# Simple Rate Expression for Catalyzed Ammonia Decomposition for Fuel Cells

## Abstract

**:**

_{3}decomposition rates based on a literature-proven six-step elementary catalytic (Ni-BaZrO

_{3}) mechanism valid for 1 × 10

^{5}Pa pressure in a 650–950 K range. The rates are generated using a hypothetical continuous stirred tank catalytic reactor model running the literature mechanism. Excellent correlations are then obtained by fitting these rates to a simple overall kinetic expression based on an assumed slow step, with the remaining steps in fast pseudo-equilibria. The robust overall simple rate expression is then successfully demonstrated in various packed bed reactor applications. This expression facilitates engineering calculations without the need for a complex, detailed mechanism solver package. The methodology used in this work is independent of the choice of catalyst. It relies on the availability of a previously published and validated elementary reaction mechanism.

## 1. Background

_{4}or pure H

_{2}) are preferred. Compressed natural gas for transportation fleets is well known. The popularity and performance of battery-powered pure electric vehicles are growing despite limited charging stations and relatively long battery recharging times.

_{3}is attracting increasing attention [1]. It combines the advantages of easy storage and transport without producing greenhouse gases. Anhydrous liquid NH

_{3}is typically stored at 9.6 × 10

^{5}Pa absolute at room temperature. “Green” NH

_{3}can be catalytically produced from N

_{2}and H

_{2}using solar power [2]. Ammonia as a H

_{2}source is gaining considerable attention [3].

_{3}combustion produces H

_{2}O and N

_{2}as major products, temperatures can easily exceed 1000 K, resulting in pollutant nitrogen oxides [4]. To avoid these temperatures, fuel cells are preferred for the oxidation of the H

_{2}from NH

_{3}decomposition [2,5]. Ammonia-powered fuel cells are now being tested for freight-hauling vehicles, including large trucks [6] and locomotives [7].

_{2}from NH

_{3}can be accomplished in various ways, including catalytic decomposition from either gaseous or solution NH

_{3}[3]. Various metals (e.g., Ru, Ir, Ni, and Rh) and supports have been investigated as catalysts for the decomposition of gaseous NH

_{3}[3,8]. While Ru shows excellent activity, its high cost and limited availability make Ni more attractive [8]. An alternative approach is the decomposition of NH

_{3}in solution using electrocatalysts [3].

_{2}produced from gaseous NH

_{3}decomposition yields protons at the anode, thus releasing electrons into the external load circuit. The protons diffuse across a membrane to the cathode, where they react with O

_{2}(typically from air) to form H

_{2}O vapor, and the electron circuit is completed. In ceramic fuel cells, temperatures can exceed 1000 K due to H

_{2}oxidation exothermicity. At this level, the kinetics of the endothermic NH

_{3}decomposition are sufficiently fast that NH

_{3}can directly feed the fuel cell [9]. However, the risk of NO

_{x}formation still exists. Therefore, care is needed for NH

_{3}decomposition catalyst and fuel cell designs to avoid these temperatures.

_{3}decomposition over a Ni-BaZrO

_{3}catalyst. This model is calibrated over a relevant temperature range using reaction rates obtained from a detailed published elementary reaction mechanism [9]. The performance of this model is then compared to that of the detailed mechanism in simulations of packed bed (plug flow PBR) ideal adiabatic reactors. The utility of the engineering model is further demonstrated with simulations of PBRs with heat transfer and H

_{2}diffusion. This simple model is intended to facilitate quick engineering calculations and screening studies for fuel cell design.

## 2. Equilibrium

^{5}Pa). Figure 1 shows the equilibrium conversion over a wide temperature range at 1 × 10

^{5}Pa pressure for pure NH

_{3}feed, with complete conversion above 650 K. The calculation uses an online equilibrium calculator [10], with results consistent with published species thermodynamic properties [11]. The temperature range of this study is 650–950 K at 1 × 10

^{5}Pa pressure. Therefore, the decomposition is not thermodynamically limited under the conditions of this study.

