# Intraparticle Model for Non-Uniform Active Phase Distribution Catalysts in a Batch Reactor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Uniform: the active phase if homogeneously distributed on 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. Reaction Rate Expressions

_{ref}= 323.15 K (4).

#### 2.2. Mass and Energy Balances

_{m}, in the liquid phase balance, represents the liquid–solid mass transfer coefficient, a

_{sp}is the specific surface area, calculated according to Equation (7), and ε represents the volumetric ratio between the phases, obtained from the ratio between the particle density (ρ

_{P}) and the catalyst bulk density (ρ

_{Bulk}) as shown in Equation (8).

_{P}is the catalyst porosity, D

_{eff,i}is the effective molecular diffusivity, while s is the shape factor calculated assuming that the catalyst particles have a spherical shape.

_{k}= 0), from the active one (Ω

_{k}= 1), where it the chemical reaction is also present.

_{p,i}and c

_{p,P}, represent the specific heat of the fluid component and of the catalytic particle, respectively, while k

_{T,P}is the particle heat conductivity.

_{T,l}) and the mass transfer coefficients (12) [11].

_{A}represents the external thermal resistance, while T

_{j}is the external temperature.

#### 2.3. Distribution Function

_{P}. Step functions (Ω

_{k}) were used including the following parameters: smoothing factor b, useful for evaluating the slope of the curve and inflection points a

_{s}and a

_{st}, for the definition of the distribution ranges. The functions used are reported in Equations (15)–(17) [13].

_{1}, while it is equal to a

_{2}for the egg yolk catalyst. Regarding the distribution of egg white, two coordinates are needed for the catalyst; therefore, the thickness of the active phase can be defined by a

_{31}-a

_{32}. The dimensions of the catalytic thickness are the same in all distributions.

#### 2.4. Numerical Methods

## 3. Results and Discussion

#### 3.1. Model Validation: Standard Simulation

_{m}), on the other hand, is sufficiently high to make the resistance to diffusion in the stagnant fluid film negligible. The same evaluation can be made with respect to the coefficient of thermal resistance at the liquid–solid interface (h), which has a value high enough to make its effects negligible. Moreover, it is important to consider that under standard conditions, the system was considered adiabatic (UA = 0), thus the worse possible situation. In real cases, heat exchange is normally provided, leading to a consequent smoothing of the temperature increase.

#### 3.2. Parametric Investigation

#### 3.2.1. Reaction Enthalpy

_{1}case, since the second reaction is slower than the first (k

_{ref}

_{2}<< k

_{ref}

_{1}) and produces C in smaller quantities than B, the energy contribution will certainly be reduced compared to the previous simulations.

_{2}obtained starting from the same values used in the study of ΔH

_{2}, led to completely different results. Both the solid-side and liquid-side profiles overlap almost completely in all the cases showing slight deviations from the standard behavior only in the production of C in the EW and EY cases. As anticipated, the energy contribution provided is very limited due to the low concentration of C produced. In cases where the catalytically active phase is positioned in the innermost parts of the particle, the reactive thrust is displayed in a slightly more evident way, but the effect of the parameter remains extremely limited.

#### 3.2.2. Reaction Rate Constants

_{Bl}is not observed in the EW and EY cases. The second chemical reaction is less affected than the ΔH

_{1}study. In these simulations, in the cases of EW and EY, the consumption of A is clearly limited by internal diffusion resistance.

_{Cref2}, is under standard conditions, four orders of magnitude lower than the first chemical reaction; consequently, its further lowering involves a decrease in the quantity of the final product C. In all the cases, it is possible to observe that both the liquid and solid side profiles tend to overlap the reference simulation with small exceptions.

_{Cref1}, it is observed an increase in the production of C. For the first time, there is a beginning of a descending phase in the c

_{Bl}profile in the ES. In the EW and EY cases the concentration of the intermediate in the liquid is the lowest ever observed, also preventing the accumulation of the component inside the particle.

#### 3.2.3. Effective Molecular Diffusivity

_{eff,A}causes significant changes in the concentration profiles of the solid and liquid sides.

_{eff,A}generates diametrically opposite effects; much faster reactions are observed, due to less diffusion resistance within the solid phase. It is interesting to note that for the reactive process causes, for the first time, the achievement of an intra-particle concentration of the intermediate is higher than 1 mol m

^{−3}.

_{Cl}obtained is very low when the mass transfer of A is less limiting.

#### 3.2.4. Particle Radius

_{P}is placed both in the internal diffusive term, altering the transport resistance, and in the external diffusive term by modifying the contact surface between the two phases, identified in the relationship between the surface of the particle and its volume (a

_{sp}).

_{P}is opposite to what is visible in the simulations involving D

_{eff,A}. It is important to remember that, unlike the previous case, where the effect was focused only on the main reacting species, in these simulations, the effect is related to all the species involved.

_{Bl}obtained in all cases is very high to the detriment of the by-product.

_{P}in the liquid side balance is less evident than that observed in the solid side, being the external diffusive term much less sensitive to its small alterations.

