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

: 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”. The following student exercises provide suggestions for further exploration of the concepts discussed in the main text. They become progressively more challenging, and answers are provided. Exercises 1 and 2 review the computation, using tabulated values of the standard enthalpy of formation and entropy, of the free energy and enthalpy changes occurring in the production and combustion of C1 synthetic fuels and of the equilibrium constants for methanol and methane production. Exercises 3 – 5 treat the state of thermodynamic equilibrium for the hydrogenation of CO 2 to form methane, and Exercises 6 and 7 deal with introductory aspects of a kinetic model of methane formation. Exercise 8 poses a question for thought regarding the thermodynamics and kinetics of CO 2 hydrogenation. The numerical solution of the differential equations describing the operation of a plug flow reactor to produce methane is the subject of Exercises 9 and 10. Finally, Exercise 11 quantitatively determines the chemical paths followed by CO 2 during its conversion to CH 4 . numerical


Free energy of formation and enthalpy of oxidation of hydrocarbon fuels.
For standard conditions (T=25°C, P=1 bar), compute the free energy of formation by hydrogenation reduction ΔGred and the enthalpy of oxidation ΔHoxid per mole of formic acid, formaldehyde, methanol and methane, and compare them with the Latimer-Frost diagram in Fig. 2. Make use of the following standard enthalpies of formation and entropies [Swaddle T. W. Inorganic Chemistry -An Industrial and Environmental Perspective, Academic Press: San Diego, 1997

Equilibrium constants for methanol and methane production reactions.
Using the thermodynamic data in Table 1, compute the equilibrium constants K 1−3 eq for the gas phase chemical reactions in Figures 3a and 3b. For methanol production, assume a reaction temperature of 230°C, and for methane, take T=400°C. Compare the values with the plot in Figure 4, and confirm in both cases that K 1 eq × K 2 eq = K 3 eq .
Answer: Modify the Equations (3-10) to describe the case of methane formation, instead of methanol formation, according to the reactions 2 and 3 in Figure 3b. Answer:

Equilibrium degree of conversion for methanation II.
Express the "reaction quotients" Q2 and Q3, from the previous exercise, in terms of the "degrees of reaction completion" ξ2 and ξ3. Answer:

Equilibrium degree of conversion for methanation III.
At a reaction temperature T=800°C, the equilibrium constants for methanation have the values: K 2 eq = 1.562 and K 3 eq = 0.1034 . Assume a reactor pressure of P=10 bar and an (ideal) initial stoichiometry number of SN=3. In order to (numerically) solve the two equations in the two unknown degrees of reaction completion, ξ2 and ξ3, form the function to minimize: Show that a 2-dimensional plot of 1/F as a function of ξ2 and ξ3 has a maximum at the ξ2 and ξ3 values we are searching for, and compute the corresponding equilibrium reduced partial pressures pj=PNj/PoNtot of CO, CO2, H2, H2O and CH4. (P0 is the pressure 1 bar.)

bar
Note that we use here the reciprocal of the equilbrium constants appearing in the work of Xu, et al. This is because we assume a reversed direction for the chemical reactions. The fitted Arrhenius parameters for the kinetic rates are: In Exercise 5, we found for the reactor conditions: T=800°C, P=10 bar, SN=3, the equilibrium reduced partial pressures pi of the 5 chemical species in the methanation reactor. Compute from these the reaction rates ri and the component creation rates Rj, per mol/kg s, for the initial conditions ( = = = =0, =2, =8) and for the conditions at thermodynamic equilibrium. Answer: Note: because there is initially no CO present, reaction rate r1 = 0, and at equilibrium, all reaction and component creation rates vanish.

a) b)
initial conditions:

Equilibrium condition on the kinetic rate expressions.
We saw in the previous exercise that at thermodynamic equilibrium, all kinetic reaction rates vanish. Show, both for the case of methanol production, Eqs (12-15), and of methanation, Exercise 6, that the condition for zero reaction rate ri implies the equality between the equilibrium constant and the reaction quotient: K i eq = Q i . Recall that the reduced partial pressure pj for each chemical component is related to its molar concentration Nj by pj=PNj/PoNtot.

