# Revisiting Isothermal Effectiveness Factor Equations for Reversible Reactions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Development of Analytical Expressions for Effectiveness Factors

#### 2.1. Derivation of Generalized Thiele Modulus Expressions

#### 2.2. Guide for Applying the Generalized Thiele Modulus Analytical Expressions

- Identify the reaction type in Table 1;
- Gather the required input data, namely:
- Operating conditions: temperature and surface concentrations of all components (${C}_{js}$);
- Rate law parameters: kinetic constant ($k)$ and equilibrium constant (${K}_{C}$) at the operating temperature;
- Catalyst properties: particle characteristic dimension ($L$, ${R}_{p}$ or $V/S$), density (${\rho}_{p}$), porosity ($\epsilon $), and tortuosity ($\tau $);
- Effective diffusivities of all components in the reaction mixture (${D}_{\mathrm{e}\mathrm{f},j}^{\mathrm{m}\mathrm{i}\mathrm{x}}$) at the operating conditions;

- Calculate the equilibrium concentration of component $A$, ${C}_{{A}_{\mathrm{e}\mathrm{q}}}$;
- Calculate the effective diffusivities of all components in the porous catalyst, ${D}_{\mathrm{e}\mathrm{f},\mathrm{j}}$;
- Calculate the constants ${F}_{0},{F}_{1},{F}_{2}$ and ${F}_{3}$ given by the specific expressions in Table 2;
- Calculate ${\varphi}_{g}$ by Equation (15), using $L$ for the slab, $L={R}_{p}$ for the sphere, or $L=V/S$ for any other catalyst shape;
- Calculate $\eta $ using Equation (2) for slab ($L$) or any geometry (with $L=V/S$) or Equation (3) for spherical particles.

## 3. Numerical Validation of the Analytical Calculations of the Effectiveness Factors

## 4. Case Studies

#### 4.1. Case 1: Esterification of Acetic Acid with Ethanol (Type I Reaction)

#### 4.1.1. Process Description and Data Compilation

#### 4.1.2. Reactor Modelling and Effectiveness Factor Calculation

^{6}mol

^{−1}min

^{−1}g

_{cat}

^{−1}; (ii) the concentration equilibrium constant is ${K}_{C}=2.67$. The calculated equilibrium concentration of component A is ${C}_{{A}_{\mathrm{e}\mathrm{q}}}=3.24$ mol/dm³ (details can be found in Section SM2 of the Supplementary Material).

#### 4.1.3. Impact of Effective Diffusivity Calculations on Effectiveness Factor Results

#### 4.2. Case 2: Acetal Synthesis (Type VII Reaction)

#### 4.2.1. Process Description and Data Compilation

#### 4.2.2. Reactor Modelling and Effectiveness Factor Calculation

#### 4.2.3. Impact of Effective Diffusivity Calculations on Effectiveness Factor Results

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

AARD | average absolute deviation |

${a}_{j}$ | activity of component $j$ |

${C}_{j}$ | concentration of component $j$, mol/dm³ |

${C}_{t}$ | total concentration, mol/dm³ |

${D}_{\mathrm{e}\mathrm{f},j}$ | intraparticle effective diffusivity of component $j$, dm²/min |

${D}_{\mathrm{e}\mathrm{f},j}^{\mathrm{m}\mathrm{i}\mathrm{x}}$ | effective diffusivity of component $j$ in reaction medium, dm²/min |

Exp. | experiment |

${K}_{\mathrm{e}\mathrm{q}}$ | thermodynamic equilibrium constant |

${K}_{C}$ | equilibrium constant in terms of concentrations |

${K}_{x}$ | constant defined in Equation (23) |

${K}_{\gamma}$ | constant defined in Equation (23) |

$k$ | kinetic constant in terms of concentrations, dm^{6}/(mol g_{cat} min) |

${k}_{\mathrm{d}\mathrm{i}\mathrm{r}}$ | kinetic constant in terms of activities, mol/(g_{cat} min) |

