynamic data will be discussed in Sect. 3.

2.2 Reactor Model

The focus of this paper is on analyzing the performance of a gradientless isothermal CSTR corresponding to the lab-scale micro Berty reactor described in ¹¹. Under these assumptions, the model equations follow from the overall material balances of the different species in both phases according to:

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with

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Species to be considered are CH₃OH, CO₂, CO, H₂, H₂O, and N₂ for the cases with inert nitrogen. Assuming further a constant total pressure p, the molar outflow follows from the overall material balances (see ¹⁸ for a detailed discussion). Process parameters used in this contribution are given in Tab. 2.

In these equations, the right-hand side reflects the rates of changes due to inflow, outflow, and chemical reaction. On the left-hand side, and represent the mole numbers of component i in the gas and the solid phase, respectively. is the mole fraction of component i in the gas phase, and is the total coverage of component i at the different active centers of the solid phase. As already mentioned in the previous section, adsorption equilibrium is assumed between the solid and the gas phase. Under these assumptions, the coverages follow from the adsorption isotherms depending on the gas phase composition. Hereby, q_sat is the saturation adsorption capacity of the solid phase.

The evaluation of in Eq. 13 requires careful consideration and is described in detail in ¹⁸. The contribution of on the left-hand side of the material balance (13) is often neglected and will be considered in some more detail in the remainder. Obviously, it does not play a role at steady state where the time derivative on the left-hand side is equal to zero. Therefore, it affects only the dynamics. The strength of this effect depends essentially on the saturation capacity q_sat. This effect is illustrated in Fig. 1 for different values of q_sat and compared to the dynamic experimental data from ¹¹, where the carbon feed starts with a pure CO feed (y_CO = 12.6 %, y_H2 = 72.2 %, and y_N2 = 16 %) to establish an initial steady state.

After 140 min it is switched to pure CO₂ (y_CO2 = 11.9 %, y_H2 = 71.5 %, and y_N2 = 16.6 %) again to pure CO at 210 min and back at 270 min. Constant conditions over the whole run are the temperature (T = 523 K), the pressure (50 bar), and the space velocity w.r.t the inlet flow (240 mL min⁻¹). In this figure, ϕ is the fraction of reduced centers on the catalyst surface ¹⁴, ¹⁵. If is neglected (i.e., q_sat = 0 mol kg_cat⁻¹), the gas phase mole fractions change almost instantaneously in response to feed changes due to the small storage capacity of the gas phase. In contrast to this, for finite values of q_sat a significant lag is observed. A value of q_sat = 0.98 mol kg_cat⁻¹ was found to fit the experimental data best. For comparison also a storage value of q_sat = 5 mol kg_cat⁻¹, reflecting a catalyst material with very high storage capacity and therefore a slower response, is given. The effect of the saturation capacity on forced periodic operation will be further discussed below in Sect. 3.

2.3 Nonlinear Frequeny Response (NFR) Method

There are many ways of operating a system periodically, e.g., it is possible to periodically modulate different inputs or input combinations, using various forcing parameters, i.e., forcing frequency, input amplitude(s), and phase differences between the inputs. This richness of possible forcing strategies also produces a challenge to find which forcing strategy is to be used in order to achieve the highest possible improvement (see, e.g., ⁵, ^19-21). One of the answers to this challenge is to apply the NFR method, which offers a systematic approach to the search for the best scenario. The NFR method is an approximate, but reliable analytical tool for evaluating possible improvements, which should be used in early stages of process development before rigorous numerical simulation and before experimental investigation ¹⁶, ¹⁷, ²².

The NFR method is a theoretical and powerful tool, mathematically based on Volterra series, generalized Fourier transforms, and the concept of higher-order frequency response functions (FRFs) ⁶, ²², ²³. In practice, the nonlinear model of the investigated system is replaced by a set of frequency response functions of different orders. The derivation procedure of these FRFs is standardized ²⁴. Although these analytical derivations can be tedious and time-consuming, especially for complex systems, the numerical burden associated with the application of the NFR method is practically negligible in comparison to classical numerical approaches. The recent introduction of a software application for automatic derivation of the FRFs, the so-called computer-aided nonlinear frequency response (cNFR) method ²⁵, substantially facilitates the derivation of the needed FRFs.