## 3. Kinetics and Mechanism

_{3}decomposition on a Ni-based catalyst consists of the following six elementary reversible steps:

_{3}catalyst over ~650–950 K range at 1 × 10

^{5}Pa pressure using experimental data from Okura et al. [8]. The kinetic parameters are presented there for both the forward and reverse steps. This eliminates the need for any estimates of thermodynamics for surface-adsorbed species that would be needed if kinetic parameters were provided for only the forward steps.

_{3}, N

_{2}, and H

_{2}) are achieved if a time-dependent (kinetic) ammonia decomposition PBR flow reactor simulation is carried out for a sufficiently long time. These kinetic parameters are used in this study as described below. This was demonstrated by Karakaya et al. [12] with an elementary catalytic mechanism for methane oxidative coupling.

_{3}decomposition are performed using continuously stirred tank reactor (CSTR, perfectly mixed) calculations together with the detailed mechanism shown above. These provide rates of NH

_{3}decomposition as functions of NH

_{3}, H

_{2}, and N

_{2}partial pressures. These rates are then used to calibrate a proposed single overall kinetic expression for NH

_{3}decomposition derived from a fast pseudo-equilibrium analysis applied to the detailed six-step mechanism shown above. In the second portion, the calibrated single kinetic rate expression is used in simple packed bed reactor (PBR, perfect plug flow) simulations to demonstrate its utility in engineering and screening calculations.

## 4. Simplified Overall Rate Expression

^{®}[13]. It is often desirable to reduce such mechanisms to a simple kinetic rate expression for relatively quick engineering calculations. Such a reduction is applied here using the fast pseudo-equilibrium approach based on the Langmuir–Hinshelwood algorithm [14].

#### 4.1. Derivation by Langmuir–Hinshelwood Algorithm

_{3}decomposition catalyst:

_{S}is the concentration of vacant sites. Step 4 can be used to estimate ${C}_{N\xb7S}$ by applying the FPE:

_{S}derives from the “site balance”: ${C}_{T}={C}_{S}+{C}_{NH3\xb7S}+{C}_{NH2\xb7S}+{C}_{NH\xb7S}+{C}_{N\xb7S}+{C}_{H\xb7S}$. Bell and Torrente-Murciano [15] suggest that adsorbed N atoms are the dominant adsorbed species. Therefore, the site balance simplifies to: ${C}_{T}\approx {C}_{S}+{C}_{N\xb7S}$, from which:

#### 4.2. Preparation for Calibration of Overall Rate Expression

_{3}decomposition:

^{®}CSTR

^{®}simulations run with the detailed NH

_{3}decomposition mechanism [8] shown above. Available online [13] with numerous applications, the Detchem

^{®}software package is widely used [16,17] and quite versatile. This package executes detailed material, energy, and momentum balances using a user-supplied elementary reaction mechanism and required reactor input and parameter information.

_{NH3}vs. temperature data of Okura et al. [8], calculations with it can be used as the “data” source against which the overall rate expression (Equation (6)) is calibrated. The same catalyst, temperature, and pressure range as Okura et al. [8] are used.

^{®}PBED

^{®}and CSTR

^{®}applications, respectively. The governing PBED

^{®}and CSTR

^{®}equations are described in the Detchem

^{®}manual [13] and listed elsewhere [18].

^{®}CSTR

^{®}application is run over a wide range of pure NH

_{3}feed rates F

_{NH3,o}to generate gaseous reactor effluent mole fractions, y

_{j}. Any pressure drop across the reactor is assumed to be small enough to ignore. The exit mole fractions are used to calculate the corresponding partial pressures, P

_{j}. Knowing P

_{j}, Equation (9) is used to determine the corresponding NH

_{3}conversion, X

_{NH}

_{3}:

_{H2}and P

_{N2}are used to test for X

_{NH3}consistency:

_{NH3}data in hand, Equation (7) is used to generate rate ${r}_{NH3}^{\u201d}$ data. At the given temperature, the partial pressure, conversion, and rate $\left({P}_{j}{,X}_{NH3},{r}_{NH3}^{\u201d}\right)$ data sets, based on exit mole fraction data from the Detchem

^{®}CSTR

^{®}runs and Equations (7), (10), and (11), are used below to calibrate the overall rate expression (Equation (6)) through the lumped constants k and $\widehat{k}$.