$L$ | catalyst characteristic dimension, dm |

${N}_{A}$ | molar flux of component A, mol/(dm² min) |

$NP$ | number of points |

$n$ | order of forward reaction |

${Q}_{x}$ | quotient of mole fractions out of equilibrium |

${Q}_{\gamma}$ | quotient of activity coefficients out of equilibrium |

${R}_{p}$ | particle radius, $\mathsf{\mu}$m |

$r$ | rate of reaction, mol/(g_{cat} min) |

${r}_{s}$ | rate of reaction at catalyst surface conditions, mol/(g_{cat} min) |

${r}_{obs}$ | observed reaction rate, mol/(g_{cat} min) |

$S$ | external surface area of the catalyst particle, dm² |

$T$ | temperature, °C |

$t$ | time, min |

$V$ | catalyst volume, dm³ |

${V}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ | reactor mixture volume, dm³ |

${x}_{j}$ | liquid phase mole fraction |

${w}_{\mathrm{c}\mathrm{a}\mathrm{t}}$ | mass of catalyst, g_{cat} |

$z$ | position coordinate inside slab, dm |

Greek Letters | |

${\gamma}_{j}$ | activity coefficient of component $j$ |

$\epsilon $ | particle porosity |

$\eta $ | effectiveness factor |

${\nu}_{j}$ | stoichiometric coefficient of component $j$ |

${\rho}_{p}$ | particle density, g_{cat}/dm³ |

$\tau $ | particle tortuosity |

$\varphi $ | Thiele modulus, dimensionless |

${\varphi}_{g}$ | generalized Thiele modulus, dimensionless |

Subscripts | |

0 | catalyst center |

$\mathrm{d}\mathrm{i}\mathrm{r}$ | direct |

$\mathrm{e}\mathrm{f}$ | effective |

$\mathrm{e}\mathrm{q}$ | equilibrium |

$\mathrm{i}\mathrm{n}$ | initial conditions |

$j$ | arbitrary component in the mixture |

$\mathrm{s}$ | conditions at the catalyst surface or mixture bulk |

$t$ | total |

Superscripts | |

ID | refers to mixture effective diffusivities computed with the model by Bird et al. [38] |

$\mathrm{m}\mathrm{i}\mathrm{x}$ | mixture |

NID | refers to mixture effective diffusivities computed with a non-ideal model [37] |

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**Figure 1.**Schematic diagram for the analytic calculation of effectiveness factors for isothermal porous catalyst particles. ($L$ is the slab thickness, the sphere radius, or $V/S$ for any other shape).

**Figure 2.**Numerical (points) and analytical (line) effectiveness factors for reaction $A+B\rightleftharpoons C+D$ (Type I; $n=2)$ in (

**a**) slab, and (

**b**) spherical porous catalyst particles. Figures were generated as function of $\varphi $ for ${C}_{\mathrm{A}\mathrm{s}}={C}_{\mathrm{B}\mathrm{s}}=8.53$ mol/dm

^{3}, ${C}_{\mathrm{C}\mathrm{s}}={C}_{\mathrm{D}\mathrm{s}}=0$ mol/dm

^{3}, and ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{B}}=1.440$, ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{C}}=1.579$, ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{D}}=0.861$ (See details in Section 4.1).

**Figure 3.**Numerical (points) and analytical (line) effectiveness factors for reaction $2A+B\rightleftharpoons C+D$ (type VII, $n=2$ ) in (

**a**) slab, and (

**b**) spherical porous catalyst particles. Figures were generated as function of $\varphi $ for ${C}_{\mathrm{A}\mathrm{s}}=14.703$ mol/dm

^{3}, ${C}_{\mathrm{B}\mathrm{s}}=7.247$ mol/dm

^{3}, ${C}_{\mathrm{C}\mathrm{s}}={C}_{D\mathrm{s}}=0$ mol/dm

^{3}, and ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{B}}=0.5068$, ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{C}}=1.057$, ${D}_{\mathrm{e}\mathrm{f},\mathrm{A}}/{D}_{\mathrm{e}\mathrm{f},\mathrm{D}}=0.6129$. (See details in Section 4.2).

**Figure 4.**Experimental (points) and modeling (curves) results for the catalytic esterification of acetic acid (A) with ethanol in a batch reactor for the operating conditions of Table 3. Mass and average diameter of Amberlyst-15 particles: Δ Exp. 1 (5.0058 g and $744$ μm); ○ Exp. 2 (10.0134 g and $744$ μm); and ✶ Exp. 3 (5.0024 g and $463$ μm).

