# Accurate Effective Diffusivities in Multicomponent Systems

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## Abstract

**:**

## 1. Introduction

## 2. New Effective Diffusivity Model

#### 2.1. Derivation of Model Equations

#### 2.2. Calculation Procedure

- Using tabulated experimental data or empirical correlations, collect the binary diffusion coefficients at infinite dilution of all $ij$ pairs of components, ${D}_{ij}^{\xb0}$. These are equal to the infinite dilution binary Maxwell–Stefan (MS) diffusion coefficients, ${\xd0}_{ij}^{\xb0}={D}_{ij}^{\xb0}$.
- Compute the MS diffusion coefficients, ${\xd0}_{ij}$, for the specific mixture composition using the following mixing rule:$${\xd0}_{ij}={({\xd0}_{ij}^{\xb0})}^{(1-{x}_{i}+{x}_{j})/2}\xb7{({\xd0}_{ji}^{\xb0})}^{(1-{x}_{j}+{x}_{i})/2}$$
- Calculate the elements of the $\left[B\right]$ matrix, via Equation (2b,c), and compute its inverse, ${\left[B\right]}^{-1}$.
- Compute the $[\Gamma ]$ matrix by applying Equation (2d), which requires an appropriate thermodynamic model to describe the nonideal behavior of the mixture. The partial derivatives can be computed numerically using, for instance, central finite differences. The increments in the mole fraction of a component j are absorbed by negative increments in the nth component in order to maintain the sum of all mole fractions equal to 1. For instance:$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \frac{\partial ln{\gamma}_{i}}{\partial {x}_{j}}{|}_{P,T,{x}_{k}[k=1,2,\dots ,n-1,k\ne j]}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \frac{1}{2h}\left[ln{\gamma}_{i}(P,T,{x}_{1},\dots ,{x}_{j}+h,\dots ,{x}_{n}-h)-ln{\gamma}_{i}(P,T,{x}_{1},\dots ,{x}_{j}-h,\dots ,{x}_{n}+h)\right]\hfill \end{array}$$
- Obtain matrix $\left[D\right]={\left[B\right]}^{-1}[\Gamma ]$ and its inverse ${\left[D\right]}^{-1}$.
- Calculate the effective diffusivity, ${D}_{i,\mathrm{eff}}$, with Equations (11) and (14). The method to obtain the molar diffusion fluxes ratios depends on the complexity of the chemical reaction(s) and is a function of the stoichiometric coefficients of the species that participate in the reaction(s).

#### 2.3. Effective Diffusivity for Ideal Mixtures

## 3. Examples of Application

#### 3.1. Liquid Phase Reaction: Ethyl Acetate Synthesis

- Obtain K or ${K}_{x}$ from the literature.
- Make an initial guess for the extent of reaction at equilibrium, ${\xi}_{eq}$.
- Compute equilibrium compositions (${x}_{i,eq}$) for the assumed ${\xi}_{eq}$ (via Equation (23)) and then the respective activity coefficients, ${\gamma}_{i,eq}$.
- Calculate the equilibrium constant via Equation (24), ${K}^{calc}$.
- Compute the square of the deviation ${({K}^{calc}-K)}^{2}$.
- Repeat steps 2–5 until the squared error is below a predetermined tolerance.

_{2}O for this reaction). The graphical comparison in terms of ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$ (with ${D}_{i,\mathrm{eff}}^{Ideal}$ given by Equation (6)) and ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$ (with ${D}_{i,\mathrm{eff}}^{W}$ given by Equation (5)) is illustrated in Figure 1.

#### 3.2. High-Pressure Gas Phase Reaction: Methanol Synthesis

- For a given temperature, pressure and initial mixture composition calculate the final values of the extents of reaction, ${\xi}_{eq}^{I}$ and ${\xi}_{eq}^{II}$, as described in Section 3.1.
- Calculate the corresponding ${\xi}_{eq}^{II}$ for each ${\xi}^{I}$ in the span of $[0,{\xi}_{eq}^{I}]$ by solving Equation (38) numerically:$$\frac{{K}^{II,calc}}{{K}^{II}}-1=0\iff \frac{{y}_{{H}_{2}O}\xb7{y}_{CO}}{{y}_{C{O}_{2}}\xb7{y}_{{H}_{2}}}\times \frac{{\widehat{\varphi}}_{{H}_{2}O}\xb7{\widehat{\varphi}}_{CO}}{{\widehat{\varphi}}_{C{O}_{2}}\xb7{\widehat{\varphi}}_{{H}_{2}}}\times \frac{1}{{K}^{II}}-1=0$$
- Once ${\xi}_{eq}^{II}$ has been determined as function of ${\xi}^{I}$ (over $[0,{\xi}_{eq}^{I}]$), the derivatives $\frac{d{\xi}_{eq}^{II}}{d{\xi}^{I}}$ can be calculated numerically using finite differences, for instance.
- The effective diffusivities can then be evaluated following the procedure delineated in Section 3.1.