## 5. Calibration of the Simplified Overall Rate Expression

^{®}CSTR

^{®}simulations with a pure NH

_{3}feed over a wide temperature range was performed. Details appear in Table 1. The runs were isothermal at constant pressure. The feed rate range for each temperature was chosen such that the X

_{NH3}ranged from near zero to almost 1. The gas volume and catalytic surface area values listed are consistent with the PBED

^{®}simulations described later.

_{3}feed, all $\left({P}_{j}{,X}_{NH3},{r}_{NH3}^{\u201d}\right)$ data sets were used to calibrate the Equation (6) overall rate expression through the constants k and $\widehat{k}$. It should be noted that the Langmuir–Hinshelwood algorithm was used to derive different overall rate expressions, such as Equation (6), each based on a different assumed slow step. All expressions were subjected to testing with the generated $\left({P}_{j}{,X}_{NH3},{r}_{NH3}^{\u201d}\right)$ data sets. These results are not shown here because the only statistically acceptable rate form was Equation (6), which is based on an assumed slow step of N adatom recombination, which is consistent with Bell and Torrente-Murciano [15].

#### 5.1. Calibration Results

_{3}conversion. Sample results are shown in Figure 2 (650 K) and Figure 3 (950 K). The simple overall rate expression (Equation (6)) does an excellent job for all pure NH

_{3}feed cases. It is satisfying that the overall rate expression shows the inflections in the curves at higher X

_{NH3s}.

_{3}feed, the rate parameters k and $\widehat{k}$ as functions of temperature are presented in Figure 4. The non-Arrhenius curves are each regressed to the form:

^{®}CSTR

^{®}simulations, wherein 25% of the NH

_{3}feed is replaced with H

_{2}or N

_{2}at the same total feed rate. The H

_{2}cases result in lower NH

_{3}decomposition rates, as suggested by Equation (6). The impact is more pronounced at lower temperatures. The impact of adding N

_{2}in this role is less clear. Equation (6) suggests little impact beyond a dilution effect on H

_{2}and NH

_{3}partial pressures. In both of these tests, the overall simplified rate expression (Equation (6)) calibrated on pure NH

_{3}feed runs did well in predicting the detailed mechanism-based rate results.

## 6. PBR Species and Energy Balances Used with Overall Rate Expression

_{3}decomposition in the packed bed reactor found in an idealized ammonia fuel cell. The species balances used are shown in Equations (13) and (14). The PBR assumes perfect plug flow.

_{3}decomposition can occur within the anode structure of the fuel cell or in the catalytic zone outside the fuel cell itself. If inside, the “diffusion” term on the right side of Equation (14) crudely approximates the passage of H

_{2}through the fuel cell membrane to the cathode, where it is oxidized. If outside, then k

_{c}= 0. The PBR energy balance is:

_{2}conducts through the fuel cell membrane to the anode to satisfy some of the NH

_{3}decomposition endothermicity, the heat transfer rate can be crudely approximated by:

_{p,j}and $\mathsf{\Delta}{H}_{r,j})$ are presented in the Appendix A. Constant pressure and ideal gas are assumed throughout this study. Equations (17) provide additional values:

#### 6.1. Packed Bed Reactor Simulations

^{®}PBED

^{®}package. Table 3 presents the data for the simulations. Both high and low pure NH

_{3}feed rates were used, together with a range of feed temperatures. These runs produce the data against which the single rate expression engineering model will be tested.

^{®}ODE package [19] solved the engineering model for the packed bed reactor.

