# Molybdenum Disulphide Precipitation in Jet Reactors: Introduction of Kinetics Model for Computational Fluid Dynamics Calculations

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) is a transition-metal dichalcogenide (TMD) and a valuable 2D-nanomaterial used in a wide range of industrial applications. The main reason for such a wide range of applications of molybdenum disulphide in various fields is its 2D terrace structure and its crystalline properties. One promising application is the use of MoS

_{2}in several catalytic reactions such as hydrogen evolution reactions (HERs), hydrodesulphurization, oxygen reduction reactions (ORRs), and methane conversion [16,17,18,19,20]. In addition, MoS

_{2}has been used for many years as a dry lubricant and an oil additive. New research also shows the possibility of improving MoS

_{2}dispersion by synthesizing MoS

_{2}particles on the surface of graphene materials [21]. These applications could be essential in future environmental projects involving sustainable energy sources, especially because the environmental impact of wet chemical synthesis is acceptable [22]. Owing to the many possible MoS

_{2}applications, the MoS

_{2}synthesis reaction kinetics must be accurately described, and the impact of the process conditions on the final product must be accurately modelled [23]. Most of the modern applications of molybdenum disulphide require a chemically pure material with reproducible properties. This wide range of applications makes molybdenum disulphide one of the most frequently studied materials in terms of its synthesis and properties. Particle size is of particular importance for catalysts and lubricants, but it is also important for applications to improve the properties of polymeric materials due to the necessity of achieving suitable dispersions. Recrystallisation or other processes related to the processing of the raw material are more commonly used in the production of electronic materials, but, even then, a substance with a known particle size distribution and morphology is essential. This type of material can be obtained by precipitation, also known as reactive crystallisation, in which the driving force behind the process is the supersaturation resulting from a chemical reaction.

_{2}particles have previously been published. Therefore, this study presents the first such attempt and draws conclusions from the obtained results. The study findings shed new light on MoS

_{2}precipitated using wet chemical synthesis, which is particularly important for optimally designing reactor geometries to carry out the described process. This will be crucial for the application of this method to produce nanoparticles with the desired properties for industrial application.

## 2. Materials and Methods

_{2}in impinging jet reactors (Figure 1), using ammonium heptamolybdate (HMA) and ammonium sulphide (AS) as reaction substrates in citric acid (CA), which acts as a catalyst and allows the reaction pH to be set. To enable reproducibility of the test, information on substrates and their preparation is described in the publications [24,25,26]. It should be noted that the aqueous solution of ammonium sulphide is unstable and degrades in contact with air, eventually leading to precipitation of free sulphur, which can completely distort the results, therefore clean and, above all, properly stored ammonium sulphide should always be used for the process. The model predicts only the particle nucleation and growth kinetics and does not allow for particle aggregation and agglomeration, which also occur extensively during the process. The reaction and particle morphology analysis were described in more detail in previous works [24,25,26]. Previously published experimental data were also used to validate the model [24,25].

_{2}precipitation. The share functions for the HMA concentration are shown in Figure 2.

## 3. CFD Modeling

_{1}and $1-{F}_{1}$ mixing functions, respectively, and the equations from both models are summed. F

_{1}= 1 near the wall and 0 elsewhere in the region.

_{2}and, therefore, do not exhibit typical MoS

_{2}lubricating properties. The mixture viscosity was calculated using Kendall and Monroe’s formula as follows:

_{2}particle sizes very well for higher Reynolds numbers. Those conditions were assumed to be close to ideal mixing—for which the correction factor is equal or close to unity. Therefore, the correction factor was defined as the quotient of the local Reynolds number to the Reynolds number obtained under near-ideal mixing conditions. The correction factor is plotted as functions of the AS inlet Reynolds number in Figure 6, and the maximum correction factor was unity. The average local Reynolds number was calculated for mesh cells in which the energy dissipation rate was 10 times higher than the average in the computational domain, which corresponds to the collision zone area. Therefore, the growth rate is defined as follows:

## 4. Results and Discussion

_{2}precipitation were compared for both reactor types.