**Figure 5.**Relative deviations between effectiveness factors computed using ideal and non-ideal effective diffusivities (i.e., ${\eta}^{NID}$ and ${\eta}^{ID}$ ), for reaction $A+B\rightleftharpoons C+D$ (Type I; $n=2)$ in (

**a**) slab and (

**b**) spherical porous catalyst particles. Figures were generated as function of $\varphi $ at initial conditions (solid line) and near equilibrium (dashed line).

**Figure 6.**Experimental data taken from Gandi et al. [40] (points) and modeling (curves) results for the catalyzed synthesis of acetal in batch reactor for the operating conditions of Table 6. Average diameter of Amberlyst-15 particles: ○ Exp. 1 ($335$ μm); Δ Exp. 2 ($510$ μm); and ✶ Exp. 3 (800 μm).

**Figure 7.**Relative deviations between effectiveness factors computed using ideal and non-ideal effective diffusivities (i.e., ${\eta}^{NID}$ and ${\eta}^{ID}$ ), for reaction $2A+B\rightleftharpoons C+D$ (Type VII; $n=2)$ in (

**a**) slab, and (

**b**) spherical porous catalyst particles. Figures were generated as function of $\varphi $ at initial conditions (solid line) and near equilibrium (dashed line).

**Table 1.**List of reversible reactions and corresponding rate law equations for which generalized Thiele modulus equations were developed in this work.

Reaction Type | Reaction | Rate Law |
---|---|---|

I | $A+B\rightleftharpoons C+D$ | $r=k\left({C}_{A}{C}_{B}-\frac{1}{{K}_{C}}{C}_{C}{C}_{D}\right)$ |

II | $2A\rightleftharpoons C+D$ | $r=k\left({C}_{A}^{2}-\frac{1}{{K}_{C}}{C}_{C}{C}_{D}\right)$ |

III | $A+B\rightleftharpoons 2C$ | $r=k\left({C}_{A}{C}_{B}-\frac{1}{{K}_{C}}{C}_{C}^{2}\right)$ |

IV | $A\rightleftharpoons C+D$ | $r=k\left({C}_{A}-\frac{1}{{K}_{C}}{C}_{C}{C}_{D}\right)$ |

V | $A+B\rightleftharpoons C$ | $r=k\left({C}_{A}{C}_{B}-\frac{1}{{K}_{C}}{C}_{C}\right)$ |

VI | $A\rightleftharpoons C$ | $r=k\left({C}_{A}-\frac{1}{{K}_{C}}{C}_{C}\right)$ |

VII | $2A+B\rightleftharpoons C+D$ | $r=k\left({C}_{A}{C}_{B}-\frac{1}{{K}_{C}}\frac{{C}_{C}{C}_{D}}{{C}_{A}}\right)$ |

**Table 2.**Expressions for the constants ${F}_{0},{F}_{1},{F}_{2},$ and ${F}_{3}$ necessary to compute the generalized Thiele modulus, Equation (15), for the reversible reactions presented in Table 1.

Case | ${\mathit{F}}_{0}$ | ${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ |
---|---|---|---|---|

1 | $0$ | ${C}_{As}^{2}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | ${C}_{As}\left(\frac{{C}_{Bs}}{{C}_{As}}-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}\right)+\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)+\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}-\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}$ |

2 | $0$ | ${C}_{As}^{2}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{1}{2}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{1}{2}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{2D}_{\mathrm{e}\mathrm{f},C}}\right)+\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $1-\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}$ |

3 | $0$ | ${C}_{As}^{2}{\left(\frac{{C}_{Cs}}{{C}_{As}}+2\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)}^{2}$ | ${C}_{As}\left(\frac{{C}_{Bs}}{{C}_{As}}-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}\right)+4\frac{{C}_{As}}{{K}_{C}}\cdot \frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\cdot \left(\frac{{C}_{Cs}}{{C}_{As}}+2\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)$ | $\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}-\frac{1}{{K}_{C}}\cdot {\left(2\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)}^{2}$ |

4 | $0$ | ${C}_{As}^{2}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $1+\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)+\frac{{C}_{As}}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $-\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},D}}$ |

5 | $0$ | ${C}_{As}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)$ | ${C}_{As}\left(\frac{{C}_{Bs}}{{C}_{As}}-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}\right)+\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}$ | $\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},B}}$ |

