Understanding Catalysis—A Simplified Simulation of Catalytic

chemengineering

Article Understanding Catalysis—A Simpliﬁed Simulation of Catalytic Reactors for CO2 Reduction

Jasmin Terreni 1,2, Andreas Borgschulte 1,2,* , Magne Hillestad 3 and Bruce D. Patterson 1,4

1 Laboratory for Advanced Analytical Technologies, Empa, CH-8600 Dübendorf, Switzerland; [email protected] (J.T.); [email protected] (B.D.P.) 2 Department of Chemistry, University of Zurich, CH-8057 Zurich, Switzerland 3 Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway; [email protected] 4 Department of Physics, University of Zurich, CH-8057 Zurich, Switzerland * Correspondence: [email protected]; Tel.: +41-58-765-46-39

Received: 11 August 2020; Accepted: 10 November 2020; Published: 20 November 2020

Abstract: The realistic numerical simulation of chemical processes, such as those occurring in catalytic reactors, is a complex undertaking, requiring knowledge of chemical thermodynamics, multi-component activated rate equations, coupled flows of material and heat, etc. A standard approach is to make use of a process simulation program package. However for a basic understanding, it may be advantageous to sacrifice some realism and to independently reproduce, in essence, the package computations. Here, we set up and numerically solve the basic equations governing the functioning of plug-flow reactors (PFR) and continuously stirred tank reactors (CSTR), and we demonstrate the procedure with simplified cases of the catalytic hydrogenation of carbon dioxide to form the synthetic fuels methanol and methane, each of which involves five chemical species undergoing three coupled chemical reactions. We show how to predict final product concentrations as a function of the catalyst system, reactor parameters, initial reactant concentrations, temperature, and pressure. Further, we use the numerical solutions to verify the “thermodynamic limit” of a PFR and a CSTR, and, for a PFR, to demonstrate the enhanced efficiency obtainable by “looping” and “sorption-enhancement”.

Keywords: CO2 reduction; methanol; methane; thermodynamics; kinetics; reactor design

1. Introduction Serious catalytic reactor design is a complex task involving the coupled phenomena of chemical thermodynamics, multi-step chemical reactions, hydrodynamic ﬂowand the generation, conduction, and dissipation of heat. Sophisticated computer software packages [1,2] are widely used to aid the reactor designer, but it has been argued that the “black-box” results they provide may obscure fundamental relationships, which are important for a more basic understanding. Quoting Reference [3], “A potential pedagogical drawback to simulation packages such as HYSYS and ASPEN is that it might be possible for students to successfully construct and use models without really understanding the physical phenomena within each unit operation. ... . Care must be taken to insure that simulation enhances student understanding, rather than providing a crutch to allow them to solve problems with only a surface understanding of the processes they are modeling”. Most chemical engineering textbooks [4–9] treat the general principles of catalytic reactor operation in terms of a set of coupled diﬀerential equations describing the creation and annihilation of chemical components. These equations are generally highly non-linear, hence requiring numerical techniques for their solution. Textbooks then either treat particularly simple reaction schemes, which do allow analytical solution, or plot numerically

ChemEngineering 2020, 4, 62; doi:10.3390/chemengineering4040062 www.mdpi.com/journal/chemengineering ChemEngineering 2020, 4, 62 2 of 16 computed results which the average student is unable to reproduce. Only in exceptional cases does a textbook assist the student in generating a suitable computer program to solve non-linear equations [7]. In this work, we demonstrate how one may simulate the simplified operation of a catalytic reactor using basic thermodynamic data, a kinetic model for the multi-step reactions, and numerical solutions of reactor-specific differential equations describing the evolution from reactant to product chemical species.ChemEngineering We focus 2020, 4 our, x FOR attention PEER REVIEW on the two archetypes of continuous-flow reactors: 2 of 17 the plug flow reactor (PFR) and the continuously stirred tank reactor (CSTR) [5]. To simplify the discussion, exceptional cases does a textbook assist the student in generating a suitable computer program to we assume constantsolve non-linear and equations uniform [7]. reaction temperature and pressure and we neglect the issues of heat flow and pressureIn this work, drop. we Furthermore,demonstrate how weone may assume simulate that the all simplified reactant operation and productof a catalytic species behave as ideal gases.reactor In theusing main basic thermodynamic text, we explain data, a kinetic how mo thedel relevant for the multi-step equations reactions, are and set numerical up to describe the solutions of reactor-specific differential equations describing the evolution from reactant to product evolution of chemicalchemical species. concentrations We focus our attention in the reactoron the two and archetypes we plot of continuous-flow and discuss reactors: their numericalthe plug solutions. Student exercisesflow reactor presented (PFR) and in the the continuously Supplementary stirred tank Materials reactor (CSTR) instruct [5]. To simplify the reader the discussion, in the creation of a we assume constant and uniform reaction temperature and pressure and we neglect the issues of heat computer program,flow and based pressure on drop. a variable-step Furthermore, we Runge–Kutta assume that all reactant integration and product method species [10 behave], to solve as non-linear differential equations.ideal gases. In the main text, we explain how the relevant equations are set up to describe the A schematicevolution diagram of chemical of the concentrations simulation in the procedure reactor and we is plot shown and discuss in Figure their numerical1. Once solutions. the overall relevant Student exercises presented in the Appendix instruct the reader in the creation of a computer chemical reactionsprogram, have based been on a defined,variable-step basic Runge–Kutta thermodynamics integration method dictates [10], the to solve Gibbs non-linear free energy change and hence thedifferential equilibrium equations. constants. From these, the temperature- (T) and pressure- (P) dependent equilibrium stateA is schematic determined diagram in of the simulati thermodynamicon procedure is shown limit—i.e., in Figure after 1. Once infinite the overall elapsed relevant reaction time. chemical reactions have been defined, basic thermodynamics dictates the Gibbs free energy change Dynamic reactorand simulationhence the equilibriu requiresm constants. knowledge From these, of thethe temperature- concentration-, (T) and Tpressure--, and P(P-dependent) dependent production rates of the individualequilibrium state chemical is determined species. in the thermodynamic This information limit—i.e., after is typically infinite elapsed contained reaction time. in a published Dynamic reactor simulation requires knowledge of the concentration-, T-, and P-dependent “kinetic model”,production which rates depends of the individual on the catalyst chemical used.species. OnceThis information the reactor is typically type (PFRcontained or CSTR)in a is defined, coupled differentialpublished equations “kinetic model”, describing which depends the on chemical the catalyst component used. Once the flowsreactor type are (PFR set up.or CSTR) These equations are then numericallyis defined, solvedcoupled differential to yield equations the time describin or position-dependentg the chemical component concentration flows are set up. These of each chemical equations are then numerically solved to yield the time or position-dependent concentration of each component. Achemical useful component. check of Athe useful computation check of the comput procedure,ation procedure, including including the the kinetic kinetic model, model, is is to extend the dynamic simulationto extend the todynamic infinite simulation elapsed to time, infini whichte elapsed should time, which reproduce should thereproduce thermodynamic the limit obtained earlier.thermodynamic limit obtained earlier.

Figure 1. Schematic diagram illustrating the catalytic reactor simulation procedure. Orange boxes Figure 1. Schematic diagram illustrating the catalytic reactor simulation procedure. Orange boxes indicate the externalindicate the input external of input information. of information.