_{3}feed (1 × 10

^{−5}mole/s) at 800 K show that the engineering model does an excellent job predicting the detailed mechanism axial profiles for both NH

_{3}content and reactor temperature. Similar excellent results are shown in Figure 6 over a range of feed temperatures at a higher feed rate (1 × 10

^{−4}mol/s). The fits of the NH

_{3}mole fractions are excellent, while the temperatures are predicted within 3 K. With the credibility of the engineering model with single overall rate expression now established, this model was applied for varied applications for which Detchem

^{®}PBED

^{®}is not applicable.

Bed radius: 5 × 10^{−3} m | Bed length: 0.1 m |

Catalyst area/bed volume: 2.19 × 10^{5} m^{−1} | Total catalyst area: 0.172 m^{2} |

Bed porosity: 0.38 | Particle diameter: 2 × 10^{−5} m |

Feed: pure NH_{3} | Total feed rates: 1 × 10^{−5}, 1 × 10^{−4} mole/s |

Pressure: 1 × 10^{5} Pa (constant) | Feed temperature range: 650–950 K |

Diffusion coefficient k_{c} (Equation (5)): 5 × 10^{−7} mol/s-m-Pa | (used for Figure 7, Figure 8, Figure 9 and Figure 10 only) |

Heat transfer factor f (Equation (7)): 0.12 | (used for Figure 7, Figure 8, Figure 9 and Figure 10 only) |

#### 6.2. Extended PBR Simulations

_{3}decomposition occurs in an adiabatic packed bed reactor external to any fuel cell. However, this decomposition can occur as part of the anode structure of the fuel cell [9]. In this case, H

_{2}diffusion through the membrane for exothermic oxidation at the cathode accompanies the NH

_{3}breakdown. In addition, heat transfer from the cathode across the membrane to the anode can supply some of the decomposition endothermicity. Additional PBR calculations were performed to crudely simulate these cases with only the engineering model (Equations (13)–(17)) and simplified rate expression (Equations (6) and (12)). The Polymath

^{®}ODE solver [19] was used.

_{2}diffusion off (k

_{c}= 0) and adiabatic (f = 0), diffusion on (k

_{c}= 5 × 10

^{−7}mol/s-m-Pa) and adiabatic, and diffusion on with heat transfer on (f = 0.12). The models of H

_{2}diffusion and heat transfer are crude, but they illustrate the utility of the engineering model with the overall rate expression.

**Figure 7.**Impact of H

_{2}diffusion and heat transfer on axial temperature in PBR; 950 K feed temperature; pure NH

_{3}feed = 1 × 10

^{−4}mol/s (see Table 3 for remaining conditions). Calculations based on engineering model with overall decomposition rate expression Equations (6) and (12).

_{3}pyrolysis, as shown in Figure 7. This is accompanied by only modest conversion (Figure 8). Loss of H

_{2}by diffusion has little impact on the bulk flow temperature, even though the drop in H

_{2}flow rate is large, as shown in Figure 10. Figure 9 shows that the NH

_{3}flow rate has not dropped much, as is also evident in Figure 8, with only an approximately 23% conversion.

_{3}rate (Figure 9) and hence a larger NH

_{3}conversion (Figure 8). Even more H

_{2}is available for diffusion (Figure 10). The N

_{2}flow rate also increases due to the higher NH

_{3}conversion.

**Figure 8.**Impact of H

_{2}diffusion and heat transfer on NH

_{3}conversion in PBR; 950 K feed temperature; pure NH

_{3}feed = 1 × 10

^{−4}mol/s (see Table 3 for remaining conditions). Calculations are based on engineering model with overall decomposition rate expression (Equations (6) and (12)).

**Figure 9.**Impact of H

_{2}diffusion and heat transfer on axial NH

_{3}flow rate in PBR; 950 K feed temperature; pure NH

_{3}feed = 1 × 10

^{−4}mol/s (see Table 3 for remaining conditions). Calculations use engineering model with single overall decomposition rate expression (Equations (6) and (12)).