_{3}OH and H

_{2}O. In order to account for such limitations effective diffusivities are also evaluated with PC-SAFT EoS. Although this equation can also incorporate binary interaction parameters fitted to experimental data, due to its theoretical foundations in statistical mechanics it tends to be highly reliable in its predictive capability as shown for several binary systems in the work by Gross and Sadowski [17].

_{2}they are: 12.14% CO, 0.12% CH

_{3}OH, 70.94% H

_{2}, 0.16% H

_{2}O, 14.90% CH

_{4}(inert) and 1.74% CO

_{2}. Fugacity coefficients from PR EoS are computed with a self-developed program. The code developed by Ángel Martín is used to compute fugacity coefficients with PC-SAFT EoS [19,20].

_{4}at equilibrium (−48.8%) and in second place for CO

_{2}at the beginning of the reaction (24.2%). For PC-SAFT EoS (Figure 2b), the largest and second largest deviations are for the same components, although with lower deviations for CH

_{4}(−23.1%) and CO

_{2}(22.4%) both at ${\xi}^{I}=0$. Other components exhibit significant differences (above 10%) with the exception of CO, whose deviation is less than 1% in magnitude. Thus, only the results for CO are in good agreement with the ideal model for this reaction system at the aforementioned temperature and pressure conditions. The average absolute deviation across all components is 13.1% and 9.0% for PR EoS and PC-SAFT EoS, respectively, with the largest average deviation being for CH

_{4}(PR EoS: −45.8%, PC-SAFT EoS: −18.2%), followed by CO

_{2}(PR EoS: 13.0%, PC-SAFT EoS: 11.1%).

_{4}, which reaches 933.8% at ${\xi}^{I}={\xi}_{eq}^{I}$, followed by CO with −36.6% at ${\xi}^{I}=0$. The component that exhibits the lowest ratio is methanol, reaching a maximum of 5.9%. The same pattern is observed with PC-SAFT EoS (Figure 3b and Table 3) but in lower percentages (716.1% for CH

_{4}, −34.8% for CO). Although the CO deviation reduces as the first reaction approaches equilibrium, that of H

_{2}remains high throughout the entire range of ${\xi}^{I}$ resulting in a higher average deviation for this component (PR EoS: −32.1%, PC-SAFT EoS: −30.5%) than for CO. However, once again the highest average deviation is for CH

_{4}(PR EoS: 604.6%, PC-SAFT EoS: 494.2%). The overall absolute averages are 138.3% and 114.8% for PR EoS and PC-SAFT EoS, respectively, with the numbers being skewed toward high values due to CH

_{4}.

_{4}, while slight deviations are achieved for CO, CH

_{3}OH, and H

_{2}. The ${D}_{i,\mathrm{eff},\mathrm{PR}}^{New}/{D}_{i,\mathrm{eff},\mathrm{PC-SAFT}}^{New}$ ratio for H

_{2}O decreases as the extent of reaction increases, exhibiting a maximum value of 7.0% at ${\xi}^{I}=0$. Such deviations are likely caused by the presence of dipole-dipole and hydrogen bonding interactions that are not accurately accounted by the PR EoS. Figure 4 emphasizes the distinct performance of both equations of state since very different fugacity coefficients are calculated for water and ethanol, notwithstanding their similar trends along reaction progression.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Activity |