#### 3.2.5. Catalyst Bulk Density

_{P}, for the same particle radius, consists in a modification of the mass of catalyst present in the reactive system. This parameter directly affects the generative terms of intraparticle mass and energy balances. Finally, it also intervenes in the liquid side mass balance in the term ε.

_{P}entails a slowdown in the entire chemical process due to the lower contribution of the generative terms. In fact, the accumulation of reagent A in the intraparticle profiles is observed, unlike the simulations with higher density values.

#### 3.2.6. Initial Bulk Concentration of A

_{A}

_{0,l}. In all the cases, the reaction times, for obtaining the desired conversion, are similarly decreasing moderately with the increment of the available reagent. ES is the catalytic case most influenced by the parameter, probably due to the immediate availability of the catalytic zone, unlike the other two cases.

#### 3.2.7. Liquid–Solid Mass Transfer Coefficient

_{m}, the influence on the ES catalyst seems to be more significant than in the other two cases.

_{Cl}at the end of the reaction, comparable if not higher than the other two cases.

#### 3.2.8. External Thermal Resistance

_{As}profiles show a considerable accumulation of the reagent given the low reaction rates, which are functions of operation temperature. Finally, it is interesting to notice that when the desired conversion is achieved, the bulk concentration of the final product C is extremely low in all the cases.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ES | Egg Shell |

EW | Egg White |

EY | Egg Yolk |

l | Liquid domain |

P | Particle |

s | Solid domain |

Symbols | |

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 [-] |

a_{sp} | Specific surface area [m^{2}/m^{3}] |

b | Smoothing factor [-] |

c_{i0}_{,}_{l} | Initial concentration in liquid phase [mol/m^{3}] |

c_{i}_{,}_{l} | Concentration in liquid phase [mol/m^{3}] |

c_{i}_{,}_{s} | Concentration in solid phase [mol/m^{3}] |

c_{p}_{,}_{i} | Specific heat of the fluid component [J/(mol K)] |

c_{p}_{,}_{l} | Specific heat of the liquid phase [J/(mol K)] |

c_{p}_{,}_{P} | Specific heat of the particle [J/(mol K)] |

D_{eff}_{,}_{i} | Effective molecular diffusivity [m^{2}/s] |

Ea_{j} | Activation energy of reaction j [J/mol] |

h | Thermal resistance solid-liquid interface [W/(m^{2} K)] |

k_{j} | Catalytic rate constant [m^{3}/(kg s)] |

k_{refj} | Catalytic reference rate constant [m^{3}/(kg s)] |

k_{m} | Fluid–solid mass transfer coefficient [m/s] |

k_{T}_{,}_{l} | Fluid heat conductivity [W/(m K)] |

k_{T}_{,}_{P} | Particle heat conductivity batch reactor [W/(m K)] |

MW_{i} | Molecular weight [g/mol] |

N | Number of component [-] |

r_{j} | Catalytic reaction rate [mol/(kg s)] |

R_{g} | Ideal gas constant [J/(K mol)] |

R_{P} | Particle radius [m] |

s | Shape factor [-] |

S_{B} | Selectivity B [-] |

t | Time [s] |

T_{j} | External temperature [K] |

T_{l} | Liquid phase temperature [K] |

T_{s} | Solid phase temperature [K] |

T_{ref} | Reference temperature [K] |

U_{A} | External thermal resistance [W/(m^{2} K)] |

x | Dimensionless radial coordinate of the spherical particle [–] |

X_{A} | Conversion degree of A [–] |

y_{B} | Yield B [-] |

Δ_{r}H_{j} | Reaction enthalpy [J/mol] |

ε | Volumetric ratio between the solid and the liquid phases batch reactor [-] |

ε_{p} | Catalyst porosity [-] |

η | Efficiency reaction [-] |

ν_{ij} | Stoichiometric matrix [-] |

ρ_{i} | Component density [kg/m^{3}] |

ρ_{l} | Liquid phase density [kg/m^{3}] |

ρ_{P} | Particle density [kg/m^{3}] |

ρ_{Bulk} | Catalyst bulk density (mass of catalyst/volume) [kg/m^{3}] |

Ωk | Distribution function [-] |

Subscripts | |

A | Initial reagent |

B | Intermediate reaction component |

C | final reaction product |

i | Component |

j | Reaction |

k | Distribution type |

ref | Reference |

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**Figure 1.**Intraparticle profiles for ES catalyst. Calculated profiles for: cA (