6 | $0$ | ${C}_{As}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}\right)$ | $1+\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{{D}_{\mathrm{e}\mathrm{f},C}}$ | $0$ |

7 | ${C}_{As}^{2}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}\right)\phantom{\rule{0ex}{0ex}}\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | $-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}{C}_{As}\left(\frac{{C}_{Cs}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}\right)-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}{C}_{As}\left(\frac{{C}_{Ds}}{{C}_{As}}+\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}\right)$ | ${C}_{As}\left(\frac{{C}_{Bs}}{{C}_{As}}-\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},B}}\right)-\frac{1}{{K}_{C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},C}}\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},D}}$ | $\frac{{D}_{\mathrm{e}\mathrm{f},A}}{2{D}_{\mathrm{e}\mathrm{f},B}}$ |

**Table 3.**Catalyst properties, rate law constants, and experimental conditions used to study the liquid phase esterification of acetic acid with ethanol in batch reactor.

$\mathbf{Initial}\mathbf{Concentrations}:{\mathit{C}}_{{\mathit{j}}_{\mathbf{i}\mathbf{n}}}$ (mol/dm^{3})
| |||
---|---|---|---|

Acetic acid ($A$) | Ethanol ($B$) | Ethyl acetate ($C$) | Water ($D$) |

${C}_{{A}_{\mathrm{i}\mathrm{n}}}=8.53$ | ${C}_{{B}_{\mathrm{i}\mathrm{n}}}=8.53$ | ${C}_{{C}_{\mathrm{i}\mathrm{n}}}=0$ | ${C}_{{D}_{\mathrm{i}\mathrm{n}}}=0$ |

Catalyst (Amberlyst-15 wet) properties | |||

${\rho}_{p}$ (g_{cat}/dm³) [34] | 600 | ||

$\epsilon $ (*) | 0.489 | ||

$\tau $ [35] | 1.3 | ||

Operating conditions: | T = 78 °C (isothermal) | ${V}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ = 0.162 dm^{3} | Batch reactor, 900 rpm |

Exp. 1 | Exp. 2 | Exp. 3 | |

${w}_{\mathrm{c}\mathrm{a}\mathrm{t}}$ (g_{cat}) | 5.0058 | 10.0134 | 5.0024 |

$2\times {R}_{p}$ (μm) | 644 | 644 | $463$ |

Rate law constants (Type I in Table 1) in terms of concentrations at 78 °C [33] | |||

For Amberlyst-15: $k=4.35\times {10}^{-5}$ dm^{6} mol^{−1} min^{−1} g_{cat}^{−1} | |||

Equilibrium constant: ${K}_{C}=2.67$ |

**Table 4.**Calculated effectiveness factors for the liquid phase Amberlyst-15 catalyzed esterification of acetic acid with ethanol for the operating conditions described in Table 3.

Catalyst Diameter (μm) | AARD (%) | $\mathit{\eta}$ at Initial Conditions | $\mathit{\eta}$ Near Equilibrium | |
---|---|---|---|---|

Exp. 1 | 744 | 1.12 | 0.9626 | 0.9783 |

Exp. 2 | 744 | 1.65 | 0.9626 | 0.9785 |

Exp. 3 | 463 | 2.21 | 0.9853 | 0.9916 |

**Table 5.**Effective diffusivities in solution calculated for the initial conditions of reactor (see Table 3).

${\mathit{D}}_{\mathbf{e}\mathbf{f},\mathit{j}}^{\mathbf{m}\mathbf{i}\mathbf{x}}$ (dm^{2}/min)
| $\mathbf{Acetic}\mathbf{Acid}\left(\mathit{A}\right)$ | $\mathbf{Ethanol}\left(\mathit{B}\right)$ | $\mathbf{Ethyl}\mathbf{Acetate}\left(\mathit{C}\right)$ | $\mathbf{Water}\left(\mathit{D}\right)$ |
---|---|---|---|---|

Non-ideal model, Rios et al. [37] | 3.17 $\times {10}^{-5}$ | 2.20 $\times {10}^{-5}$ | 2.01 $\times {10}^{-5}$ | 3.68 $\times {10}^{-5}$ |

Ideal model, Bird et al. [38] | 2.75 $\times {10}^{-5}$ | 3.28 $\times {10}^{-5}$ | 2.87 $\times {10}^{-5}$ | 3.11 $\times {10}^{-5}$ |

**Table 6.**Catalyst properties, rate law constants and experimental conditions for Amberlyst-15 catalyzed acetal synthesis reaction in batch reactor [40].