**Figure 10.**Impact of H

_{2}diffusion and heat transfer on axial N

_{2}and H

_{2}flow rates in PBR; 950 K feed temperature; pure NH

_{3}feed = 1 × 10

^{−4}mol/s (see Table 3 for remaining conditions). Calculations use engineering model with overall decomposition rate expression (Equations (6) and (12)).

## 7. Conclusions

_{2}) with five remaining elementary steps in fast pseudo-equilibria, has been derived to accurately simulate the decomposition of NH

_{3}over a Ni-BaZrO

_{3}catalyst at 1 × 10

^{5}Pa at 650–950 K. The expression was calibrated using decomposition rates calculated with a six-step elementary mechanism from the literature proven independently against experimental data. The overall rate expression, with two temperature-dependent parameters, and its implementation in an engineering model successfully predicted NH

_{3}decomposition performance in a packed bed reactor as calculated with the detailed mechanism. The utility of the engineering model with the overall rate expression was further demonstrated with simulations of NH

_{3}decomposition in a packed bed reactor allowing for H

_{2}diffusion and heat transfer in an approximation of a fuel cell anode feeding NH

_{3}. The methodology used in this work is independent of the choice of catalyst. It relies on the availability of a previously published and validated elementary reaction mechanism.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | first of three fitted parameters for lumped kinetic constants (Equation (12) |

A_{c} | packed bed reactor cross section (m^{2}) |

A_{CSTR} | catalytic surface area for CSTR (m^{2}) |

a_{v} | catalytic surface area/packed bed volume (m^{2}/m^{3}) |

b | second of three fitted parameters for lumped kinetic constants (Equation (12) |

c | last of three fitted parameters for lumped kinetic constants (Equation (12) |

${C}_{j\xb7S}$ | surface concentration of adsorbed species j (mol/m^{2}) |

c_{p,j} | heat capacity of species j (J/mol-K) |

C_{S} | concentration of vacant surface sites (mole/m^{2}) |

C_{T} | total concentration of surface sites (mole/m^{2}) |

f | assumed fraction of H_{2} oxidation heat transferring from cathode to anode |

${F}_{j}$, F_{jo}, F_{T} | molar rate of species j (mol/s) - subscript “o” for feed, “T” for total |

k | lumped constant (mol-Pa/s-m^{2}) |

$\widehat{k},\widehat{K}$ | lumped constants (Pa^{0.5}) |

k_{i,-i} | rate constants (forward/reverse) for reaction “i” in decomposition mechanism |

K_{i} | equilibrium constant for reaction “i” in decomposition mechanism |

k_{c} | mass transfer coefficient for H_{2} diffusion across cell membrane (mol/s-m-Pa) |

${P}_{j}$ | partial pressure of species j (Pa) - subscript “o” indicates feed |

$\dot{q}$ | local external heat transfer rate in packed bed reactor (J/s-m) |

${r}_{j}^{\u201d}$ | catalytic surface-based reaction rate of species j (mol/s-m^{2}) |

T | temperature (K) - subscript “o” indicates feed |

${X}_{NH3}$ | fractional conversion of NH_{3} |

y_{j} | gaseous species j mole fraction |

z | packed bed reactor axial length (m) |

$\delta $ | net change in moles per mole of key reactant (NH_{3}) as per reaction stoichiometry |

$\u2206{H}_{r,j}$ | heat of reaction (J/mol of j) |

$\u03f5$ | product of feed mole fraction of key reactant and net change in moles by reaction |

${\theta}_{j}$ | ratio of feed flow rate of species j to feed rate of key reactant (NH_{3}) |

${\nu}_{j}$ | stoichiometric coefficient (+/−) assuming key reactant coefficient (NH_{3}) is unity |

## Appendix A

cp,j (J/mol-K) | |

H_{2} | 29.66 |

N_{2} | 31.42 |

NH_{3} | 0.02815 T + 28.64 |

$\u2206{H}_{r,j}$ (J/mol) | |

NH_{3} = 0.5N_{2} + 1.5H_{2} | 45,900. (per mol NH_{3}) at 298 K * |

H_{2} + 0.5O_{2} = H_{2}O | −241,830. (per mol H_{2}O) |

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**Figure 1.**Equilibrium NH

_{3}conversion for pure NH

_{3}feed at 1 × 10

^{5}Pa pressure. Calculation based on NASA CEA equilibrium code [10].

