# Intraparticle Modeling of Non-Uniform Active Phase Distribution Catalyst

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
**Uniform**: the active phase is homogeneously distributed in the support;- (2)
**Egg-shell**: the active phase is located in the outer surface of the support;- (3)
**Egg-white**: the active phase is included in a region between the outer-shell and the inner-core;- (4)
**Egg-yolk**: the active phase is present in the inner-core of the support.

## 2. Mathematical Model

#### 2.1. Intraparticle Mass and Energy Balances

_{k}) have been chosen, whose co-domains vary from 0 to 1. Defining a smoothing factor (b) (needed to modify the function steepness), a

_{s}and a

_{st}are the coordinates of the distribution domains, Equations (2)–(4),

_{31}and a

_{32}can be correctly calculated as in Equations (6) and (7).

_{1}, while it is equal to a

_{2}for the egg-yolk catalyst.

_{1}= 0.6, a

_{2}= 0.4, a

_{31}= 0.7, a

_{32}= 0.3, centering the catalytic layer for the egg-white case in the middle of the particle.

_{p}/R

_{p}.

_{k}x; thus, it is possible to discriminate the inert zone (Ω

_{k}x = 0, (no reaction) and a catalytically active zone (Ω

_{k}x ≠ 0, where both reaction and diffusion take place. The boundary conditions (BCs) needed to solve the partial differential equation are listed in Equation (9),

#### 2.2. Reaction Network

_{1}was considered, as A→B is the desired reaction. This calculation does not account for the specific distribution.

## 3. Results and Discussion

#### 3.1. Model Validation

_{1}≤ x ≤ 1 (Figure 3a); C

_{B}shows a maximum inside the particle, since the intermediate B is formed in reaction 1 and consumed in reaction 2 (Figure 3b). For 0 ≤ x < a

_{1}, each profile is described by the diffusive mechanism only, as no active phase is present in the inner part of the particle. The final product C is formed in the outer shell of the catalyst particle by the progress of the second reaction, then it diffuses in the inner core of the particle (Figure 3c). A temperature rise (4 K) under steady-state conditions was calculated (Figure 3d).

#### 3.2. Egg-Shell: Sensitivity Study

_{1}, is varied from 0 to 1, simulating the cases ranging from a uniform distribution to an inert particle. Figure 4 shows the steady state intraparticle profiles of component A conversion, yield and selectivity to B for a

_{1}= [0, 0.2, 0.4, 0.6, 0.8, 1].

_{1}: an increase in the amount of the active phase leads to a corresponding increase in the conversion of component A (Figure 4a). The selectivity to B is equal to 1 for an inert particle, since, considering an infinitesimal active-phase thickness, A reacts to give the only the intermediate, thus a fictitious unitary selectivity is computed. It is interesting to observe that the yield to B is zero for an inert particle and a finite value, not the maximum one, for a uniform distribution (Figure 4b). Temperature profiles at steady state show a maximum temperature gradient of roughly 5 K for the uniformly distributed active phase catalyst (Figure 4c). This value decreases when a lower amount of the active phase is present in the catalyst particle. Maximizing the yield to B is surely possible with the developed model, as the simulation results suggest the possibility of optimizing the steady state yield of the desired product by varying the active-phase thickness. The plot y

_{B}vs. a

_{1}is depicted in Figure 5.

_{ref2}/k

_{ref1}is defined for each simulation; intraparticle profiles under steady-state conditions for σ = [0.01, 0.05, 0.1] are displayed in Figure 6.

_{ref}

_{2}(Figure 6b); besides, σ does not affect the conversion degree, as k

_{ref}

_{1}is constant (Figure 6a). Moreover, by increasing σ, a corresponding increase in the temperature inside the particle is predicted (Figure 6c).

_{T,p}is shown, with k

_{T,p}= [0.01, 0.05, 0.1] W m

^{−1}K

^{−1}.

_{m}, by a factor of 100 (k

_{m}= 1 m s

^{−1}), and the heat transfer coefficient h was calculated from Equation (12).

_{i}≠ C

_{i}

_{0}. This fact strongly influences the concentration gradients, as demonstrated.

_{eff,i}= [1 × 10

^{−7}, 1 × 10

^{−8}, 1 × 10

^{−9}] m

^{2}s

^{−1}. The steady state results are reported in Figure 9.

#### 3.3. Egg-Yolk, Egg-White and Comparisons

_{1}; it is equal to a

_{2}for the egg-yolk catalyst. No comparisons with the egg-white distribution are possible because δ, in this case, requires two coordinates to be determined.

_{B}vs. δ for different distributions are depicted in Figure 11. It must be noted that, when δ = 1, the active phase is uniformly distributed within the catalyst. Thus, Figure 10 and Figure 11 show identical values of y