If one or more inputs of a weakly nonlinear systems is/are periodically modulated around a previously established steady state, the frequency response of the system output is a complex periodic function ²⁶, obtained as a sum of the output steady-state value (y_s), the basis harmonic (y_I), an infinite number of higher harmonics (y_II, y_III,…) and a non-periodic (DC) term (y_DC) ⁶, ²², ²³:

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Based on the NFR method, the change of the reactor performances caused by forced periodic operations can be evaluated from the non-periodic (the so-called DC) component of the frequency response of the reactor.

Using the concept of higher-order FRFs, the DC component can be written as the following infinite series ²⁷:

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In Eq. 17, is the asymmetrical second-order (ASO) FRF, the asymmetrical fourth-order FRF, etc.

For weakly nonlinear systems, the significance of different terms in Eq. 17 decreases with the increase of the corresponding FRF order. As a consequence, the DC component can be approximated with its dominant term, which is proportional to the asymmetrical second-order function and the square of the input amplitude ⁶:

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Eq. 18 is the foundation of the NFR method for evaluating periodic operations with one modulated input. Therefore, for single input modulations of, e.g., input x, the DC component is proportional to the second-order asymmetrical frequency response function (ASO FRF) relating the output of interest (y) and the modulated input (x), meaning that for single input forced periodically operated chemical reactors it is enough to derive and analyze only one FRF ¹⁶.

For simultaneous modulation of two inputs, the DC component is evaluated based on three FRFs, where two of them are correlating the output of interest to the two modulated inputs separately and one cross FRF which correlates the output to both modulated inputs. It should be pointed out that the cross effect of two inputs strongly depends on the phase shift of simultaneous modulation of those two inputs and that it is possible to achieve high improvement for simultaneous modulation of two inputs even for the cases when separate input modulations would not lead to improvement at all. The influence of the forcing parameters (frequency, amplitudes, and phase shift of the input modulations) on the possible improvement can also be determined by the NFR method ²³.

The DC component of an output y, for the case when two inputs (e.g., x and z) are periodically modulated, can be given as a sum of the contributions of the DC component related to the single inputs (x and z) separately and the contribution of the DC component originating from the cross-effect of both inputs ⁷:

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For co-sinusoidal modulations of inputs x and z, with equal frequencies ω, input amplitudes A_x and A_z, respectively, and phase difference between them, the separate contributions of the two inputs to the DC component can be approximately evaluated from the corresponding asymmetrical second-order FRFs, in the same way as explained above:

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while the contribution of the cross-effect can be approximately evaluated in the following way ⁷, ²³:

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is the so-called cross ASO term, which correlates the output y with both modulated inputs (x and z). It is a function of both frequency and phase difference between the two modulated inputs, and is evaluated based on the cross asymmetrical second-order FRF (), in the following way:

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The overall DC component of the output y could be written as follows:

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and should be calculated for a chosen set of forcing parameters (forcing frequency, forcing amplitudes, and phase difference) ⁷, ²⁴. In principle, it is possible to find a set of forcing parameters resulting the periodic operation with highest improvement ⁷, ²³, ²⁴.

The phase difference is a crucial parameter for a periodic operation with simultaneous modulations of two inputs, considering that by choosing the optimal phase difference the cross-effect can be maximized ⁷, ¹⁰, ²³, ²⁴.

Results for methanol synthesis specific for the gradientless isothermal, isobaric, lab-scale micro Berty reactor will be discussed in Sect. 3.

2.4 Optimization Methods

For a rigorous evaluation of the potential of forced periodic operation compared to conventional steady state operation, one should compare the best possible steady states with the best possible forced periodic operation. This is done in this paper using multi-objective optimization. Objectives to be considered are the average methanol flow rate at the reactor outlet and methanol yield based on the total carbon feed.

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The evaluation of the objective functions for the NFR method is given in the Appendix. For the computation of Pareto optimal steady- state and forced periodic solutions, a benchmark problem with given temperature, pressure, volume, average volumetric feed flow rate F, catalyst mass m_cat and, for Sect. 3.1, average inert nitrogen concentration in the feed according to Tab. 2 is considered. The reactor dimension and the catalyst mass correspond to the experimental study of ¹¹, which is the starting point for a parallel ongoing new experimental study.

In order to compute the Pareto front (see, e.g., Fig. 2 as an example), in a first step, two single objective optimization problems are solved to generate the left (maximum methanol flow rate) and the right (maximum methanol yield based on total carbon feed) boundary points of the Pareto front using a multi-start heuristic. In the second step, the Pareto front between these extreme points is traced out using the ε-constraint approach ⁸, ⁹ for multi-objective optimization as described in some more detail in ¹⁸ as follows:

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Here, the solution of the previous point is used as an initial guess for the current point. If the solver fails to converge, the calculation is redone by perturbing the initial values until convergence is achieved.