**Figure 2.**Calculated (Detchem

^{®}, Equation (6) overall rate expression with 2 parameters) NH

_{3}decomposition rates at 650 K for both pure NH

_{3}and mixed (25 mole% H

_{2}or N

_{2}, bal. NH

_{3}) feeds (see Table 1). N H

_{3}conversion varies as a function of total feed rate.

**Figure 3.**Calculated (Detchem

^{®}, Equation (6) overall rate expression with 2 parameters) NH

_{3}decomposition rates at 950 K for both pure NH

_{3}and mixed (25 mole% H

_{2}or N

_{2}, bal. NH

_{3}) feeds (see Table 1). NH

_{3}conversion varies as a function of total feed rate.

**Figure 4.**Temperature dependencies (point values and regressions) of the two parameters of overall kinetic rate expression (Equation (6)). Units: k (mol-Pa/s-m

^{2}); khat = $\widehat{k}\left({\mathrm{Pa}}^{0.5}\right)$.

**Figure 5.**Comparison of engineering model (EM) with overall kinetic rate expression (Equation (6)) vs. Detchem

^{®}with detailed mechanism for axial profiles in adiabatic packed bed reactor feeding pure NH

_{3}at 1 × 10

^{−5}mol/s and 800 K (see Table 3 for remaining conditions).

**Figure 6.**Comparison of engineering model (EM) with overall kinetic rate expression (Equations (6) and (12)) vs. Detchem

^{®}with detailed mechanism for adiabatic PBR feeding pure NH

_{3}at 1 × 10

^{−4}mol/s (see Table 3 for remaining conditions).

Feed Rate (mol/s) | ||
---|---|---|

Temperature (K) | Lowest | Highest |

650 | 1 × 10^{−11} | 1 × 10^{−7} |

725 | 1 × 10^{−10} | 1 × 10^{−5} |

800 | 1 × 10^{−8} | 1 × 10^{−4} |

875 | 1 × 10^{−7} | 1 × 10^{−3} |

950 | 1 × 10^{−6} | 1 × 10^{−4} |

^{5}Pa (constant); Catalyst area: 1.92 × 10

^{−2}m

^{2}; Reactor gas volume: 3.33 × 10

^{−8}m

^{3}; Feed: pure NH

_{3}, or 25 mol% H

_{2}or N

_{2}(balance NH

_{3}) at same total molar rate for selected cases.

**Table 2.**Rate constants for 2-parameter kinetic rate expression (Equation (6)). Form: $ln\left(kor\widehat{k}\right)=a+b\left(1/T\right)+c{\left(1/T\right)}^{2}$ where T (K).

For k (mol-Pa/s-m^{2}) | $\mathbf{For}\widehat{\mathit{k}}({\mathrm{Pa}}^{0.5})$ | |
---|---|---|

a | −5.996 | −6.181 |

b | 4.344 × 10^{4} | 2.849 × 10^{4} |

c | −2.610 × 10^{7} | −1.287 × 10^{7} |

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**MDPI and ACS Style**

Barat, R.B.
Simple Rate Expression for Catalyzed Ammonia Decomposition for Fuel Cells. *Molecules* **2023**, *28*, 6006.
https://doi.org/10.3390/molecules28166006

**AMA Style**

Barat RB.
Simple Rate Expression for Catalyzed Ammonia Decomposition for Fuel Cells. *Molecules*. 2023; 28(16):6006.
https://doi.org/10.3390/molecules28166006

**Chicago/Turabian Style**

Barat, Robert B.
2023. "Simple Rate Expression for Catalyzed Ammonia Decomposition for Fuel Cells" *Molecules* 28, no. 16: 6006.
https://doi.org/10.3390/molecules28166006