B | Coefficients defined by Equation (2b,c), s/cm^{2} |

${C}_{T}$ | Total concentration, mol/cm^{3} |

D | Diffusion coefficient, cm^{2}/s |

Ð | Maxwell–Stefan diffusion coefficient, cm^{2}/s |

EoS | Equation of state |

h | Finite difference step size |

$\overline{J}$ | Molar diffusion flux, mol/(cm^{2} s) |

K | Equilibrium constant |

${k}_{ij}$ | Binary interaction parameter |

MS | Maxwell–Stefan |

$\overline{N}$ | Molar flux, mol/(cm^{2} s) |

n | Number of moles, mol, or number of components in a mixture |

P | Pressure, MPa |

PC-SAFT | Perturbed-Chain Statistical Associating Fluid Theory |

PR | Peng–Robinson |

r | Reaction rate, mol/(cm^{3} s) |

T | Temperature, K |

x | Mole fraction in the liquid phase |

y | Mole fraction in the gas phase |

Greek Letters | |

${\Gamma}_{ij}$ | Element of $[\Gamma ]$ matrix as defined by Equation (2d) |

$\gamma $ | Activity coefficient |

$\delta $ | Kronecker function |

$\nu $ | Stoichiometric coefficient |

$\xi $ | Extent of reaction |

$\varphi $ | Solvent association factor of Wilke–Chang equation |

$\widehat{\varphi}$ | Fugacity coefficient |

Subscripts | |

0 | Initial condition |

eff | Effective |

eq | Equilibrium |

ij | Refers to the pair of components i and j |

i, j, k, n | Arbitrary component identification |

T | Total |

Superscripts | |

${}^{\xb0}$ | Infinite dilution or Standard State |

$I,II$ | Reaction identification |

calc | Calculated value |

$BK$ | Burghardt and Krupiczka effective diffusivity model |

$Ideal$ | Ideal (Bird et al. [4]) effective diffusivity model |

$inv$ | Element of inverse matrix |

K | Kubota et al. [5] effective diffusivity model |

$Kato$ | Kato et al. [6] effective diffusivity model |

$New$ | New effective diffusivity model |

W | Wilke effective diffusivity model |

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**Figure 1.**Effective diffusivities ratios from $\xi =0$ up to equilibrium (${\xi}_{eq}=0.931$) for an initial equimolar reactants mixture at 78 ${}^{\xb0}$C: (

**a**) ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$, and (

**b**) ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$.

**Figure 2.**Effective diffusivities ratios ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$ as function of extent of reaction I, up to equilibrium, using (

**a**) Peng–Robinson EoS (${\xi}_{eq}^{I}=7.5068$; ${\xi}_{eq}^{II}=0.2994$), and (

**b**) PC-SAFT EoS (${\xi}_{eq}^{I}=7.3797$; ${\xi}_{eq}^{II}=0.2894$).

**Figure 3.**Effective diffusivities ratios ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$ as function of extent of reaction I, up to equilibrium, using (

**a**) Peng–Robinson EoS (${\xi}_{eq}^{I}=7.5068$; ${\xi}_{eq}^{II}=0.2994$), and (

**b**) PC-SAFT EoS (${\xi}_{eq}^{I}=7.3797$; ${\xi}_{eq}^{II}=0.2894$).

**Figure 4.**Fugacity coefficients as function of extent of reaction I computed with (

**a**) Peng–Robinson EoS and (

**b**) PC-SAFT EoS.

**Table 1.**Effective diffusivities ratios ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Model}$ calculated at beginning ($\xi =0$; values enclosed in parentheses) and equilibrium (${\xi}_{eq}=0.931$) for an initial equimolar reactants mixture at 78 ${}^{\xb0}$C using UNIFAC model to estimate the activity coefficients.

Component | CH_{3}COOH | CH_{3}CH_{2}OH | CH_{3}COOCH_{2}CH_{3} | H_{2}O | |
---|---|---|---|---|---|

Initial mole fractions | 0.500 | 0.500 | 0.000 | 0.000 | |

Calculated equilibrium mole fractions | 0.190 | 0.190 | 0.310 | 0.310 | |

Ratio | Reference Model | ||||

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$ | Ideal (Bird et al.) [4], Equation (6) | (1.379) 1.124 | (0.737) 0.904 | (1.000) 0.809 | (1.000) 1.384 |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$ | Wilke [3], Equation (5) | (1.536) 1.212 | (1.066) 1.029 | (1.000) 0.694 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{BK}$ | Burghardt and Krupiczka [2], Equation (8) | (1.254) 1.188 | (0.793) 0.999 | (1.0000) 0.650 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Kato}$ | Kato et al. [6], Equation (9) | (1.086) 1.048 | (0.780) 1.039 | (1.000) 1.625 | - |

**Table 2.**Effective diffusivities ratios ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Model}$ calculated at beginning (${\xi}^{I}=0$; ${\xi}_{eq}^{II}=0.1014$; values enclosed in parentheses) and equilibrium (${\xi}_{eq}^{I}=7.5068$; ${\xi}_{eq}^{II}=0.2994$), using Peng–Robinson EoS at 573.15 K and 10 MPa.

Component | CO | CH_{3}OH | H_{2} | H_{2}O | CH_{4} | CO_{2} | |
---|---|---|---|---|---|---|---|

Initial mole fractions | 0.1224 | 0.0012 | 0.7084 | 0.0026 | 0.1490 | 0.0164 | |

Calculated equilibrium mole fractions | 0.0580 | 0.0897 | 0.6545 | 0.0054 | 0.1753 | 0.0170 | |

Ratio | Reference Model | ||||||

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$ | Ideal (Bird et al.) [4], Equation (6) | (1.011) 0.9961 | (0.9994) 0.9612 | (0.9436) 0.8708 | (0.8817) 0.9465 | (0.5557) 0.5118 | (1.2423) 1.0446 |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$ | Wilke [3], Equation (5) | (0.6340) 0.8241 | (0.9992) 0.9414 | (0.6826) 0.6808 | (1.3416) 1.1576 | (5.2983) 10.3318 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{K}$ | Kubota et al. [5], Equation (7) | (0.7529) 0.8717 | (1.0018) 1.1376 | (0.2793) 0.3119 | (1.2530) 1.0454 | - | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{BK}$ | Burghardt and Krupiczka [2], Equation (8) | (0.8063) 0.8912 | (1.0015) 1.1119 | (0.7403) 0.6922 | (1.3480) 1.1658 | (6.8448) 12.9857 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Kato}$ | Kato et al. [6], Equation (9) | (0.7775) 0.8782 | (1.0023) 1.1822 | (0.7162) 0.7011 | (1.3473) 1.1656 | (6.7167) 12.7846 | - |