We demonstrate the usefulness of this approach to reactor simulation and hopefully motivate further explorationChemEngineering by 2020 the, 4, x; reader doi: FOR PEER by REVIEW examining the enhanced www.mdpi.com/journal/chemengineering efficiency of two modifications of the PFR, which effectively shift the thermodynamic equilibrium: product separation and the removal/recycling of unreacted species in a “looped” reactor [7], and product removal by selective absorption (“sorption enhancement”) [11]. As mentioned, a series of progressively more challenging student exercises which review and develop the concepts treated in the main text is included, with answers, as an Supplementary Materials. ChemEngineering 2020, 4, 62 3 of 16

2. Materials and Methods Equilibrium constants for the chemical reactions considered were either computed from thermodynamic data on the Gibbs free energy change [12] or taken from the literature [13,14], and the kinetic rate factors were obtained from published models of experimental data [13–15]. The numerical computations were performed with Wolfram Mathematica, Version 11.3 [16].

3. Results and Discussion

3.1. CO2 Hydrogenation to Methanol and Methane In order to demonstrate our simulation of catalytic reactors by the setup and numerical solution of kinetic equations, we have chosen as chemical processes the reduction of carbon dioxide by hydrogenation to form the synthetic fuels methanol and methane [17,18]. Note that the production of methane in this fashion is also called “CO2 methanation”. Carbon-based fossil fuels remain the most important energy source worldwide due to the high chemical stability of their combustion product, carbon dioxide [19]. However, carbon dioxide is a major greenhouse gas causing global warming. Therefore viable alternatives to the burning of fossil fuels have to be found, and one option is the use of synthetic carbon-based fuels, produced using renewable energy, and carbon dioxide which has been recycled from natural or industrial processes [17,20]. The CO2 may then be converted to a fuel by catalytic hydrogenation [18,21,22]. The simplest carbon-based synthetic fuels are the C1 species formic acid (HCOOH), formaldehyde (HCHO), methanol (CH3OH), and methane (CH4) (Figure2). We note that CO2 can also be converted to higher hydrocarbons and alcohols by Fischer–Tropsch synthesis [18,23]. From Figure2, we can see that methanol and methane are promising candidates for synthetic carbon-based fuels. Note that the production reactions of both methanol and methane are spontaneous under standard conditions (∆Gred < 0) [24]. While methanol has the advantage of being a liquid at room temperature and hence has a high volumetric energy storage density, the gas methane oﬀers high gravimetric energyChemEngineering storage 2020, 4, [x25 FOR]. PEER REVIEW 4 of 17

Figure 2. AFigure Latimer–Frost-type 2. A Latimer–Frost-type diagram diagram [ 26[26]] showing,showing, as asa function a function of the degree of the of degreehydrogen of hydrogen reduction n(Hreduction2) and n(H at2)standard and at standard temperature temperature and and pre pressure,ssure, the change the in change Gibbs free in energy Gibbs ΔGred free upon energy ∆Gred production by CO2 hydrogenation and the change in enthalpy ΔHoxid upon combustion in oxygen for upon production by CO2 hydrogenation and the change in enthalpy ∆Hoxid upon combustion in the C1 chemicals formic acid (HCOOH), formaldehyde (HCHO), methanol (CH3OH), and methane oxygen for the C1 chemicals formic acid (HCOOH), formaldehyde (HCHO), methanol (CH OH), (CH4). The negative values of ΔGred for methanol and methane formation imply spontaneous 3 and methaneproduction (CH4). reactions, The negative and the large values values of of ΔH∆oxidG implyred for a high methanol capacity for and chemical methane energy storage. formation imply The thermodynamic data are from references [12,27]. spontaneous production reactions, and the large values of ∆Hoxid imply a high capacity for chemical

energy storage.The Thereduction thermodynamic of carbon dioxide data to are methanol from references is usually carried [12,27 ].out over a copper-zinc oxide catalyst at temperatures of approximately 200–300 °C and a pressure of several tens of bars [13,14,21]. Nickel is a practical catalyst for CO2 methanation, and the reaction is carried out at a few bars of pressure and temperatures of approximately 250–450 °C [15,21,28,29]. The important overall chemical reactions [14,15] involved in the gas phase hydrogenation of CO2 to CH3OH and CH4 are shown in Figure 3. In both cases, the production can either be direct (reaction 3) or can proceed via carbon monoxide as an intermediate (reactions 2 and 1); the reverse water gas shift (RWGS) reaction 2 competes for CO2 with direct hydrogenation. From thermodynamic arguments, we can conclude the following: (1) In contrast to the CO2 and CO hydrogenation reactions, the RWGS reaction 2 is endothermic. Therefore, increasing the reaction temperature will lead to an increase in the formation of CO and will consequently hinder the direct hydrogenation of CO2 to CH3OH or CH4. (2) Since, in both cases, reaction 3 involves a reduction in the number of moles, an increase in pressure will facilitate the direct hydrogenation of CO2 to CH3OH or CH4. These predictions follow from Le Chatelier’s principle [24].

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The reduction of carbon dioxide to methanol is usually carried out over a copper-zinc oxide catalyst at temperatures of approximately 200–300 ◦C and a pressure of several tens of bars [13,14,21]. Nickel is a practical catalyst for CO2 methanation, and the reaction is carried out at a few bars of pressure and temperatures of approximately 250–450 ◦C[15,21,28,29]. The important overall chemical reactions [14,15] involved in the gas phase hydrogenation of CO2 to CH3OH and CH4 are shown in Figure3. In both cases, the production can either be direct (reaction 3) or can proceed via carbon monoxide as an intermediate (reactions 2 and 1); the reverse water gas shift (RWGS) reaction 2 competes for CO2 with direct hydrogenation. From thermodynamic arguments, we can conclude the following: (1) In contrast to the CO2 and CO hydrogenation reactions, the RWGS reaction 2 is endothermic. Therefore, increasing the reaction temperature will lead to an increase in the formation of CO and will consequently hinder the direct hydrogenation of CO2 to CH3OH or CH4. (2) Since, in both cases, reaction 3 involves a reduction in the number of moles, an increase in pressure will facilitate the direct hydrogenation of CO2 to CH3OH or CH4. These predictions follow ChemEngineeringfrom Le Chatelier’s 2020, 4, x principle FOR PEER [ REVIEW24]. 5 of 17