$\mathbf{Initial}\mathbf{Concentrations},{\mathit{C}}_{{\mathit{j}}_{\mathbf{i}\mathbf{n}}}$ (mol/dm^{3})
| |||
---|---|---|---|

Methanol (A) | Acetic acid (B) | Acetal (C) | Water (D) |

${C}_{{A}_{\mathrm{i}\mathrm{n}}}=14.703$ | ${C}_{{B}_{\mathrm{i}\mathrm{n}}}=7.247$ | ${C}_{{C}_{\mathrm{i}\mathrm{n}}}=0$ | ${C}_{{D}_{\mathrm{i}\mathrm{n}}}=0$ |

Catalyst (Amberlyst-15 dry) properties (*) | |||

${\rho}_{p}$ (g_{cat}/dm³) | 1205 | ||

$\epsilon $ | 0.36 | ||

$\tau $ (*) | 1.79 | ||

Operating conditions: | |||

T = 20 °C (isothermal) | ${w}_{\mathrm{c}\mathrm{a}\mathrm{t}}$ = 0.79 g_{cat} | ${V}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ = 0.600 dm^{3} | Batch reactor, 600 rpm |

Exp. 1 | Exp. 2 | Exp. 3 | |

$2\times {R}_{p}$ (μm) | 335 | 510 | 800 |

Rate law constants (Type VII in Table 1) in terms of activities at 20 °C | |||

${k}_{\mathrm{d}\mathrm{i}\mathrm{r}}$ (mol min^{−1} g_{cat}^{−1}) | 9.13 | ||

$K$ | 21.934 | ||

${K}_{x}$ | 5.353 |

**Table 7.**Calculated effectiveness factors for the liquid phase Amberlyst-15 catalyzed acetal synthesis for the conditions described in Table 6.

Catalyst Diameter (μm) | AARD (%) | $\mathit{\eta}$ at Initial Conditions | $\mathit{\eta}$ Near Equilibrium |
---|---|---|---|

335 | 1.23 | 0.2701 | 0.2265 |

510 | 3.38 | 0.1840 | 0.1507 |

800 | 2.43 | 0.1208 | 0.0981 |

**Table 8.**Effective diffusivities in solution calculated for the initial conditions described in Table 6.

${\mathit{D}}_{\mathbf{e}\mathbf{f},\mathit{j}}^{\mathbf{m}\mathbf{i}\mathbf{x}}$ (dm^{2}/min)
| Methanol (A) | Acetic Acid (B) | Acetal (C) | Water (D) |
---|---|---|---|---|

Non-ideal model, Rios et al. [37] | 1.74 $\times {10}^{-5}$ | 3.43 $\times {10}^{-5}$ | 1.64 $\times {10}^{-5}$ | 2.83 $\times {10}^{-5}$ |

Ideal model, Bird et al. [38] | 1.82 $\times {10}^{-5}$ | 2.93 $\times {10}^{-5}$ | 1.64 $\times {10}^{-5}$ | 3.17 $\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Rios, W.Q.; Antunes, B.; Rodrigues, A.E.; Portugal, I.; Silva, C.M.
Revisiting Isothermal Effectiveness Factor Equations for Reversible Reactions. *Catalysts* **2023**, *13*, 889.
https://doi.org/10.3390/catal13050889

**AMA Style**

Rios WQ, Antunes B, Rodrigues AE, Portugal I, Silva CM.
Revisiting Isothermal Effectiveness Factor Equations for Reversible Reactions. *Catalysts*. 2023; 13(5):889.
https://doi.org/10.3390/catal13050889

**Chicago/Turabian Style**

Rios, William Q., Bruno Antunes, Alírio E. Rodrigues, Inês Portugal, and Carlos M. Silva.
2023. "Revisiting Isothermal Effectiveness Factor Equations for Reversible Reactions" *Catalysts* 13, no. 5: 889.
https://doi.org/10.3390/catal13050889