_{B}when δ = 1.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Notation

A | reactant |

a_{1} | inflection point for egg-shell profile, [-] |

a_{2} | inflection point for egg-yolk profile, [-] |

a_{31} | inflection point 1 for egg-white profile, [-] |

a_{32} | inflection point 2 for egg-white profile, [-] |

B | intermediate |

b | smoothing factor, [-] |

C | product |

C_{i0} | concentration of species i in bulk phase at time = 0, [mol m^{−3}] |

C_{i} | concentration of species i, [mol m^{−3}] |

c_{p,i} | specific heat of the fluid component, [J mol^{−1} K^{−1}] |

c_{p,p} | specific heat of the particle, [J mol^{−1} K^{−1}] |

D_{eff,i} | effective diffusivity, [m^{2} s^{−1}] |

E_{a,j} | activation energy, [J mol^{−1}] |

h | heat exchange coefficient, [W m^{−2} K^{−1}] |

k_{j} | kinetic constant, [m^{3} kg^{−1} s^{−1}] |

k_{m} | mass transfer coefficient, [m s^{−1}] |

k_{ref},_{j} | reference kinetic constant, [m^{3} kg^{−1} s^{−1}] |

k_{T,f} | fluid thermal conductivity, [W m^{−1} K^{−1}] |

k_{T,p} | particle thermal conductivity, [W m^{−1} K^{−1}] |

MW_{i} | molecular weight, [-] |

N_{c} | number of components, [-] |

r_{j} | reaction rate j, [mol kg^{−1} s^{−1}] |

R_{p} | particle radius, [m] |

r_{p} | radial coordinate, [m] |

T | temperature, [K] |

t | time, [s] |

T_{0} | surface temperature at time = 0, [K] |

x | dimensionless radial coordinate, [-] |

x_{A} | conversion degree, [-] |

y_{B} | yield in B, [-] |

Subscripts | |

f | fluid |

i | component |

j | reaction |

k | distribution type |

p | particle |

s | coordinate domain |

t | coordinate domain |

Greek letters | |

∆_{r} H_{j} | reaction enthalpy j, [J mol^{−1}] |

δ | active-phase thickness, [-] |

ε | void degree, [-] |

ϕ_{B} | selectivity to B, [-] |

φ | Thiele modulus, [-] |

η | efficiency factor, [-] |

ν_{ij} | stoichiometric matrix, [-] |

ρ_{p} | particle density, [kg m^{−3}] |

σ | kinetic constant ratio, [-] |

Ω_{k} | distribution function, [-] |

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**Figure 3.**Intraparticle profiles for egg-shell catalyst. Calculated profiles for: (

**a**) C

_{A}; (

**b**) C

_{B}; (

**c**) C

_{C}; (

**d**) T. Color-bar of each plot is located on the right-hand-side.

**Figure 4.**Intraparticle profiles calculated for the egg-shell catalyst varying the active phase content: (

**a**) A conversion; (

**b**) yield to B; (

**c**) temperature.

**Figure 5.**Yield to B (y

_{B}) vs. active phase inert fraction coordinate (a

_{1}) for the egg-shell catalyst.

**Figure 6.**Intraparticle profiles computed varying the ratio between the kinetic constants at the reference temperature of reactions 2 and 1 for the egg-shell catalyst: (

**a**) A conversion; (

**b**) yield to B; (

**c**) temperature.

**Figure 7.**Intraparticle profiles: k

_{T,p}sensitivity study. (

**a**) A conversion; (

**b**) yield to B; (

**c**) temperature.

**Figure 8.**Intraparticle concentration and temperature profiles; k

_{m}= 1 m s

^{−1}; h = 4218 W m

^{−1}K

^{−1}. (

**a**) concentration of A; (

**b**) concentration of B. Color-bar of each plot is located on the right-hand-side.

**Figure 9.**Intraparticle profiles: D

_{eff,i}sensitivity study. (

**a**) A concentration; (

**b**) B concentration; (

**c**) temperature.

PARAMETER | VALUE |
---|---|

R_{p} [m] | 1 × 10^{−2} |

a_{1} [-] | 6 × 10^{−1} |

a_{2} [-] | 4 × 10^{−1} |

a_{31} [-] | 7 × 10^{−1} |

a_{32} [-] | 3 × 10^{−1} |

b [-] | 1 × 10^{−5} |

T_{0} [K] | 2.98 × 10^{2} |

ρ_{p} [kg m^{−3}] | 4 × 10^{3} |

k_{m} [m s^{−1}] | 1 × 10^{2} |

ε [-] | 5 × 10^{−1} |

k_{T,p} [W m^{−1} K^{−1}] | 1 × 10^{−1} |

k_{T,f} [W m^{−1} K^{−1}] | 0.25 |

h [W m^{−2} K^{−1}] | 8 × 10^{2} |

c_{p,p} [J kg^{−1} K^{−1}] | 6 × 10^{2} |

A | B | C | |
---|---|---|---|

MW_{i} [kg kmol^{−1}] | 30 | 30 | 30 |

D_{eff,i} [m^{2} s^{−1}] | 1 × 10^{−7} | 1 × 10^{−7} | 1 × 10^{−7} |

C_{i}_{0} [mol m^{−3}] | 1 | 0 | 0 |

c_{p,i} [J mol^{−1} K^{−1}] | 30 | 30 | 30 |

Reaction | 1 | 2 |
---|---|---|

∆_{r}H_{j} [J mol^{−1}] | −5 × 10^{4} | −5 × 10^{4} |

E_{a,j} [J mol^{−1}] | 8 × 10^{4} | 8 × 10^{4} |

k_{ref,j} [m^{3} kg^{−1} s^{−1}] | 1 × 10^{−5} | 5 × 10^{−7} |

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**MDPI and ACS Style**

Russo, V.; Mastroianni, L.; Tesser, R.; Salmi, T.; Di Serio, M.
Intraparticle Modeling of Non-Uniform Active Phase Distribution Catalyst. *ChemEngineering* **2020**, *4*, 24.
https://doi.org/10.3390/chemengineering4020024

**AMA Style**

Russo V, Mastroianni L, Tesser R, Salmi T, Di Serio M.
Intraparticle Modeling of Non-Uniform Active Phase Distribution Catalyst. *ChemEngineering*. 2020; 4(2):24.
https://doi.org/10.3390/chemengineering4020024

**Chicago/Turabian Style**

Russo, Vincenzo, Luca Mastroianni, Riccardo Tesser, Tapio Salmi, and Martino Di Serio.
2020. "Intraparticle Modeling of Non-Uniform Active Phase Distribution Catalyst" *ChemEngineering* 4, no. 2: 24.
https://doi.org/10.3390/chemengineering4020024