In contrast to ¹⁸, the underlying single objective optimization problems were solved using a simultaneous approach by discretizing the model in time using finite differences with 500 equidistant time steps per period.

Parameters to be optimized for steady-state optimization are the feed composition y_i,0,SS. Parameters to be optimized for forced periodic operation are the mean values in the feed y_i,0,SS, the forcing amplitudes of CO A_CO and the volumetric feed flow rate A_F, the forcing frequency ω, and the phase shift Δϕ between the periodic forcing of the two inputs.

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For evaluating operation with an inert gas, the concentration of N₂ in the feed was also varied periodically in a countercurrent way to compensate the variation of CO and to meet the summation condition at any time t according to .

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For the industrially relevant operation without inert nitrogen this compensation was done by countercurrent periodic forcing of H₂.

It is important to note that the mean values y_i,0,SS for forced periodic operation are also optimized and are therefore not identical with the corresponding values for steady-state optimization. For the optimization without inert nitrogen the constraint for y_H2,0,SS is relaxed to 0.35 ≤ y_H2,0,SS ≤ 1.

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The single objective optimization problems were solved with Julia ²⁸, applying the JuMP ²⁹, a domain-specific modeling language for mathematical optimization. One of the major benefits of using JuMP is the capability of employing (forward mode) automatic differentiation (AD), which outperforms other (non-AD) algorithms in speed and accuracy ³⁰. The applied nonlinear solver is Ipopt 3.13.4.

In summary, it is found that the improved optimization methodology applied in this paper is much more robust and efficient than the methodology applied earlier in ¹⁸.

3.1 Optimization with Inert Nitrogen

Using first the NFR method it was found in preliminary studies that single input modulation does not provide potential for significant improvement of reactor performance in terms of the objectives (methanol flow rate and yield) introduced in the previous section ¹⁶. The best potential for improvement was found for simultaneous modulation of the flow rate and the CO feed concentration ¹⁷. Improvements of up to 33.51 % of methanol production (single objective) were reported and further studied with multi-objective optimization. The results of the approximate NFR method were validated with numerical simulation of the full blown model described above for selected operating points in ¹⁷.

Further comparison between multi-objective optimization with the NFR and the full model is depicted in Fig. 2 for the conditions described in ¹⁷. In this figure, optimal steady-state operation is compared to optimal forced periodic operation as predicted by the two different approaches. Again, very good agreement between both approaches is found together with a significant potential for improvement of the methanol production rate in the left part of Fig. 2a. The periodic time required is about 18 s and the optimal phase shift between flow rate and CO inlet concentration is close to zero as shown in Fig. 2b. Consequently, there is periodic change between high CO feed flow rate, and afterwards the flow rate is reduced and CO is reacted away. In contrast to this, on the right side of the diagram (high yield region) not much improvement is found ¹⁷, ¹⁸.

In the remainder, results are extended step by step using numerical optimization of the full model. In a first step, a constraint on the volumetric outflow of the system is added. This is required since methanol synthesis is volume reducing and the feed flow rate at the input becomes zero at the minimum for large amplitudes of A_F = 1 in Fig. 2b. Results are presented in Fig. 3a.

At steady state, outlet volumetric flow rate non-negative was always satisfied, so that the steady-state Pareto front marked by the black crosses does not change. The effect on the Pareto front of forced periodic operation is also relatively small compared to the previous Fig. 2. Again, big improvements in terms of the methanol flow rate are found in the left part of Fig. 3, and minor improvements in the right part. Due to the additional constraint, improvements are reduced a little with a maximum reduction of about 2 % on the left side of the figure compared to Fig. 2. Due to the additional constraint of a non-negative outlet volumetric flow rate, the corresponding optimal forcing amplitude A_F is now restricted to values lower than 1 in Fig. 3b. The operating conditions in Fig. 3a essentially correspond to the base case which was considered in ¹⁸.

However, more recent calculations with the improved optimization methodology presented in this paper reveal that there exists an additional branch of the Pareto front which also predicts significant improvements compared to steady-state operation in the high yield part of the Pareto plot. This second branch is illustrated in Fig. 4a in red. The existence of the second branch of the Pareto front can be explained with the presence of multiple local minima of the present nonconvex optimization problem (see, e.g., ³¹). Depending on the initial guesses either the green or the red branch in Fig. 4a is found.