**a**), cB (

**b**), cC (

**c**) and Ts (

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

**Figure 2.**Intraparticle profiles for EW catalyst. Calculated profiles for: cA (

**a**), cB (

**b**), cC (

**c**) and Ts (

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

**Figure 3.**Intraparticle profiles for EY catalyst. Calculated profiles for: cA (

**a**), cB (

**b**), cC (

**c**), and Ts (

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

**Figure 4.**(

**a**) Bulk profiles over time for ES, EW, and EY. Calculated profiles for: X

_{A}and S

_{B}. (

**b**) Bulk profiles over time for EY, EW, and EY. Calculated profiles for: T

_{l}. (

**c**) Catalytic effectiveness factor profiles over time for ES, EW, and EY.

**Figure 5.**ES ΔrH

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 6.**EW ΔrH

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 7.**EY ΔrH

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 8.**ES ΔrH

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 9.**EW ΔrH

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 10.**EY ΔrH

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 11.**ES k

_{Cref}

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 12.**EW k

_{Cref}

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 13.**EY k

_{Cref}

_{1}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 14.**ES k

_{Cref}

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 15.**EW k

_{Cref}

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 16.**EY k

_{Cref}

_{2}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 17.**ES D

_{eff,A}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 18.**EW D

_{eff,A}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 19.**EY D

_{eff,A}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 20.**ES R

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 21.**EW R

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 22.**EY R

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 23.**ES ρ

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 24.**EW ρ

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 25.**EY ρ

_{P}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 26.**ES c

_{A}

_{0,l}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 27.**EW c

_{A}

_{0,l}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 28.**EY c

_{A}

_{0,l}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 29.**ES k

_{m}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 30.**EW k

_{m}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 31.**EY k

_{m}investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 32.**ES UA investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 33.**EW UA investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

**Figure 34.**EY UA investigation. (

**a**) Bulk profiles over time of X

_{A}and S

_{B}. (

**b**) Bulk profile of T

_{l}over time.

Parameter | Value | Unit |
---|---|---|

a_{1} | 0.60 | - |

a_{2} | 0.40 | - |

a_{31} | 0.20 | - |

a_{32} | 0.60 | - |

b | 1.00·10^{−5} | - |

c_{A}_{0}_{,}_{l} | 1.00 | mol m^{−3} |

c_{B}_{0,l} | 0 | mol m^{−3} |

c_{C}_{0,l} | 0 | mol m^{−3} |

cp_{i} | 30.0 | J (mol K)^{−1} |

cp_{P} | 6.00·10^{2} | J (kg K)^{−1} |

D_{eff}_{,}_{i} | 1.00·10^{−6} | m^{2} s^{−1} |

Ea_{j} | 8.00·10^{4} | J mol^{−1} |

k_{Cref}_{1} | 2.00·10^{−4} | m^{3} (kg s)^{−1} |

k_{Cref}_{2} | 5.00·10^{−8} | m^{3} (kg s)^{−1} |

k_{m} | 1.00·10^{2} | m s^{−1} |

k_{T}_{,}_{l} | 0.60 | J (s m K)^{−1} |

k_{T}_{,}_{P} | 0.10 | W (m K)^{−1} |

MW_{i} | 30.0 | g mol^{−1} |

R_{p} | 5.00·10^{−3} | m |

UA | 0 | W (m^{3} K)^{−1} |

|ΔrH_{j}| | 5.00·10^{4} | J mol^{−1} |

ε_{P} | 0.50 | - |

ρ_{B} | 1.00 | kg m^{−3} |

ρ_{i} | 1.00·10^{3} | kg m^{−3} |

ρ_{P} | 4.00·10^{3} | kg m^{−3} |

Value | |||
---|---|---|---|

Parameter | Lower | Higher | Unit |

|ΔrH_{1}| | 3.00·10^{4} | 7.00·10^{4} | J mol^{−1} |

|ΔrH_{2}| | 3.00·10^{4} | 7.00·10^{4} | J mol^{−1} |

k_{Cref}_{1} | 7.00·10^{−5} | 7.00·10^{−4} | m^{3} (kg s)^{−1} |

k_{Cref}_{2} | 5.00·10^{−10} | 5.00·10^{−6} | m^{3} (kg s)^{−1} |

D_{eff}_{,}_{A} | 5.00·10^{−7} | 5.00·10^{−6} | m^{2} s^{−1} |

R_{p} | 2.50·10^{−3} | 1.00·10^{−2} | m |

ρ_{B} | 0.50 | 2.00 | kg m^{−3} |

c_{A}_{0,l} | 0.50 | 2.00 | mol m^{−3} |

k_{m} | 1.00·10^{−3} | - | m s^{−1} |

UA | - | 1.00·10^{3} | W (m^{3} K)^{−1} |

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**MDPI and ACS Style**

Salucci, E.; Russo, V.; Salmi, T.; Di Serio, M.; Tesser, R.
Intraparticle Model for Non-Uniform Active Phase Distribution Catalysts in a Batch Reactor. *ChemEngineering* **2021**, *5*, 38.
https://doi.org/10.3390/chemengineering5030038

**AMA Style**

Salucci E, Russo V, Salmi T, Di Serio M, Tesser R.
Intraparticle Model for Non-Uniform Active Phase Distribution Catalysts in a Batch Reactor. *ChemEngineering*. 2021; 5(3):38.
https://doi.org/10.3390/chemengineering5030038

**Chicago/Turabian Style**

Salucci, Emiliano, Vincenzo Russo, Tapio Salmi, Martino Di Serio, and Riccardo Tesser.
2021. "Intraparticle Model for Non-Uniform Active Phase Distribution Catalysts in a Batch Reactor" *ChemEngineering* 5, no. 3: 38.
https://doi.org/10.3390/chemengineering5030038