A question to ponder.
In our discussion of thermodynamic equilibrium, we considered only two of the three reactions in Figure 3, invoking the "degrees of completion", ξ2 and ξ3. We justified this by stating that only 2 of the 3 reactions are "independent". But in our models for the kinetics of methanol and methane production, we took account of all three reactions. Why are two reactions sufficient for thermodynamics but three are necessary for kinetics? Answer: As mentioned in a footnote in the main text, the "degrees of completion", ξ2 and ξ3, are in reality only book-keeping devices, which serve to guarantee that the stoichiometry, i.e., the number of atoms of each element, is preserved. The equilibrium condition, at a given pressure P, temperature T and initial stoichiometry number SN, defines a particular combination of molar concentrations N CO , N CO 2 , N H 2 , N H 2 O , N CH 3 OH /CH 4 , representing 5 unknowns. This takes account of three stoichiometry constraints, one for each element C, H and O. In order to determine the equilibrium state, two additional constraints are thus required, and we took these to be the two equations K 2 eq = Q 2 and K 3 eq = Q 3 . Note that the third equation of this type, K 1 eq = Q 1 , is simply a linear combination of the other two. The equilibrium state only gives us the corresponding molar concentrations. Thermodynamics cannot tell us how we got there, e.g., whether the methanol or methane we obtained was produced directly from CO2 (reaction 3) or via CO from the reverse water gas shift reaction (reactions 2 and 1). For this information, we need to know the individual kinetic reaction rates of all three reactions. (See Exercise 11 below.)

PFR simulation I.
Based on the computation scheme in Figure 6, we want to build our own computer routine to numerically simulate the operation of a simplified plug flow reactor (PFR) for methane production. As in the main text, we assume all chemical components are ideal gases, at a uniform temperature T and pressure P.
In a first step, set up a routine which, given the molar flow rates For this, it will be necessary, from the flow rates, to compute the component-specific reduced partial (1), 88 -96.], and the reactor parameters above, find the numerical values of these three vector quantities. Answer:

PFR simulation II.
For the PFR reactor, in order to compute the degree of conversion of the initial CO2 to product species, we must numerically integrate the PFR equation: beginning at x=0 with the initial molar flow rates  N initial , until we reach the end of the reactor at x=Ltube. This is an example of coupled first-order ordinary differential equations with given initial conditions, and a standard algorithm for the numerical integration is the Runge- For a simulation of the plug flow reactor, where rapid changes occur at the beginning of the reactor and much slower changes occur at the end, the standard Runge-Kutta procedure, with a fixed step size h, is insufficient. We need to extend the method to dynamically adjust the step size, as required. The "quality control" Runge-Kutta procedure RKQC accomplishes this by comparing the effect of making two small RK steps with that of making a single step and adjusting the step size accordingly. An additional input parameter, ε 0 , specifies the required relative accuracy of the dependent variables.
Implement the RK and RKQC procedures and use them, for the methanation PFR parameters given in Exercise 9, to compute the degree of conversion from CO2 to CH4,  for a sequence of reactor temperatures, T = 200, 300, 400, 500, 600, 700 and 800°C, and compare it with the corresponding values ξ 3 from equilibrium thermodynamics (Exercise 5). Use ε 0 = 0.001 .
Compare with Figure 7b. Answer: As discussed in Exercise 8, in order to determine how much of the product CH4 is produced directly from the hydrogenation of CO2 and how much indirectly via CO from the reverse water gas shift reaction, we need to consider the reaction kinetics in the plug flow reactor. To this end, we introduce modified expressions to the PFR equations:

T [°C] Equilibrium
While the 5-component reduced partial pressure vector p is unchanged, we have added two CH4 components, specific to reactions 1 and 3, to the molar flow and component creation rate vectors.
Use the Runge-Kutta routines from Exercise 10 to integrate the 7-component PFR equation, and plot the resulting CO2 to CH4 degrees of conversion as a function of reactor temperature. Discuss the results. Answer: We note the following two features: a) At the lowest reaction temperatures, the exothermic reaction 1 is principally responsible for the CH4 production, b) At very high temperatures, the conversion via reaction 1 is negative, implying that the CH4 produced by reaction 3 is "backconverted" to CO via the inverse of reaction 1.