## 5. Conclusions

_{2}particles synthesized using confined impinging jet reactors. Because the improved model can predict particle size distributions with satisfactory accuracy, it can provide a valuable engineering tool for designing chemical reactors to produce MoS

_{2}particles. Parity plots of characteristic particle sizes clearly show that the model can accurately predict particle size distributions regardless of the apparatus geometry.

- The model enables kinetic constants to be determined for other complex chemical reactions, even with limited knowledge of the reaction mechanism, and can be applied with CFD to various reaction processes;
- The effect of the mixing conditions on the chemical reaction was determined using the SST k–ω model combined with the developed kinetic model to calculate results close to the experimental ones. In addition, the modelling and experimental results deviated more markedly at higher concentrations and lower flow rates, which may be because the reactant mixing was worse and, consequently, deviated even further from the ideal mixing conditions. Under these conditions the viscosity is higher due to the higher concentration of citric acid, what strongly affects the flow parameters, and thus the mixing-limited reaction;
- The concentration and velocity contour plots indicate that the V-type reactor exhibited superior fluid mixing than the T-type one at the same flow rate. However, although this is associated with a more marked pressure drop, the particles were better mixed and nucleated over a much larger area. Fluid mixing is also affected by reagent concentration because fluid viscosity increases with increasing reagent concentration, which affects the Reynolds number.

_{2}, which will enable the development of a more physical nucleation and growth model. Additionally, particle aggregation, agglomeration, and aggregate and agglomerate disintegration should be investigated in the future to determine their effect on the precipitation process and possible separation of particles from the suspension.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## Nomenclature

$a$ | $-$ | share function constant |

${a}_{he}$ | $-$ | heterogenous nucleation rate constant for sulfur particles |

${a}_{he}^{\prime}$ | $-$ | heterogenous nucleation rate constant for MoS_{2} particles |

${a}_{ho}$ | $-$ | homogenous nucleation rate constant for sulfur particles |

${a}_{ho}^{\prime}$ | $-$ | homogenous nucleation rate constant for MoS_{2} particles |

${b}_{he}$ | $-$ | heterogenous nucleation rate constant for sulfur particles |

${b}_{he}^{\prime}$ | $-$ | heterogenous nucleation rate constant for MoS_{2} particles |

${b}_{ho}$ | $-$ | homogenous nucleation rate constant for sulfur particles |

${b}_{ho}^{\prime}$ | $-$ | homogenous nucleation rate constant for MoS_{2} particles |

${B}_{N}$ | ${\mathrm{m}}^{-4}{\text{}\mathrm{s}}^{-1}$ | birth rate |

${C}_{AS}$ | ${\mathrm{mol}\text{}\mathrm{m}}^{-3}$ | ammonium sulphide concentration |

${C}_{HMA}$ | ${\mathrm{mol}\text{}\mathrm{m}}^{-3}$ | ammonium heptamolybdate concentration |

${D}_{N}$ | ${\mathrm{m}}^{-4}{\text{}\mathrm{s}}^{-1}$ | death rate |

${F}_{1}$ | $-$ | auxiliary function in equation for turbulent viscosity in SST k–ω model |

${F}_{2}$ | $-$ | auxiliary function in equation for turbulent viscosity in SST k–ω model |

$G$ | ${\mathrm{m}\text{}\mathrm{s}}^{-1}$ | total particle growth rate |

${G}_{{\mathrm{MoS}}_{2}}$ | ${\mathrm{m}\text{}\mathrm{s}}^{-1}$ | particle growth rate for MoS_{2} particles |

${G}_{S}$ | ${\mathrm{m}\text{}\mathrm{s}}^{-1}$ | particle growth rate for sulfur particles |

$k$ | $-$ | turbulence kinetic energy |

${k}_{a}$ | $-$ | surface shape factor |

${k}_{r}$ | ${\mathrm{m}}^{5}{\text{}\mathrm{s}}^{-1}{\text{}\mathrm{mol}}^{-2}$ | linear growth rate constant for sulfur particles |

