# Distinctions of the Emergence of Convective Flows at the “Diffusion–Convections” Boundary in Isothermal Ternary Gas Mixtures with Carbon Dioxide

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}− N

_{2}. The mixing process in the system under study was modeled at different initial carbon dioxide contents. To carry out a numerical experiment, a mathematical algorithm based on the D2Q9 model of lattice Boltzmann equations was used for modeling the flow of gases. We show that the model presented in the paper allows one to study the occurrence of convective structures at different heavy component contents (carbon dioxide). It has been established that in the system under study, the instability of the mechanical equilibrium occurs when the content of carbon dioxide in the mixture is more than 0.3 mole fractions. The characteristic times for the onset of convective instability and the subsequent creation of structural formations, the values of which depend on the initial content of carbon dioxide in the mixture, have been determined. Distributions of concentration, pressure and kinetic energy that allow one to specify the types of mixing and explain the occurrence of convection for a situation where, at the initial moment of time, the density of the gas mixture in the upper part of the diffusion channel is less than in the lower one, were obtained.

## 1. Introduction

## 2. Mathematical Formulation of the Problem and Numerical Methods

_{1}contains gas 1 (with the minimum molar mass M

_{1}) and gas 2 (which has the highest molecular weight M

_{2}) diffusing into gas 3 (with an intermediate molar mass M

_{3}) located in the lower part of channel S

_{2}. The condition M

_{2}> M

_{3}> M

_{1}is satisfied for the molecular weights of the components. We consider a two-dimensional region of the cross-section of the cylindrical region H x d in the Cartesian coordinate system (Figure 1a).

_{N}is the number density, t is the time, c

_{i}is the concentration of the ith component, ${\overrightarrow{\mathrm{j}}}_{\mathrm{i}}$ is the density of the diffusion flux of the i-th component and ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}^{*}$ is the practical diffusion coefficients.

_{i}> taken as the reference point (<c

_{i}> >> ${\mathrm{c}}_{\mathrm{i}}^{/}$), ${\mathsf{\beta}}_{\mathrm{i}}=\frac{1}{{\mathsf{\rho}}_{0}}{\left(\frac{\partial \mathsf{\rho}}{\partial {\mathrm{c}}_{\mathrm{i}}}\right)}_{\mathrm{p},\mathrm{T},{\mathrm{c}}_{\mathrm{j}}}$; ρ

_{0}is the average value of mixture density.

_{1}, n

_{2}) is the outer normal to the boundary of the computational domain.

_{i}is the concentrations of components in the upper S

_{1}and lower S

_{2}regions.

^{2}/ν is the time, ${\mathrm{D}}_{22}^{*}$/H is the velocity, A

_{i}H is the concentrations of the i-th component and ρ

_{0}ν${\mathrm{D}}_{22}^{*}$/H

^{2}is the pressure. The system of Equation (4) in dimensionless quantities is transformed into the following equations:

_{i}is the dimensionless initial concentration gradient of the i-th component.

_{i}, h

_{i,α}are the velocity and concentration distribution functions of the α-component, ${\overrightarrow{\mathrm{e}}}_{\mathrm{i}}$ is the discrete lattice velocity in the i direction, ${\mathsf{\tau}}_{\mathrm{f}},{\mathsf{\tau}}_{\mathrm{h},\mathsf{\alpha}}$ are the relaxation times, F

_{i}is the external force component, Q

_{i,α}is responsible for the source q

_{α}, ∆t is the lattice time step, and ${\mathrm{f}}_{\mathrm{i}}^{\mathrm{e}\mathrm{q}},{\mathrm{h}}_{\mathrm{i},\mathsf{\alpha}}^{\mathrm{e}\mathrm{q}}$ are the equilibrium distribution function of the velocity and concentration of the α-component, respectively.

_{i}in the directions i, which are included in the expressions for the equilibrium functions (7), have the following values:

_{0}= (0, 0), e

_{1}= (1, 0), e

_{2}= (0, 1), e

_{3}= (−1, 0), e

_{4}= (0, −1),

e

_{5}= (1, 1), e

_{6}= (−1, 1), e

_{7}= (−1, −1), e

_{8}= (1, −1).

_{α}are approximated using the following formula:

_{∗}.

_{H}= Δx

_{phy}, Ct = Δt

_{phy}, where Δx

_{phy}, Δt

_{phy}represent physical steps through space and time. The index “phy” will denote physical quantities.