Figure 3. Gas phase CO2 reduction to methanol (a) and methane (b)[14,15]. Reaction 3, in each case is Figure 3. Gas phase CO2 reduction to methanol (a) and methane (b) [14,15]. Reaction 3, in each case the direct hydrogenation of CO2. Reaction 2 is the reverse water gas shift (RWGS) reaction, and reaction is the direct hydrogenation of CO2. Reaction 2 is the reverse water gas shift (RWGS) reaction, and 1 is the hydrogenation of carbon monoxide. Standard enthalpies of formation are from Swaddle [12]. reaction 1 is the hydrogenation of carbon monoxide. Standard enthalpies of formation are from SwaddleThe equilibrium [12]. state in the conversion of carbon dioxide to methanol or methane is deﬁned by thermodynamics and can be determined using the temperature-dependent equilibrium constants Keq.The These equilibrium are, in turn, state determined in the conversion by the change of carbon in Gibbsdioxide free to methanol energy at or a given methane temperature is defined [ 24by]. thermodynamics and can be determined using the temperature-dependent equilibrium constants Keq For example, for the direct conversion of CO2 to CH3OH in reaction 3 in Figure3, we have: . These are, in turn, determined by the change in Gibbs free energy at a given temperature [24]. For 0 0 0 " ∆H0 ∆H0 +3∆H0 +∆H0 # example, for theeq direct conversion∆G of CO2∆ toH +CHT∆S3OH in reactionCH 3OH in FigureH O 3, weH have:CO ( ) = − 3 = − 3 3 = − 3 − 2 2 2 K3 T exp RT exp RT exp RT ∆ ∆ ∆ ∆ ∗ ∆ ∆ ∆ 𝐾 (𝑇) =exp =exp" 0 0 =𝑒𝑥𝑝0 0 # ∗ (1) S +S 3S S CH3OH H2O− H2 − CO2 exp R , (1) 𝑒𝑥𝑝 , 0 0 wherewhere TT isis the the absolute absolute temperature, temperature, R isis the the gas gas constant, constant, and and Δ∆H0 andand SS0 areare the the standard standard enthalpy enthalpy andand entropy entropy of of formation formation of of the the correspondin correspondingg reactant reactant or product species (Table 1 1).).

Table 1. Standard enthalpy of formation and entropy of involved species for CO2 reduction to form methanol and methane [12].

ΔH0 [kJ/mol] S0 [J/mol K] CO −110.52 197.67 CO2 −393.51 213.74 H2 0 130.68 H2O −241.82 188.82 CH3OH −200.66 239.81 CH4 −74.81 186.26

In this way, we arrive at the equilibrium constants for the reactions of Figure 3, as plotted as a function of inverse temperature in Figure 4 for methanol formation (in red) and for methane formation (in blue). The equilibrium constant for the reverse water gas shift reaction (reaction 2) is plotted in green.

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Table 1. Standard enthalpy of formation and entropy of involved species for CO2 reduction to form methanol and methane [12].

∆H0 [kJ/mol] S0 [J/mol K] CO 110.52 197.67 − CO 393.51 213.74 2 − H2 0 130.68 H O 241.82 188.82 2 − CH OH 200.66 239.81 3 − CH 74.81 186.26 4 −

In this way, we arrive at the equilibrium constants for the reactions of Figure3, as plotted as a function of inverse temperature in Figure4 for methanol formation (in red) and for methane formation ChemEngineering 2020, 4, x FOR PEER REVIEW 6 of 17 (in blue). The equilibrium constant for the reverse water gas shift reaction (reaction 2) is plotted in green.

Figure 4. Equilibrium constants for methanol (red) and methane (blue) production reactions as a functionFigure of4. theEquilibrium inverse temperature. constants for The methanol green line (red) shows and the methane equilibrium (blue) constant production of the reversereactions water as a gasfunction shift reaction. of the inverse Note the temperature. substantially The higher green values line forshows methane the equilibrium production constant compared of to the those reverse for methanolwater gas and shift that reaction. the endothermic Note the RWGSsubstantially reaction hi isgher enhanced values withfor methane increasing production temperature. compared to those for methanol and that the endothermic RWGS reaction is enhanced with increasing Becausetemperature. the three reactions in each set are coupled, only two of the three equilibrium constants are independent: eq eq eq Because the three reactions in eachK ( setT) areK couple(T) = Kd, only(T). two of the three equilibrium constants (2) 1 ∗ 2 3 are independent: 3.2. Thermodynamic Equilibrium 𝐾 (𝑇) ∗𝐾 (𝑇) = 𝐾 (𝑇). (2) At a given temperature and pressure, the thermodynamic yield of a reaction is the equilibrium result that is approached after an inﬁnite elapsed time. Because of their coupling (Equation (2)), 3.2. Thermodynamic Equilibrium we need only to consider two of the three reactions. The thermodynamic yield is determined by equatingAt a the given equilibrium temperature constant and pressure, with the the corresponding thermodynamic “reaction yield quotient”of a reaction [24 is]. the In theequilibrium case of methanolresult that synthesis, is approached we consider after an reactions infinite 2elapsed and 3 of time. Figure Because3 to arrive of their at the coupling following (Equation expressions: (2)), we need only to consider two of the three reactions. The thermodynamic yield is determined by equating N NH o the equilibrium constant with the correspondingeq( ) = CO “r∗eaction2 quotient” [24]. In the case of methanol K2 T , (3) NH N synthesis, we consider reactions 2 and 3 of Figure 32 to∗ arriveCO2 at the following expressions: ∗ 𝐾 (𝑇) = , (3) ∗

∗ ∗ ∗ 𝐾(𝑇) = . ∗ ∗ (4)

The reaction quotients are determined by the reduced partial pressures pj or molar concentrations Nj of the reactant and product species j, the reaction pressure P, and the atmospheric pressure P0. The molar concentrations, in turn, are related to the degrees of completion ξi of the individual reactions i. For the case of methanol synthesis, the relationships between the equilibrium molar concentrations Nj and the degrees of completion ξi are given by [30]:

𝑁 =𝜉 ∗𝑁, (5)

( ) 𝑁 = 1−𝜉 −𝜉 ∗𝑁, (6)

( ) 𝑁 = 𝑆𝑁+1−𝜉 −3𝜉 ∗𝑁, (7)

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2 2 eq NCH3OH NH2O N P K (T) = ∗ ∗ tot ∗ 0 . (4) 3 3 2 N NCO P H2 ∗ 2 ∗ The reaction quotients are determined by the reduced partial pressures pj or molar concentrations Nj of the reactant and product species j, the reaction pressure P, and the atmospheric pressure P0. The molar concentrations, in turn, are related to the degrees of completion ξi of the individual reactions i. For the case of methanol synthesis, the relationships between the equilibrium molar concentrations Nj and the degrees of completion ξi are given by [30]:

0 NCO = ξ2 N , (5) ∗ CO2 0 NCO = (1 ξ2 ξ3) N , (6) 2 − − ∗ CO2 ChemEngineering 2020, 4, x FOR PEER REVIEW 7 of 17 0 NH = (SN + 1 ξ2 3ξ3) N , (7) 2 − − ∗ CO2 0 NH O𝑁= (ξ=2 +(𝜉ξ 3+𝜉) N) ∗𝑁, , (8)(8) 2 ∗ CO2 0 N = N CH3𝑁OH ξ3=𝜉 CO∗𝑁, (9) ∗ 2 , (9) 0 Ntot = NCO + NCO + NH + NH O + NCH OH = (SN + 2 2ξ3) N . (10) 2 2 2 3 ( − ∗) CO2 𝑁 =𝑁 +𝑁 +𝑁 +𝑁 +𝑁 = 𝑆𝑁 + 2 − 2𝜉 ∗𝑁. (10) Here, Ntot is the total molar concentration and SN is the “stoichiometric number” [31], defined as Here, Ntot is the total molar concentration and SN is the “stoichiometric number” [31], defined as the ratio between the difference of the initial molar concentrations of hydrogen and carbon dioxide the ratio between the difference of the initial molar concentrations of hydrogen and carbon dioxide and the sum of the initial concentrations of CO and CO2: and the sum of the initial concentrations of CO and CO2: 0 0 N N 𝑆𝑁H2 − = CO2. SN = . (11)(11) N0 + N0 CO CO2 In the present work, we assume no initial concentration of carbon monoxide (𝑁= 0). For ideal 0 conditions,In the present SN = work,2 for CO we2 assumereduction no to initial methanol concentration and SN = of 3 carbonfor the monoxidereduction to (N methane.CO= 0). For ideal conditions,The SNspecification= 2 for CO of2 reductionSN, T, and to P methanol allows us and to numericallySN = 3 for the solve reduction Equations to methane. (5)–(10) for the two unknowns,The specification ξ2 and ξ of3, SNwhere, T, andξ3 representsP allowsus the to degree numerically of conversion solve Equations of CO2 (5)–(10)to CH3OH. for theA similar two unknowns,procedureξ 2canand beξ 3applied, where ξto3 representstreat CO2 reduction the degree to of methane conversion (see of Exercises CO2 to CH 3–5).3OH. The A similarresulting procedureequilibrium can conversions be applied tofor treatthe CO CO2 2reductionreduction to tomethanol methane and (see methane, Exercises with 3–5). the ideal The resultingSN values, equilibriumare shown conversions as a function for of theT and CO P2 reductionin Figure 5. to methanol and methane, with the ideal SN values, are shown as a function of T and P in Figure5.