${k}_{r}^{\prime}$ | ${\mathrm{m}}^{5}{\text{}\mathrm{s}}^{-1}{\text{}\mathrm{mol}}^{-2}$ | linear growth rate constant for MoS_{2} particles |

${K}_{sp}$ | ${\mathrm{mol}}^{3}{\text{}\mathrm{dm}}^{-3}$ | solubility index |

$L$ | m | particle size |

${L}_{10}$ | m | characteristic dimension |

${L}_{32}$ | m | characteristic dimension |

${L}_{30}$ | m | characteristic dimension |

${L}_{43}$ | m | characteristic dimension |

$M$ | ${\mathrm{kg}\text{}\mathrm{mol}}^{-1}$ | molar mass |

m_{0} | ${\mathrm{m}}^{-3}$ | moment 0 |

m_{1} | ${\mathrm{m}\text{}\mathrm{m}}^{-3}$ | moment 1 |

m_{2} | ${\mathrm{m}}^{2}{\text{}\mathrm{m}}^{-3}$ | moment 2 |

m_{3} | ${\mathrm{m}}^{3}{\text{}\mathrm{m}}^{-3}$ | moment 3 |

m_{4} | ${\mathrm{m}}^{4}{\text{}\mathrm{m}}^{-3}$ | moment 4 |

$n$ | $-$ | moment index |

${R}_{M}$ | ${\mathrm{mol}\text{}\mathrm{m}}^{-3}{\text{}\mathrm{s}}^{-1}$ | substrate consumption rate |

${R}_{N}$ | ${\mathrm{s}}^{-1}$ | total nucleation rate |

${R}_{{\mathrm{MoS}}_{2}}$ | ${\mathrm{s}}^{-1}$ | nucleation rate for MoS_{2} particles |

${R}_{S}$ | ${\mathrm{s}}^{-1}$ | nucleation rate for sulfur particles |

S | $-$ | supersaturation |

t | s | time |

${u}_{pi}$ | ${\mathrm{m}\text{}\mathrm{s}}^{-1}$ | particle velocity component |