_{phy}= H/N. According to the stability conditions of LBM algorithms, the characteristic lattice velocity is u

_{lbm}< 0.4 for τ ≥ 0.55. In this work, the lattice velocity is set to u

_{lbm}= 0.2. The conversion coefficient for velocity will be determined through C

_{u}= u

_{phy}/u

_{lbm}, where the physical value of velocity is equal to ${\mathrm{u}}_{\mathrm{p}\mathrm{h}\mathrm{y}}=\sqrt{\mathrm{g}\mathrm{H}}$. Then, the physical time step is equal to Δt

_{phy}= Δx

_{phy}/C

_{u}.

## 3. Results of the Numerical Calculation

_{1}, x

_{2}axes, respectively. Calculations were performed for the physical parameters determined experimentally [35] or calculated from the kinetic concepts [36] for given geometric characteristics of the channel (see Table 1) and are shown in Figure 3, Figure 4 and Figure 5. The length and height of the computational domain are L = 0.01 m, H = 0.01 m. Physical steps in space and time have the following values: Δx

_{phy}= 0.000125 and Δt

_{phy}= 7.986 · 10

^{−5}.

_{0}and Kp = p

_{0}/p, respectively. Here, T

_{0}= 298.0 K and p

_{0}= 0.1 MPa, and T and p are the temperature and pressure of the experiment. The density and dynamic shear viscosity of the components at the experimental parameters are calculated using the formulas ${\mathsf{\rho}}_{\mathrm{i}}={\mathsf{\rho}}_{\mathrm{i}}^{0}/\left(\mathrm{K}\mathrm{p}\times \mathrm{K}\mathrm{t}\right)$ and ${\eta}_{\mathrm{i}}={\eta}_{\mathrm{i}}^{0}\mathrm{K}{\mathrm{t}}^{1/2}$, where ${\mathsf{\rho}}_{\mathrm{i}}^{0}$ is the density of the i-th component, and ${\eta}_{\mathrm{i}}^{0}$ is the dynamic viscosity of the i-th component corresponding to the conditions T

_{0}= 298.0 K and p

_{0}= 0.1 MPa. Kinematic viscosity is calculated using the formula ${\nu}_{\mathrm{f}}={\displaystyle \sum _{\mathrm{i}}{\mathrm{C}}_{\mathrm{i}}\frac{{\eta}_{\mathrm{i}}}{{\mathsf{\rho}}_{\mathrm{i}}}}$, where C

_{i}is the concentration of the i-th component. The interdiffusion coefficients given in Table 1 at different pressures and temperatures are related to each other in the following way ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}={\mathrm{D}}_{\mathrm{i}\mathrm{j}}^{\mathrm{o}}\mathrm{K}{\mathrm{t}}^{3/2}\mathrm{K}\mathrm{p},\mathrm{i}\ne \mathrm{j}$, where ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}^{0}$ are the interdiffusion coefficients presented in Table 1. Practical diffusion coefficients ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}^{*}$ and mutual diffusion coefficients are related to each other by the following relationships [22,23]:

_{i}is the molar mass of the i-th component.

_{2}(2) − N

_{2}(3) at pressure p = 1.0 MPa and T = 298.0 K. At t = 0 s, the density of the binary mixture located in the upper part of the diffusion channel is less than the density of nitrogen, which is localized in the lower part. Diffusion takes place at the initial stage (see Figure 3a). After 0.33 s, a violation of the monotonicity in the distribution of isoconcentration lines, which increases with time (Figure 3b,c), is recorded. Such a distribution is not characteristic of diffusion mixing. It can be assumed that, starting from this time, instability in the mechanical equilibrium, which is the cause of the appearance of convection, arises in the system under study. Figure 3d shows the development of a convective cell, which begins 0.95 s after the start of mixing. At the final stage (Figure 3e–g), the convective formation begins to move in the gravity field relative to the diffusion interface. Then, but already under other initial condition, the process of structural formation begins again, but, already under other boundary conditions, i.e., in the system under study, the appearance of a drop convective mixing mode, which was recorded experimentally in various ternary mixtures [17,18,22], including those containing CO

_{2}[23], is possible.

_{2}(2) − N

_{2}(3) system given in [22], which shows that at pressures above 0.5 MPa, the conditions for the preferential transfer of carbon dioxide are realized in the system. The main reason that does not allow for more accurate quantitative comparisons is the significant difference in the characteristic mixing time in the experimental diffusion cell (several thousand seconds) and the computational region (several seconds).