Figure 5. Equilibrium molar conversion for the reduction of CO2 to methanol (a) and to methane (b), as aFigure function 5. Equilibrium of temperature molar and conversion at diﬀerent for pressures. the reduction of CO2 to methanol (a) and to methane (b), as a function of temperature and at different pressures.

As predicted by Le Chatelier’s principle, the equilibrium conversion of CO2 to methanol or methane increases with increasing pressure and decreases with increasing temperature. The results in Figure 5 are in good agreement with published studies for methanol [30] and methane [32,33] synthesis.

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As predicted by Le Chatelier’s principle, the equilibrium conversion of CO2 to methanol or methane increases with increasing pressure and decreases with increasing temperature. The results in Figure5 are in good agreement with published studies for methanol [ 30] and methane [32,33] synthesis.

3.3. Kinetic Behavior in a Continuous Flow Catalytic Reactor In a practical chemical reactor, a catalyst is used to selectively accelerate the desired reaction. It should be noted that the presence of a catalyst cannot by itself increase the reaction yield beyond that given by thermodynamics; by effectively lowering the pertinent potential energy barrier, it can only increase the rate at which a reaction proceeds [34,35]. A more realistic treatment of a chemical process than that provided by equilibrium thermodynamics requires the analysis of the kinetic behavior, which, besides the choice of catalyst, depends on the reactor geometry [4]. In the pharmaceutical industry, a “batch reactor” is often used to repeatedly process limited amounts of material. Here, we consider the kinetic behavior of CO2 hydrogenation in the two archetypical “continuous flow” reactor types: the “plug flow reactor” (PFR) and the “continuously stirred tank reactor” (CSTR). The kinetics of the reduction of CO2 to methanol over a Cu/ZnO/Al2O3 catalyst have been modeled by Graaf et al. [13,14]. By analyzing the important reaction intermediates and determining the rate-limiting steps, these authors find the following expressions for ri, the rates of the three reactions in Figure3a, and hence for R a 5-component vector giving the net production rates for the individual chemical species:      3 pCH OH   2 3  r1 = k1KCOpCO p /denom, (12)  ∗ H2 − 1 eq   P 2 K  H2 ∗ 1    pH O pCO   2 ∗  r2 = k2KCO pCO pH2 /denom, (13) 2  2 ∗ − eq  K2      3 pCH OH pH O   2 3 2  r3 = k3KCO pCO p ∗ /denom, (14) 2  2 ∗ H2 − 3 eq   p 2 K  H2 ∗ 3          1 KH O    2  2   denom = 1 + KCO pCO + Kco2 pCO p +   pH O, (15) ∗ ∗ 2 ∗  H2  1  ∗ 2    K 2   H2 R = R , R , RH , R , R = ( r + r , r r , 2r r 3r , r + r , r + r . (16) CO CO2 2 H2O CH3OH − 1 2 − 2 − 3 − 1 − 2 − 3 2 3 1 3 Like the equilibrium constants, the temperature-dependent kinetic factors in Equations (12)–(16) also have the Arrhenius form: ! bn Kn(T) = an exp . (17) ∗ R T ∗ The values presented by Graaf et al. for the Arrhenius parameters for the various factors in methanol production are given in Table2. The corresponding factors for methanation are given in Exercise 6 of the Supplementary Materials. Instead of the thermodynamically derived expressions for eq the equilibrium rate constants K j, we use for methanol synthesis the expressions from Graaf et al. [13,14] in Table2. ChemEngineering 2020, 4, 62 8 of 16

Table 2. Arrhenius parameters for the temperature-dependent factors in the model of the kinetics of methanol production of Graaf et al. [13,14].

h J i Variable a b mol eq 13 4 K 2.39 10− 9.84 10 1eq × × K 1.07 102 3.91 104 2eq × − × K 2.56 10 11 5.87 104 3 × − × k 4.89 107 mol 1.13 105 1 × kg s − × 11 mol 5 k2 9.64 10 1.53 10 × kg s − × 5 mol 5 k3 1.09 10 kg s 0.875 10 × 5 − × 5 KCO 2.16 10− 0.468 10 × 7 × 5 KCO2 7.05 10− 0.617 10 1/2 × 9 × 5 KH2O/KH2 6.37 10− 0.840 10 ChemEngineering 2020, 4, x FOR PEER REVIEW × × 9 of 17

3.3.1. Plug Flow Reactor 3.3.1. Plug Flow Reactor We first consider the plug flow reactor, where the reactants and products flow with a constant We first consider the plug flow reactor, where the reactants and products flow with a constant total mass flow rate through one or more parallel tubes filled with loosely packed catalyst. For our total mass flow rate through one or more parallel tubes filled with loosely packed catalyst. For our calculations, we make the simplifying assumptions: (1) that all reactants and products are ideal calculations, we make the simplifying assumptions: (1) that all reactants and products are ideal gases, gases, and (2) that the temperature and pressure are constant and uniform along the reactor tubes. and (2) that the temperature and pressure are constant and uniform along the reactor tubes. The The working principle and the corresponding method of numerical simulation for the PFR are given working principle and the. corresponding method of numerical simulation for the PFR are given in inFigure Figure 6. 6The. The vector vector 𝐍 Ndenotesdenotes the the molar molar flow flow rates rates (moles/s), (moles/s), the the components components of of which which are are the x position-dependent flows of CO, CO ,H ,H O, and CH OH. As the gases proceed along the reactor position-dependent flows of CO, CO22, H22, H2O, and CH33OH. As the gases proceed along the reactor tubes, the initial reactants CO and H are converted to the product species CO, H O, and CH OH. tubes, the initial reactants CO2 and H2 are converted to the product species CO, H22O, and CH3OH. The values used for thethe PFRPFR parametersparameters inin FigureFigure6 6 are are shown shown in in Table Table3. 3.

Figure 6. Computational scheme for a plug flow reactor, defining the function PFR, which relates the Figure 6. Computational scheme for a plug flow reactor, defining the function PFR, which relates the input and output component molar flow rates [5]. The component molecular weights are given by mj. input and output component molar flow rates [5]. The component molecular weights are given by mj.