**Table 3.**Effective diffusivities ratios ${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Model}$ calculated at beginning (${\xi}^{I}=0$; ${\xi}_{eq}^{II}=0.0935$; values enclosed in parentheses) and equilibrium (${\xi}_{eq}^{I}=7.3797$; ${\xi}_{eq}^{II}=0.2894$) using PC-SAFT EoS at 573.15 K and 10 MPa.

Component | CO | CH_{3}OH | H_{2} | H_{2}O | CH_{4} | CO_{2} | |
---|---|---|---|---|---|---|---|

Initial mole fractions | 0.1223 | 0.0012 | 0.7085 | 0.0025 | 0.1490 | 0.0165 | |

Calculated equilibrium mole fractions | 0.0592 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 | |

Ratio | Reference Model | ||||||

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Ideal}$ | Ideal (Bird et al.) [4], Equation (6) | (1.0100) 1.0007 | (0.9994) 0.9569 | (0.9424) 0.8695 | (0.8339) 0.9152 | (0.7685) 0.8996 | (1.2238) 1.0319 |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{W}$ | Wilke [3], Equation (5) | (0.6525) 0.8441 | (1.0330) 0.9738 | (0.6962) 0.6976 | (1.2542) 1.1845 | (4.6316) 8.1610 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{K}$ | Kubota et al. [5], Equation (7) | (0.7749) 0.8963 | (1.0357) 1.1687 | (0.2848) 0.3231 | (1.1762) 1.1586 | - | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{BK}$ | Burghardt and Krupiczka [2], Equation (8) | (0.8298) 0.9148 | (1.0354) 1.1466 | (0.7551) 0.7100 | (1.2599) 1.1928 | (5.9837) 10.2637 | - |

${D}_{i,\mathrm{eff}}^{New}/{D}_{i,\mathrm{eff}}^{Kato}$ | Kato et al. [6], Equation (9) | (0.7943) 0.8986 | (1.0362) 1.2237 | (0.7450) 0.7306 | (1.2591) 1.1923 | (5.8233) 10.0110 | - |

**Table 4.**Effective diffusivities ratios ${D}_{i,\mathrm{eff},\mathrm{PR}}^{New}/{D}_{i,\mathrm{eff},\mathrm{PC-SAFT}}^{New}$ at ${\xi}^{I}=0$ (values enclosed in parentheses) and ${\xi}^{I}=7.3797$, at 573.15 K and 10 MPa.

Component | CO | CH_{3}OH | H_{2} | H_{2}O | CH_{4} | CO_{2} |
---|---|---|---|---|---|---|

Initial mole fractions (PR) | 0.1224 | 0.0012 | 0.7084 | 0.0026 | 0.1490 | 0.0164 |

Initial mole fractions (PC-SAFT) | 0.1223 | 0.0012 | 0.7085 | 0.0025 | 0.1490 | 0.0165 |

Calculated final mole fractions (PR) | 0.0593 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 |

Calculated final mole fractions (PC-SAFT) | 0.0592 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 |

${D}_{i,\mathrm{eff},\mathrm{PR}}^{New}/{D}_{i,\mathrm{eff},\mathrm{PC-SAFT}}^{New}$ | (0.9716) 0.9722 | (0.9672) 0.9678 | (0.9805) 0.9754 | (1.0697) 0.9796 | (1.1438) 1.2379 | (0.9140) 0.9343 |

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**MDPI and ACS Style**

Rios, W.Q.; Antunes, B.; Rodrigues, A.E.; Portugal, I.; Silva, C.M.
Accurate Effective Diffusivities in Multicomponent Systems. *Processes* **2022**, *10*, 2042.
https://doi.org/10.3390/pr10102042

**AMA Style**

Rios WQ, Antunes B, Rodrigues AE, Portugal I, Silva CM.
Accurate Effective Diffusivities in Multicomponent Systems. *Processes*. 2022; 10(10):2042.
https://doi.org/10.3390/pr10102042

**Chicago/Turabian Style**

Rios, William Q., Bruno Antunes, Alírio E. Rodrigues, Inês Portugal, and Carlos M. Silva.
2022. "Accurate Effective Diffusivities in Multicomponent Systems" *Processes* 10, no. 10: 2042.
https://doi.org/10.3390/pr10102042