${u}_{{\mathrm{MoS}}_{2}}$ | $-$ | MoS_{2} precipitation share function |

${u}_{S}$ | $-$ | S precipitation share function |

$V$ | ${\mathrm{m}}^{3}$ | volume |

$x,y,z$ | m | Cartesian coordinates |

Greek symbols | ||

$\rho $ | ${\mathrm{kg}\text{}\mathrm{m}}^{-3}$ | density |

$\zeta $ | $-$ | share function |

## References

- Baldyga, J.; Bourne, J.R. Turbulent Mixing and Chemical Reactions, 1st ed.; Wiley: Hoboken, NJ, USA, 1999. [Google Scholar]
- Fox, R.O. Computational Models for Turbulent Reacting Flows; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar] [CrossRef]
- Wojtas, K.; Makowski, Ł.; Orciuch, W. Barium sulfate precipitation in jet reactors: Large eddy simulations, kinetics study and design considerations. Chem. Eng. Res. Des.
**2020**, 158, 64–76. [Google Scholar] [CrossRef] - Wojtas, K.; Orciuch, W.; Makowski, Ł. Large eddy simulations of reactive mixing in jet reactors of varied geometry and size. Processes
**2020**, 8, 1101. [Google Scholar] [CrossRef] - Battistini, A.; Cammi, A.; Lorenzi, S.; Colombo, M.; Fairweather, M. Development of a CFD—LES model for the dynamic analysis of the DYNASTY natural circulation loop. Chem. Eng. Sci.
**2021**, 237, 116520. [Google Scholar] [CrossRef] - Cheng, J.; Yang, C.; Mao, Z.S. CFD-PBE simulation of premixed continuous precipitation incorporating nucleation, growth and aggregation in a stirred tank with multi-class method. Chem. Eng. Sci.
**2012**, 68, 469–480. [Google Scholar] [CrossRef] - Metzger, L.; Kind, M. The influence of mixing on fast precipitation processes—A coupled 3D CFD-PBE approach using the direct quadrature method of moments (DQMOM). Chem. Eng. Sci.
**2017**, 169, 284–298. [Google Scholar] [CrossRef] - Sultan, M.A.; Fonte, C.P.; Dias, M.M.; Lopes, J.C.B.; Santos, R.J. Experimental study of flow regime and mixing in T-jets mixers. Chem. Eng. Sci.
**2012**, 73, 388–399. [Google Scholar] [CrossRef] - Johnson, N.K.; Richardson, L.F. Weather Prediction by Numerical Process. Math. Gaz.
**1922**, 11, 125–127. [Google Scholar] [CrossRef] [Green Version] - Anderson, J. Computational Fluid Dynamics: The Basics with Applications; McGrawhill Inc.: New York, NY, USA, 1995; pp. 1–547. [Google Scholar]
- Patankar, S.V. Numerical heat transfer and fluid flow. Nucl. Sci. Eng.
**1981**, 78, 196–197. [Google Scholar] [CrossRef] - Wojtas, K.; Orciuch, W.; Makowski, Ł. Comparison of large eddy simulations and k-ε Modelling of fluid velocity and tracer concentration in impinging jet mixers. Chem. Process Eng.
**2015**, 36, 251–262. [Google Scholar] [CrossRef] - Wojtas, K.; Orciuch, W.; Wysocki, Ł.; Makowski, Ł. Modeling and experimental validation of subgrid scale scalar variance at high Schmidt numbers. Chem. Eng. Res. Des.
**2017**, 123, 141–151. [Google Scholar] [CrossRef] - Makowski, Ł.; Bałdyga, J. Large Eddy Simulation of mixing effects on the course of parallel chemical reactions and comparison with k-ε modeling. Chem. Eng. Process. Process Intensif.
**2011**, 50, 1035–1040. [Google Scholar] [CrossRef] - Vicum, L.; Ottiger, S.; Mazzotti, M.; Makowski, Ł.; Bałdyga, J. Multi-scale modeling of a reactive mixing process in a semibatch stirred tank. Chem. Eng. Sci.
**2004**, 59, 1767–1781. [Google Scholar] [CrossRef] - Hai-Dou, W.; Bin-Shi, X.; Jia-Jun, L.; Da-Ming, Z. Characterization and anti-friction on the solid lubrication MoS
_{2}film prepared by chemical reaction technique. Sci. Technol. Adv. Mater.**2005**, 6, 535–539. [Google Scholar] [CrossRef] [Green Version] - He, B.; Chen, L.; Jing, M.; Zhou, M.; Hou, Z.; Chen, X. 3D MoS