Table 3. Plug flow reactor parameter values used in the present simulation of methanol and methane synthesis.

Parameter Methanol Synthesis Methane Synthesis Catalyst Cu/ZnO/Al2O3 Ni/MgAl2O4 Catalyst density ρcatalyst 1000 kg/m3 1000 kg/m3 Nr of parallel tubes ntubes 10,000 10 Tube diameter d 2 cm 2 cm Tube area Atube = ntube x πd2/4 3.14 m2 0.00314 m2 Tube length Ltube 3 m 1 m Initial flow velocity vflow 0.05 m/s 5 m/s Stoichiometric number SN 2 3 Temperature range T 180–340 °C 100–1000 °C Pressures P 20, 40, 60 bar 1, 10, 20 bar

The resulting temperature- and pressure-dependent degree of CO2 → CH3OH conversion is shown with solid curves in Figure 7 and is in qualitative agreement with previously published studies [36–38]. We highlight two important features: (1) at practical pressures, only a low degree of conversion to methanol (<0.25) is obtained. (2) The conversion predicted by the kinetic behavior cannot exceed the equilibrium value (dotted curves in Figures 5 and 7. We attribute the slight

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Table 3. Plug ﬂow reactor parameter values used in the present simulation of methanol and methane synthesis.

Parameter Methanol Synthesis Methane Synthesis

Catalyst Cu/ZnO/Al2O3 Ni/MgAl2O4 3 3 Catalyst density ρcatalyst 1000 kg/m 1000 kg/m Nr of parallel tubes ntubes 10,000 10 Tube diameter d 2 cm 2 cm Tube area A = n πd2/4 3.14 m2 0.00314 m2 tube tube × Tube length Ltube 3 m 1 m Initial ﬂow velocity vﬂow 0.05 m/s 5 m/s Stoichiometric number SN 2 3 Temperature range T 180–340 ◦C 100–1000 ◦C Pressures P 20, 40, 60 bar 1, 10, 20 bar

The resulting temperature- and pressure-dependent degree of CO CH OH conversion is shown 2 → 3 with solid curves in Figure7 and is in qualitative agreement with previously published studies [ 36–38]. WeChemEngineering highlight two 2020 important, 4, x FOR PEER features: REVIEW (1) at practical pressures, only a low degree of conversion 10 toof 17 methanol (<0.25) is obtained. (2) The conversion predicted by the kinetic behavior cannot exceed the overshoot of the kinetic data at 20 bar to slight inconsistencies in the parameter values of Graaf et al. equilibrium value (dotted curves in Figures5 and7. We attribute the slight overshoot of the kinetic (Table 2) [13,14]. data at 20 bar to slight inconsistencies in the parameter values of Graaf et al. (Table2)[13,14].

Figure 7. Comparison of the thermodynamic (dotted curves) and kinetic (solid curves) degrees of Figure 7. Comparison of the thermodynamic (dotted curves) and kinetic (solid curves) degrees of molar conversion, as a function of temperature and pressure, (a) for the reduction of CO2 to methanol molar conversion, as a function of temperature and pressure, (a) for the reduction of CO2 to methanol and (b) for CO2 methanation. The assumed values for the reactor parameters are given in Table3. and (b) for CO2 methanation. The assumed values for the reactor parameters are given in Table 3.

We leave as an exercise for the reader the setup of a similar kinetic simulation for CO2 methanation We leave as an exercise for the reader the setup of a similar kinetic simulation for CO2 (Exercises 6–10), using, for example, the kinetic model of Xu and Froment for a Ni/MgAl2O4 catalyst [15]. Wemethanation assume an ideal(Exercises stoichiometric 6–10), using, number for SNexampl= 3,e, again the kinetic with no model initial CO.of Xu As and indicated Froment by thefor a Ni/MgAl2O4 catalyst [15]. We assume an ideal stoichiometric number SN = 3, again with no initial equilibrium constants in Figure4, CO 2 methanation is much more rapid than the reduction of CO2 to methanol.CO. As indicated As a consequence, by the equilibr to obtainium constants the results in shownFigure in4, CO Figure2 methanation7b we have is modiﬁed much more the rapid reactor than parameterthe reduction values of fromCO2 to the methanol. methanol As case a consequence, (see Table3). to Again, obtain ourthe results kinetic shown results in are Figure in qualitative 7b we have agreementmodified withthe reactor the published parameter data values [39]. from the methanol case (see Table 3). Again, our kinetic results are in qualitative agreement with the published data [39]. By taking the tube length Ltube to be a variable, one may use the computation scheme of Figure6 to simulateBy the taking molar the ﬂow tube rates length of the Ltube individual to be a variable, chemical one components may use the as acomputation function of position scheme alongof Figure the 6 reactorto simulate tubes (Figurethe molar8). Theseflow rates position of the dependencies individual ch areemical comparable components to those as a found function previously of position [ 32 ,along40]. the reactor tubes (Figure 8). These position dependencies are comparable to those found previously [32,40].

Figure 8. Position-dependent component molar flow rates in a plug flow reactor. (a) CO2 reduction to methanol (40 bar, 230 °C, SN = 2); (b) CO2 methanation (10 bar, 400 °C, SN = 3).

In Figure 7, we observed that the molar degree of conversion achievable with a simple PFR cannot exceed the value predicted by equilibrium thermodynamics. By greatly extending the length (to impractical sizes), one should, however, approach the thermodynamic limit. In Figure 9, we show the PFR molar conversion for very long PFR tubes, and we confirm the approach to this limit.

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overshoot of the kinetic data at 20 bar to slight inconsistencies in the parameter values of Graaf et al. (Table 2) [13,14].

Figure 7. Comparison of the thermodynamic (dotted curves) and kinetic (solid curves) degrees of

molar conversion, as a function of temperature and pressure, (a) for the reduction of CO2 to methanol and (b) for CO2 methanation. The assumed values for the reactor parameters are given in Table 3.

We leave as an exercise for the reader the setup of a similar kinetic simulation for CO2 methanation (Exercises 6–10), using, for example, the kinetic model of Xu and Froment for a Ni/MgAl2O4 catalyst [15]. We assume an ideal stoichiometric number SN = 3, again with no initial CO. As indicated by the equilibrium constants in Figure 4, CO2 methanation is much more rapid than the reduction of CO2 to methanol. As a consequence, to obtain the results shown in Figure 7b we have modified the reactor parameter values from the methanol case (see Table 3). Again, our kinetic results are in qualitative agreement with the published data [39]. By taking the tube length Ltube to be a variable, one may use the computation scheme of Figure 6 to simulate the molar flow rates of the individual chemical components as a function of position along the reactor tubes (Figure 8). These position dependencies are comparable to those found previously ChemEngineering[32,40]. 2020, 4, 62 10 of 16

Figure 8. Position-dependent component molar flow rates in a plug flow reactor. (a) CO2 reduction to Figure 8. Position-dependent component molar flow rates in a plug flow reactor. (a) CO2 reduction to methanol (40 bar, 230 ◦C, SN = 2); (b) CO2 methanation (10 bar, 400 ◦C, SN = 3). methanol (40 bar, 230 °C, SN = 2); (b) CO2 methanation (10 bar, 400 °C, SN = 3). In Figure7, we observed that the molar degree of conversion achievable with a simple PFR In Figure 7, we observed that the molar degree of conversion achievable with a simple PFR cannotChemEngineering exceed the 2020 value, 4, x FOR predicted PEER REVIEW by equilibrium thermodynamics. By greatly extending the length 11 of 17 (tocannot impractical exceed sizes), the value one should, predicted however, by equilibrium approach thermodynamics. the thermodynamic By limit. greatly In Figureextending9, we the show length the(to PFR impractical molar conversion sizes), one for should, very long however, PFR tubes, approach and we the confirm thermodynamic the approach limit. to In this Figure limit. 9, we show the PFR molar conversion for very long PFR tubes, and we confirm the approach to this limit.