After careful inspection using a refined multi-start approach no further branches were found. In principle, deterministic global optimization could be applied to prove that no better solutions exist than the red branch ³¹. This, however, was not done due to the high computational effort. It is worth noting that the optimal forcing parameters along this second red branch are completely different from the previous green branch as shown in the diagram of Fig. 4b. In particular, the periodic time is now in the range of minutes with an increasing phase shift as we proceed to higher yields. Both branches merge at a yield of about 0.63.

For constant average inert nitrogen concentration applied so far, the total inert nitrogen feed flow rate as the product of total feed flow rate times concentration is not constant and in particular in the left part of Figs. 3 and 4 different from the steady-state value. Therefore, finally an additional constraint on the overall inert nitrogen feed flow as the product between the total flow rate and the concentration was added to the optimization problem and the assumption of constant average inert nitrogen concentration of N_N2,0,SS = 0.15 was relaxed.

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Results are illustrated as blue curves in Fig. 5 together with the branches from Fig. 4. It is shown that the potential for improvement in the right part of the diagram is even higher than the red curve, whereas the potential for improvement in the left part of the diagram is reduced. Periodic times are in the range of minutes, similar to the red branch. Phase shift in particular in the left part of the diagram is higher compared to the previous red and green curves. Improvements using forced periodic operation for chosen operating points can be found in Tab. 3.

3.2 Optimization without Inert Nitrogen

As mentioned above, the operating conditions considered in the previous section were inspired by the lab-scale experiments of ¹¹. In these experiments, the changing amounts of the inert nitrogen were used to quantify the contraction of the gas mixture due to the stoichiometry of the reactions. In industrial processes, however, depending on the source of the feed mixture, often only very little or no inert nitrogen is present ², ⁴. Therefore, the focus in this section is on optimal steady-state operation compared to optimal forced periodic operation without inert nitrogen in the feed. Again, periodic forcing of the CO feed concentration and the total volumetric feed flow rate is applied. To satisfy the closure condition at any time point, now H₂, instead of N₂, is used for compensation.

Results are illustrated in Fig. 6. In general, methanol production rates are higher for both, steady-state and forced periodic operation, due to more reactants in the feed than for the case with inert nitrogen discussed in the previous section. Most improvement is found in the high yield part of Fig. 6a. There, the methanol yield can be improved through forced periodic operation for a given production rate, or the methanol flow rate for a given yield. The latter effect is particularly large in the high yield region. The methanol flow rate for the highest steady yield is almost doubled. Instead for low yields, in the left part of the diagram hardly any improvement is found. This situation is qualitatively similar to Fig. 5 with inert nitrogen and an additional constraint for the inert nitrogen feed flow rate. Again, the optimal periodic time is in the range of minutes and the optimal phase shift is in the range of π/2. Improvements using forced periodic operation for chosen operating points can be found in Tab. 4, which are in the some order of magnitude as the results with inert gas.

Further insight is provided by evaluating the corresponding average molar feed rates of the three different feed components in Fig. 7. Improvement of forced periodic operation is achieved, with increased feed amounts of CO and CO₂ and a reduced amount of H₂. This seems to be particularly attractive for power-to-methanol processes (see, e.g., ¹, ³, ³², where H₂ is generated via electrolysis with regenerative electrical energy and is therefore often more expensive. For the interpretation of the results it should be emphasized that both curves, i.e., the black for steady-state operation and the green for forced periodic operation, are optimal for the given objectives and constraints.

Finally, the effect of the saturation capacity of the catalyst q_sat on forced periodic operation is discussed. Results are presented in Fig. 8. As q_sat increases, the Pareto fronts of forced periodic operation tend towards the Pareto front of steady-state operation and improvements are getting smaller and smaller. This is consistent with the observations in Sect. 2.2, where it was shown that a finite saturation capacity of the solid phase introduces an additional lag. Such a lag will lead to a dampening of the forced periodic solution bringing it closer to steady-state operation. Knowledge regarding the adsorption saturation capacity of the catalyst q_sat is therefore an important quantity for a reliable prediction of the potential for improvement through applying am optimized forced periodic operation. It appears to be essential to tune this catalyst property for an efficient periodic production of methanol.