_{2}-rGO@Mo nanohybrids for enhanced hydrogen evolution: The importance of the synergy on the Volmer reaction. Electrochim. Acta**2018**, 283, 357–365. [Google Scholar] [CrossRef] - Voiry, D.; Salehi, M.; Silva, R.; Fujita, T.; Chen, M.; Asefa, T.; Shenoy, V.B.; Eda, G.; Chhowalla, M. Conducting MoS
_{2}Nanosheets as Catalysts for Hydrogen Evolution Reaction. Nano Lett.**2013**, 13, 6222–6227. [Google Scholar] [CrossRef] - Si, K.; Ma, J.; Guo, Y.; Zhou, Y.; Lu, C.; Xu, X.; Xu, X. Improving photoelectric performance of MoS
_{2}photoelectrodes by annealing. Ceram. Int.**2018**, 44, 21153–21158. [Google Scholar] [CrossRef] - Merki, D.; Fierro, S.; Vrubel, H.; Hu, X. Amorphous molybdenum sulfide films as catalysts for electrochemical hydrogen production in water. Chem. Sci.
**2011**, 2, 1262–1267. [Google Scholar] [CrossRef] [Green Version] - Bojarska, Z.; Kopytowski, J.; Mazurkiewicz-Pawlicka, M.; Bazarnik, P.; Gierlotka, S.; Rożeń, A.; Makowski, Ł. Molybdenum disulfide-based hybrid materials as new types of oil additives with enhanced tribological and rheological properties. Tribol. Int.
**2021**, 160, 106999. [Google Scholar] [CrossRef] - Deorsola, F.A.; Russo, N.; Blengini, G.A.; Fino, D. Synthesis, characterization and environmental assessment of nanosized MoS
_{2}particles for lubricants applications. Chem. Eng. J.**2012**, 195–196, 1–6. [Google Scholar] [CrossRef] - Santillo, G.; Deorsola, F.A.; Bensaid, S.; Russo, N.; Fino, D. MoS
_{2}nanoparticle precipitation in turbulent micromixers. Chem. Eng. J.**2012**, 207–208, 322–328. [Google Scholar] [CrossRef] - Wojtalik, M.; Orciuch, W.; Makowski, Ł. Nucleation and growth kinetics of MoS
_{2}nanoparticles obtained by chemical wet synthesis in a jet reactor. Chem. Eng. Sci.**2020**, 225, 55–65. [Google Scholar] [CrossRef] - Wojtalik, M.; Bojarska, Z.; Makowski, Ł. Experimental studies on the chemical wet synthesis for obtaining high-quality MoS
_{2}nanoparticles using impinging jet reactor. J. Solid State Chem.**2020**, 285, 121254. [Google Scholar] [CrossRef] - Wojtalik, M.; Przybyt, P.; Zubańska, K.; Żuchowska, M.; Bojarska, Z.; Arasimowicz-Andres, M.; Rożeń, A.; Makowski, Ł. Production of synthetic particles of MoS
_{2}using wet chemical synthesis. Przem. Chem.**2019**, 98, 1817–1821. [Google Scholar] [CrossRef] - Al-Tarazi, M.; Heesink, A.B.M.; Versteeg, G.F. New method for the determination of precipitation kinetics using a laminar jet reactor. Chem. Eng. Sci.
**2005**, 60, 805–814. [Google Scholar] [CrossRef] - Al-Tarazi, M.; Heesink, A.B.M.; Versteeg, G.F. Precipitation of metal sulphides using gaseous hydrogen sulphide: Mathematical modelling. Chem. Eng. Sci.
**2004**, 59, 567–579. [Google Scholar] [CrossRef] [Green Version] - Bardina, J.E.; Huang, P.G.; Coakley, T.J. Turbulence Modeling Validation, Testing, and Development; Nasa Technical Memorandum; NASA: Washington, DC, USA, 1997.
- Stickel, J.J.; Powell, R.L. Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech.
**2005**, 37, 129–149. [Google Scholar] [CrossRef] - Simion, A.I.; Grigoraş, C.G.; Gavrilă, L.G. Modelling of the thermophysical properties of citric acid aqueous solutions. Density and viscosity. Ann. Food Sci. Technol.
**2014**, 1, 193–202. [Google Scholar]