ChemEngineering 2020, 4, x; doi: FOR PEER REVIEW www.mdpi.com/journal/chemengineering

Figure 9. A demonstration that extending the tube length of a plug ﬂow reactor to impractically Figure 9. A demonstration that extending the tube length of a plug flow reactor to impractically large large sizes causes the degree of molar conversion of CO2 to CH3OH to approach the limit given bysizes equilibrium causes thermodynamicsthe degree of molar (dotted conversion curves). of The CO pressure2 to CH is3OH taken to toapproach be 20 bar, the and limit the given initial by stoichiometricequilibrium numberthermodynamicsSN = 2. (dotted curves). The pressure is taken to be 20 bar, and the initial stoichiometric number SN = 2. 3.3.2. Continuously Stirred Tank Reactor 3.3.2.In anContinuously ideal continuously Stirred Tank stirred Reactor tank reactor (CSTR), the environmental conditions in the reactorIn are an everywhere ideal continuously the same, stirred since tank the contents reactor (C (reactants,STR), the catalyst,environmental and products) conditions are in constantly the reactor mixed—e.g.,are everywhere by a mechanicalthe same, since stirrer. the Thecontents numerical (reactants, simulation catalyst, scheme and products) for the CSTR are constantly reaction kinetics mixed— is givene.g., by in a Figure mechanical 10. stirrer. The numerical simulation scheme for the CSTR reaction kinetics is given in Figure 10.

Figure 10. Computational scheme for a continuously stirred tank reactor, defining the function CSTR, which relates the input and output component molar flows [5].

The following points provide additional information regarding the computation scheme in Figure 10:

• The volume of the reactor tank is given by Vtank; • The reactants enter the tank with a flow velocity vflow through an inlet aperture, the area of which is Ainlet;

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Figure 9. A demonstration that extending the tube length of a plug flow reactor to impractically large

sizes causes the degree of molar conversion of CO2 to CH3OH to approach the limit given by equilibrium thermodynamics (dotted curves). The pressure is taken to be 20 bar, and the initial stoichiometric number SN = 2.

3.3.2. Continuously Stirred Tank Reactor In an ideal continuously stirred tank reactor (CSTR), the environmental conditions in the reactor ChemEngineeringare2020 everywhere, 4, 62 the same, since the contents (reactants, catalyst, and products) are constantly mixed— 11 of 16 e.g., by a mechanical stirrer. The numerical simulation scheme for the CSTR reaction kinetics is given in Figure 10.

Figure 10. ComputationalFigure 10. Computational scheme scheme for for a continuously a continuously sti stirredrred tanktank reactor, reactor, defining deﬁningthe function the CSTR function, CSTR, which relates the input and output component molar flows [5]. which relates the input and output component molar ﬂows [5]. The following points provide additional information regarding the computation scheme in The followingFigure 10: points provide additional information regarding the computation scheme in

Figure 10: • The volume of the reactor tank is given by Vtank; • The reactants enter the tank with a flow velocity vflow through an inlet aperture, the area of which The volumeis ofAinlet the; reactor tank is given by V ; • tank The reactantsChemEngineering enter 2020 the, 4, x; tank doi: FOR with PEERa REVIEW flow velocity v through www.mdpi.com/journal/chemengineering an inlet aperture, the area of which • flow is Ainlet; The remaining variables have the same meaning as for the PFR described in Figure6; • ChemEngineering 2020, 4, x FOR PEER REVIEW 12 of 17 The solution of the CSTR equations is self-consistent and requires the component production rates • Rj to be• evaluatedThe remaining at the variables exit of have the reactorthe same [meaning41]. as for the PFR described in Figure 6; • The solution of the CSTR equations is self-consistent and requires the component production For calculatingrates R thej to be catalytic evaluated conversion at the exit of the of COreactor2 to [41]. CH 3OH in a CSTR, we again use the kinetic model of GraafFor et al.calculating [13,14 ].the In catalytic Figure conversion 11, we showof CO2 to the CH degree3OH in a of CSTR, conversion we again use for the CO kinetic2 to CH3OH in a CSTR asmodel a function of Graaf ofet al. temperature [13,14]. In Figure and 11 tank, we show volume the degree (Vtank of). conversion The pressure for CO2is to constantCH3OH in ata 20 bar; the catalyst densityCSTR as a is function again 1000of temperature kg/m3; and, and tank as for volume the methanol (Vtank). The PFRpressure case, is constant the initial at 20 volumetric bar; the flow catalyst density is again 1000 kg/m3; and,3 as for the methanol PFR case, the initial volumetric flow is is taken to be vflow Ainlet = 3.14 0.05 m /s. The red curve in Figure 11 corresponds to a CSTR tank taken to be vflow × Ainlet = 3.14 × 0.05 m3/s. The red curve in Figure 11 corresponds to a CSTR tank volume × × 2 volume equal to that of the PFR tubes (A L ), with A 2 = 3.14 m and L = 3 m. The blue, equal to that of the PFR tubes (Atubes tubes× Ltube×), withtube Atubes = 3.14 mtubes and Ltube = 3 m. The blue,tube orange, and orange, andgreen green curves curves correspond correspond to larger to tank larger volumes tank (Atubes volumes × Ltube, with (A Atubestube constantLtube but, with varyingAtube Ltubeconstant = 10, but 20, 50, and 100 m, respectively). In analogy with the results of an extra-long× PFR reactor (Figure 9), varying Ltube = 10, 20, 50, and 100 m, respectively). In analogy with the results of an extra-long PFR 2 → 3 reactor (Figurewe 9see), we from see Figure from 11 Figure that, with 11 that,increasi withng the increasing CSTR tank the volume, CSTR the tank degree volume, of CO the CH degreeOH of CO conversion increases and approaches the thermodynamic equilibrium value (dotted curve). 2 CH OH conversion increases and approaches the thermodynamic equilibrium value (dotted curve). → 3

Figure 11. CO2 CH3OH degree of molar conversion for a CSTR (P = 20 bar) as a function of the Figure→ 11. CO2 → CH3OH degree of molar conversion for a CSTR (P = 20 bar) as a function of the reactor temperaturereactor temperature and tank and volume. tank volume. In analogyIn analogy withwith the the case case for forthe extra-long the extra-long PFR (Figure PFR 9), (Figure at 9), at large valueslarge of valuesVtank theof Vtank CSTR the CSTR conversion conversion approaches approaches that predicted predicted by byequilibrium equilibrium thermodynamics thermodynamics (dotted curve).(dotted curve).