**Figure 1.**Geometries of (

**a**) tangential inlets V-type and (

**b**) coaxial inlets T-type impinging jet reactors (all dimensions are given in mm).

**Figure 4.**Domain divided into zones: (1) inlet zone of HMA and CA mixture, (2) inlet zone of AS solution, (3) reactor outlet zone, (4) transition zone between zones 1–3 and reaction zone, and (5) reaction zone.

**Figure 6.**Correction factor (ζ) plotted as functions of AS inlet Reynolds number for different reactor types and flow rates.

**Figure 7.**Inert-tracer concentration [−] distribution for 0.6 mol/dm

^{3}AS and 20 mL/min flow rate in (

**a**) V- and (

**b**) T-type reactors, (

**c**) comparison of the radial concentration distribution of the inert tracer in a T-type and V-type reactor at two different distances from the reactors bottom.

**Figure 8.**Comparison of velocity [m s

^{−1}] contour plots for 0.6 mol/dm

^{3}AS and 20 mL/min flow rate in (

**a**) V- and (

**b**) T-type reactors.

**Figure 9.**Diffusivity [kg m

^{−1}s

^{−1}] contours for 0.6 mol/dm

^{3}AS in V-type reactor at (

**a**) 20 and (

**b**) 50 mL/min flow rates and in T-type reactor at (

**c**) 20 and (

**d**) 50 mL/min flow rates.

**Figure 10.**Comparison of nucleation rate [s

^{−1}] for 0.6 mol/dm

^{3}AS and 20 mL/min flow rate in (

**a**) V- and (

**b**) T-type reactors.

**Figure 11.**Comparison of growth rate [m s

^{−1}] contour plots for 0.6 mol/dm

^{3}AS and (

**a**) 20 and (

**b**) 50 mL/min flow rates in a V-type reactor.

**Figure 12.**Comparison of growth rate [m s

^{−1}] contour plots for 0.6 mol/dm

^{3}AS and 20 mL/min flow rate in (

**a**) V- and (

**b**) T-type reactors.

**Figure 13.**Characteristic particle sizes for V-type reactor at (

**a**) 20 and (

**b**) 50 mL/min flow rates and for T-type reactor at (

**c**) 20 and (

**d**) 50 mL/min flow rates (ideal mixing refers to the simulation of a tubular reactor with ideal mixing conditions, and non-ideal mixing refers to CFD modelling that takes into account the reactor geometry and flow conditions).

**Figure 14.**Parity plots for V-type reactor at (

**a**) 20 and (

**b**) 50 mL/min flow rates and for T-type reactor at (

**c**) 20 and (

**d**) 50 mL/min flow rates.

**Figure 15.**Particle moments in function of initial AS concentration for experimental results and calculated in ideal and nonideal mixing condition (

**a**) moment 0, (

**b**) moment 1, (

**c**) moment 2, (

**d**) moment 3, (

**e**) moment 4.

$S=\sqrt[3]{\frac{7{C}_{HMA}{\left(1.068\times {10}^{-14}{C}_{AS}\right)}^{2}}{Ks}}$ | Supersaturation $(1.068\times {10}^{-14}$ = product of sulphide ion equilibrium constants) | (1) |

${R}_{{N}_{{\mathrm{MoS}}_{2}}}={{a}_{ho}}^{\prime}exp\left(\frac{-{{b}_{ho}}^{\prime}}{{\mathrm{log}}^{2}\left(S\right)}\right)+{{a}_{he}}^{\prime}exp\left(\frac{-{{b}_{he}}^{\prime}}{{\mathrm{log}}^{2}\left(S\right)}\right)$ | Rate of formation from MoS_{2} precipitation | (2) |

${R}_{{N}_{S}}={a}_{he}{C}_{AS}{}^{{b}_{he}}+{a}_{ho}{C}_{AS}{}^{{b}_{ho}}$ | Rate of formation from S precipitation | (3) |

${G}_{{\mathrm{MoS}}_{2}}={{k}_{r}}^{\prime}{S}^{{k}_{D}}$ | Linear growth rate coefficient from MoS_{2} precipitation | (4) |

${G}_{S}={k}_{r}{C}_{AS}{}^{2}$ | Linear growth rate coefficient from S precipitation | (5) |

${u}_{{\mathrm{MoS}}_{2}}=exp\left(\frac{-{C}_{HMA}}{a}\right)$ | MoS_{2} precipitation kinetics share function in model | (6) |

${u}_{S}=1-exp\left(\frac{-{C}_{HMA}}{a}\right)$ | S precipitation kinetics share function in model | (7) |

${R}_{N}={u}_{{\mathrm{MoS}}_{2}}{R}_{{N}_{{\mathrm{MoS}}_{2}}}+{u}_{S}{R}_{{N}_{S}}$ | Total formation rate | (8) |

$G={u}_{{\mathrm{MoS}}_{2}}{G}_{{\mathrm{MoS}}_{2}}+{u}_{S}{G}_{S}$ | Total linear growth rate coefficient | (9) |

${R}_{M}=\frac{\rho}{M}\frac{{k}_{a}G{m}_{2}}{2}$ | Substrate consumption rate | (10) |