3.4. Modified Plug Flow Reactor

Figure 7 illustrates that the degree of conversion in a PFR, particularly in the case of CO2 hydrogenation to CH3OH, is severely limited by thermodynamics. In an attempt to circumvent this limitation, we now simulate two modifications of a PFR—namely, “looping” [7] and “sorption- enhancement” [11]—which have the effect of shifting the thermodynamic equilibrium to allow a more efficient reactor operation. In both cases, this shift of equilibrium is achieved by the continuous removal of reaction products, thus reducing the probability of a back-reaction.

3.4.1. Looped Plug Flow Reactor with Recycling A well-established method for compensating for the low conversion in methanol synthesis is PFR “looping” [7]. In contrast to a single-pass PFR, a looped PFR recycles the unreacted H2, CO2, and CO, as illustrated in the corresponding computation scheme shown in Figure 12.

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3.4. Modiﬁed Plug Flow Reactor

Figure7 illustrates that the degree of conversion in a PFR, particularly in the case of CO 2 hydrogenation to CH3OH, is severely limited by thermodynamics. In an attempt to circumvent this limitation, we now simulate two modifications of a PFR—namely, “looping” [7] and “sorption-enhancement” [11]—which have the effect of shifting the thermodynamic equilibrium to allow a more efficient reactor operation. In both cases, this shift of equilibrium is achieved by the continuous removal of reaction products, thus reducing the probability of a back-reaction.

3.4.1. Looped Plug Flow Reactor with Recycling A well-established method for compensating for the low conversion in methanol synthesis is PFR “looping” [7]. In contrast to a single-pass PFR, a looped PFR recycles the unreacted H2, CO2, and CO, asChemEngineering illustrated in2020 the, 4, correspondingx FOR PEER REVIEW computation scheme shown in Figure 12. 13 of 17

Figure 12. Computational schemescheme ofof the the looped looped PFR. PFR. The The function function “PFR “PFR” refers” refers to theto the operation operation of the of previouslythe previously considered considered single-pass single-pass plug flowplug reactorflow reactor (Figure (Figure6). The 6). individual The individual molar flows molar are flows defined are indefined the Figure. in the Figure.

In thethe loopedlooped PFR,PFR, the the input input “make-up “make-up gas” gas” (MUG (MUG), with), with a given a given initial initial stoichiometric stoichiometric number numberSN, isSN mixed, is mixed with thewith recycled the recycled unreacted unreacted eductgases educt prior gase tos prior entering to entering the PFR. Thisthe PFR. requires This a separationrequires a of the reactor output flow, removing the products H O and CH OH from the unreacted species, CO, separation of the reactor output flow, removing the2 products H3 2O and CH3OH from the unreacted CO and H , which are then re-introduced at the PFR input. In order to adjust the looping process, species,2 CO,2 CO2 and H2, which are then re-introduced at the PFRinput. In order to adjust the looping a controlled fraction b of the recycled gas, principally H , is purged. The looping factor f effectively process, a controlled fraction bof the recycled gas, principally2 H2, is purged. The looping factor determinesfeffectively howdetermines many timeshow themany unreacted times the gases unreac are recycledted gases through are recycled the reactor. through The the looping reactor. causes The anlooping increase causes in the an overall increase conversion in the overall efficiency conversi comparedon efficiency to the single-pass compared PFR,to the as single-pass shown in Figure PFR, 13as, whereshown ainMUG Figurestoichiometric 13, where a number MUG stoichiometric of SN = 2.02 and number a looping of SN factor = 2.02 of fand= 4 a are looping used. Thefactor effi ofciency f = 4 increaseare used. in The a looped efficiency PFR increase comes atin thea looped expense PFR of co themes “temperature at the expense swing” of the required “temperature to re-heat swing” the recycled unreacted gases following product removal by condensation. required to re-heat the recycled unreacted gases following product removal by condensation.

Figure 13. Simulated CO2 to CH3OH molar conversion efficiency for a looped PFR (dashed curves), compared to that for a single-pass PFR (solid curves). For the looped PFR, the stoichiometric number SN of the incoming make-up gas (MUG) is taken to be 2.02 (with no initial CO), and the looping factor f is 4.

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Figure 12. Computational scheme of the looped PFR. The function “PFR” refers to the operation of the previously considered single-pass plug flow reactor (Figure 6). The individual molar flows are defined in the Figure.

In the looped PFR, the input “make-up gas” (MUG), with a given initial stoichiometric number SN, is mixed with the recycled unreacted educt gases prior to entering the PFR. This requires a separation of the reactor output flow, removing the products H2O and CH3OH from the unreacted species, CO, CO2 and H2, which are then re-introduced at the PFRinput. In order to adjust the looping process, a controlled fraction bof the recycled gas, principally H2, is purged. The looping factor feffectively determines how many times the unreacted gases are recycled through the reactor. The looping causes an increase in the overall conversion efficiency compared to the single-pass PFR, as shown in Figure 13, where a MUG stoichiometric number of SN = 2.02 and a looping factor of f = 4 are used. The efficiency increase in a looped PFR comes at the expense of the “temperature swing”

required to re-heat the recycled unreacted gases following product removal by condensation. ChemEngineering 2020, 4, 62 13 of 16

Figure 13. Simulated CO2 to CH3OH molar conversion efficiency for a looped PFR (dashed curves), Figure 13. Simulated CO2 to CH3OH molar conversion efficiency for a looped PFR (dashed curves), compared to that for a single-pass PFR (solid curves). For the looped PFR, the stoichiometric number SN compared to that for a single-pass PFR (solid curves). For the looped PFR, the stoichiometric number ChemEngineeringof the incoming 2020, 4 make-up, x FOR PEER gas (REVIEWMUG) is taken to be 2.02 (with no initial CO), and the looping factor f is 4. 14 of 17 SN of the incoming make-up gas (MUG) is taken to be 2.02 (with no initial CO), and the looping factor 3.4.2. Sorption-Enhanced f is 4. Plug Flow Reactor A second method of increasing the PFR conversion efficiency eﬃciency is by by “sorption “sorption enhancement” enhancement” [11]. [11].

The correspondingcorresponding simulation scheme is shown in FigureFigure 1414.. Compared to the basic PFR scheme of Figure6 6,, notenote thethe presence presence of of an an e effectivﬀectivee “sorbent“sorbent channel”channel” andand thethe additionaddition ofof HH22OO andand CHCH33OH “transfer“transferChemEngineering terms”terms” 2020 inin the,the 4, x; di differentialdoi:ﬀerential FOR PEER PFR PFR REVIEW equations. equations. www.mdpi.com/journal/chemengineering

Figure 14.14. Computational scheme of the sorption-enhanced reactor. Note the presence of a parallelparallel

“sorbent”“sorbent” channelchannel andand thethe additionaddition ofof newnew HH22O and CH33OH “transfer” terms to the PFR equations. The symbol δδjkjk is the “Kronecker delta”. The individual molar flows flows are defineddefined in the figure.figure.