$\frac{d\left(V{C}_{HMA}\right)}{dt}=-\frac{1}{7}{R}_{M}$ | HMA balance | (11) |

$\frac{d\left(V{C}_{AS}\right)}{dt}=-3{R}_{M}$ | AS balance | (12) |

$\frac{d{m}_{0}}{dt}={R}_{N}$ | Zero-moment balance | (13) |

$\frac{d{m}_{n}}{dt}=kG{m}_{n-1}$ | Higher-moment balance (for n = 1, 2, 3, or 4) | (14) |

$\frac{dV}{dt}=0$ | Volumetric change (zero for pipe reactor) | (15) |

Constant | Value | Unit |
---|---|---|

${a}_{he}$ | 5.25 × 10^{22} | $-$ |

${a}_{ho}$ | 3.97 × 10^{23} | $-$ |

${b}_{he}$ | 1.89 | $-$ |

${b}_{ho}$ | 6.58 | $-$ |

${{a}_{he}}^{\prime}$ | 3.05 × 10^{15} | $-$ |

${{a}_{ho}}^{\prime}$ | 1.31 × 10^{16} | $-$ |

${{b}_{he}}^{\prime}$ | 24.04 | $-$ |

${{b}_{ho}}^{\prime}$ | 1.04 × 10^{−11} | $-$ |

${k}_{D}$ | 0.3533 | $-$ |

${k}_{r}$ | 0.0102 | ${\mathrm{m}}^{5}{\mathrm{s}}^{-1}{\mathrm{mol}}^{-2}$ |

${{k}_{r}}^{\prime}$ | 3.21 × 10^{−9} | ${\mathrm{m}}^{5}{\mathrm{s}}^{-1}{\mathrm{mol}}^{-2}$ |

$a$ | 0.034 | ${\mathrm{mol}\mathrm{dm}}^{-3}$ |

**Table 3.**CA properties at 298.15 K according to Ref. [31].

CA Mass Concentration [−] | CA Dynamic Viscosity [Pa s] | CA Density [kg/m^{3}] |
---|---|---|

0.0000 | 8.94 × 10^{−4} | 997.0 |

0.0643 | 1.05 × 10^{−3} | 1022.5 |

0.0994 | 1.12 × 10^{−3} | 1035.3 |

0.1699 | 1.28 × 10^{−3} | 1059.4 |

0.1982 | 1.37 × 10^{−3} | 1068.5 |

0.2518 | 1.53 × 10^{−3} | 1085.0 |

0.3000 | 1.69 × 10^{−3} | 1099.0 |

0.3400 | 1.84 × 10^{−3} | 1110.1 |

0.3994 | 2.12 × 10^{−3} | 1125.7 |

Parameter | V-Type Reactor | T-Type Reactor |
---|---|---|

Number of cells | 2,030,910 | 2,018,986 |

Minimum Orthogonal Quality | 0.354 | 0.351 |

Maximum Aspect Ratio | 34.13 | 33.50 |

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**MDPI and ACS Style**

Wojtalik, M.; Wojtas, K.; Gołębiowska, W.; Jarząbek, M.; Orciuch, W.; Makowski, Ł.
Molybdenum Disulphide Precipitation in Jet Reactors: Introduction of Kinetics Model for Computational Fluid Dynamics Calculations. *Molecules* **2022**, *27*, 3943.
https://doi.org/10.3390/molecules27123943

**AMA Style**

Wojtalik M, Wojtas K, Gołębiowska W, Jarząbek M, Orciuch W, Makowski Ł.
Molybdenum Disulphide Precipitation in Jet Reactors: Introduction of Kinetics Model for Computational Fluid Dynamics Calculations. *Molecules*. 2022; 27(12):3943.
https://doi.org/10.3390/molecules27123943

**Chicago/Turabian Style**

Wojtalik, Michał, Krzysztof Wojtas, Weronika Gołębiowska, Maria Jarząbek, Wojciech Orciuch, and Łukasz Makowski.
2022. "Molybdenum Disulphide Precipitation in Jet Reactors: Introduction of Kinetics Model for Computational Fluid Dynamics Calculations" *Molecules* 27, no. 12: 3943.
https://doi.org/10.3390/molecules27123943