As in the case of the looped PFR, the idea of the sorption-enhanced reactor is that the product is constantly removed from from the the gas gas stream stream by by a asuitable suitable absorber absorber and, and, consequently, consequently, the the equilibrium equilibrium of ofthe the reaction reaction is shifted is shifted toward toward the product the product side, in side, accord in accord with Le with Chatelier’s Le Chatelier’s principle principle [11]. We [can11]. Wemodel can such model a reactor such a by reactor introducing by introducing a second a parallel second parallelchannel, channel, the “sorption the “sorption channel”, channel”, and by andintroducing by introducing terms in terms the PFR in theequations PFR equations (those proportional (those proportional to the inverse to the transfer inverse lengths transfer λk), lengths which λcausek), which a continuous cause a continuoustransfer of the transfer products of the H2 productsO (k= 4) and/or H2O( CHk = 34)OH and (k=/or 5) CHfrom3OH the ( kPFR= 5) channel from the to PFR channel to the sorption channel. We show the results, in Figure 15, for two cases respectively: the sorption channel. We show the results, in Figure 15, for two cases respectively: (a) 𝜆 = 0— (a) λCH3OH = 0—i.e., water absorption only—and (b) λCH3OH = λH2O—i.e., the simultaneous absorption i.e., water absorption only—and (b) 𝜆 = 𝜆—i.e., the simultaneous absorption of both water ofand both methanol. water andThe efficiency methanol. enhancement The eﬃciency is particularly enhancement pronounced is particularly when pronouncedboth product whenspecies both are productabsorbed. species In a practical are absorbed. sorption In aenhanced practical PFR, sorption the product enhanced absorptionoccurs PFR, the product in absorption the catalyst occurs system in itself [42]. The sorption-enhanced PFR suffers the drawback of requiring a recurring “pressure- swing”, to first desorb from and then to re-pressurize the absorbing catalyst material.

Figure 15. Simulated sorption-enhanced PFR conversion of CO2 to CH3OH: (a) Water adsorption alone, with methanol extraction from channel 1. (b) Simultaneous adsorption of water and methanol,

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3.4.2. Sorption-Enhanced Plug Flow Reactor A second method of increasing the PFR conversion efficiency is by “sorption enhancement” [11]. The corresponding simulation scheme is shown in Figure 14. Compared to the basic PFR scheme of Figure 6, note the presence of an effective “sorbent channel” and the addition of H2O and CH3OH “transfer terms” in the differential PFR equations.

Figure 14. Computational scheme of the sorption-enhanced reactor. Note the presence of a parallel

“sorbent” channel and the addition of new H2O and CH3OH “transfer” terms to the PFR equations. The symbol δjk is the “Kronecker delta”. The individual molar flows are defined in the figure.

As in the case of the looped PFR, the idea of the sorption-enhanced reactor is that the product is constantly removed from the gas stream by a suitable absorber and, consequently, the equilibrium of the reaction is shifted toward the product side, in accord with Le Chatelier’s principle [11]. We can model such a reactor by introducing a second parallel channel, the “sorption channel”, and by introducing terms in the PFR equations (those proportional to the inverse transfer lengths λk), which cause a continuous transfer of the products H2O (k= 4) and/or CH3OH (k= 5) from the PFR channel to

the sorption channel. We show the results, in Figure 15, for two cases respectively: (a) 𝜆 = 0—

i.e., water absorption only—and (b) 𝜆 = 𝜆—i.e., the simultaneous absorption of both water ChemEngineeringand methanol.2020 The, 4, 62 efficiency enhancement is particularly pronounced when both product species14 of 16 are absorbed. In a practical sorption enhanced PFR, the product absorptionoccurs in the catalyst system itself [42]. The sorption-enhanced PFR suffers the drawback of requiring a recurring “pressure- theswing”, catalyst to system first desorb itself [ 42from]. The and sorption-enhanced then to re-pressurize PFR the suﬀ absorbingers the drawback catalyst of material. requiring a recurring “pressure-swing”, to ﬁrst desorb from and then to re-pressurize the absorbing catalyst material.

Figure 15. Simulated sorption-enhanced PFR conversion of CO2 to CH3OH: (a) Water adsorption alone,Figure with 15. methanol Simulated extraction sorption-enhanced from channel PFR 1. (bconversion) Simultaneous of CO adsorption2 to CH3OH: of water (a) Water and methanol, adsorption withalone, methanol with methanol extraction extr fromaction channel from 2.channel In both 1.cases, (b) Simultaneous the pressure adsorption is 40 bar, and of water the stoichiometric and methanol, factor SN = 2. ChemEngineering 2020, 4, x; doi: FOR PEER REVIEW www.mdpi.com/journal/chemengineering 4. Summary and Conclusions This work has presented, in tutorial form, recipes for the numerical simulation of catalytic reactions. The application examples chosen are the hydrogenation reduction of carbon dioxide to form the synthetic fuels methanol and methane, and it has been assumed throughout that the reactants and products are ideal gases and that the reactor temperature and pressure are constant and uniform. Based on the change in the standard Gibbs free energy, the degrees of chemical conversion were first calculated in thermodynamic equilibrium. The results are in good agreement with the qualitative predictions of Le Chatelier’s principle. In a practical chemical reactor, a catalyst is used to selectively accelerate the reaction rate. The dynamic behaviors of the two archetypical continuous-flow reactors, the plug flow reactor (PFR) and the continuously stirred tank reactor (CSTR), were simulated by incorporating published models for the catalyst-specific, multi-step chemical kinetics into reactor-specific differential equations for the product yield. The numerical solution of these equations quantified the effects on the reaction yield of temperature, pressure, and various reactor parameters, in reasonable agreement with previous published work. In particular, it could be shown that, for a very long PFR or a very large volume CSTR, we regain the thermodynamic limit. It was also shown how the evolution from reactant to product species may be spatially followed along the length of the PFR tubes. The thermodynamic limit, in particular the low efficiency of methanol production, can be influenced by shifting the chemical equilibrium—e.g., via the continuous removal of product species. Using our numerical framework, it was demonstrated how this can facilitate methanol production for two particular cases: PFR looping (product removal and reactant recycling—involving a continuous “temperature-swing”) and sorption-enhanced PFR (product removal via absorption, perhaps on a specialized catalyst—involving repeated “pressure-swings”). All of these simulations could have been performed using a commercial software package. By laying out the fundamental concepts and constructing and solving the reactor-specific differential equations for the chemical yield, we have attempted to provide the reader with a view into the inner workings of such a “black-box” package. The student exercises in the Supplementary Materials lead the reader through the creation of a homemade computer program to numerically solve the differential equations. Our hope is that our audience will gain an understanding of the basic principles governing catalytic reactors and will be motivated to apply and adapt the presented formalism to applications of specific interest. ChemEngineering 2020, 4, 62 15 of 16

Supplementary Materials: The following are available online at http://www.mdpi.com/2305-7084/4/4/62/s1: see Supporting Information with exercises. Author Contributions: Conceptualization: J.T. and B.D.P.; Formal analysis: J.T., A.B., M.H., B.D.P.; Investigation: J.T. and B.D.P.; Supervision: B.D.P.; Writing—original draft: J.T. and B.D.P.; Writing—review and editing: J.T., A.B., M.H., B.D.P. All authors have read and agreed to the published version of the manuscript. Funding: This work was partly supported by the UZH-UFSP program LightCheC. The financial support from BFE and FOGA (SmartCat Project) and the Swiss National Science Foundation (Grant no. 200021_144120 and 172662) is acknowledged. Conflicts of Interest: The authors declare no conflict of interest.

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