_{2}concentrations, pressure and kinetic energy are similar to the distributions shown in Figure 3, Figure 4 and Figure 5.

_{1}–t

_{3}; creation of structural convective formations (Figure 3d,e), which correspond to times t

_{4}and t

_{5}; organization of the convective cell (Figure 3f,g) and its initial movement in the gravity field. It should be expected that the identified mixing stages are able to manifest themselves in a similar way with other initial compositions but with different characteristic times.

_{1}–t

_{7}) specifying diffusion, the onset and development of mechanical equilibrium instability, the occurrence of a structural formation and its subsequent evolution leading to an initial mixing in the channel due to the force gravity are listed in Table 2 for the various compositions of carbon dioxide in the ternary mixture of He + CO

_{2}− N

_{2}.

_{1,}corresponding to the initial formation of the curvature of the isoconcentration lines. However, the difference in the characteristic mixing times t

_{2}and t

_{3}for different initial compositions becomes noticeable at subsequent stages of the development of mechanical equilibrium instability. The dependence on the concentration of CO

_{2}in the initial system becomes more significant at the stage of formation of convective cells with the corresponding times t

_{4}, t

_{5}and the emergence of currents in the initial development phase with mixing times t

_{6}, t

_{7}. The identified trend shows that with an increase in the concentration of carbon dioxide in the initial mixture, more time is required for the development of mechanical equilibrium instability and the formation of convective structures. This can be explained by the fact that at high CO

_{2}contents in the initial mixture, the partial helium flux, due to its small size, does not create conditions for the diffusion mechanisms that form the inversion density layers. In this case, the emergence of gravitational convection is carried out in the traditional way [6].

## 4. Conclusions

_{2}− N

_{2}gas system at different carbon dioxide content in the mixture can be carried out on the basis of the method of lattice Boltzmann equations. For the system under consideration, despite the implementation in the initial stage of mixing of the conditions for the mixture density to decrease with height, diffusion and convective types of mixing are recorded. The presented mathematical model makes it possible to describe the process of formation of a convective structure for various compositions of the ternary mixture. The obtained isoconcentration distributions in a vertical flat channel are discussed in detail and make it possible to specify the types of mixing and explain the occurrence of convection for the situation when, at the initial moment of time, the density of the gas mixture in the upper part of the diffusion channel is less than in the lower part. The main conclusions of this study are as follows:

- The occurrence of convective instability can be associated with a significant curvature of isoconcentration distributions, which are absent during diffusion. The concentration profiles obtained are not typical for diffusion.
- The disappearance of curvature in isoconcentration lines occurs at a certain initial composition of the mixture. For these calculation conditions, this occurs when the content of carbon dioxide is less than 0.3 mole fractions and determines the diffusion mode of mixing.
- The degree of curvature of the concentration distributions depends on the content of the component with the highest molecular weight in the system. An increase in the concentration of the component with the highest molecular weight leads to a rise in the characteristic mixing times.
- Further evolution of multicomponent mixing can result in the creation of convective formations and a “drop” flow regime.
- The given distributions of pressure and average kinetic energy show the complex structure of the resulting flow. The maximum values of the distribution of pressure and kinetic energy correspond to the formation of convective structures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Symbols | ||

A_{i} | [-] | dimensionless initial concentration gradient of the i-th component |

C | [-] | component concentration |

c_{i} | [-] | concentration of the i-th component |

${\mathrm{D}}_{\mathrm{i}\mathrm{j}}^{*}$ | [m^{2}/s] | diffusion complexes |

F_{i} | [-] | external force component |

H | [m] | height |

Pr | [-] | Prandtl number |

Ra | [-] | Rayleigh number |

T | [K] | temperature |

${\overrightarrow{\mathrm{e}}}_{\mathrm{i}}$ | [-] | discrete velocities |

f_{i} | [-] | velocity distribution function |

g | [m/s^{2}] | free-fall acceleration scalar |

h_{i} | [-] | concentration distribution function |

p | [Pa] | pressure |

r | [m] | radius |

t | [s] | time |

$\overrightarrow{\mathrm{u}}$ | [m/s] | weight-average velocity vector |

$\overrightarrow{\mathsf{\upsilon}}$ | [m/s] | number-average velocity vector |

x | [-] | abscissa axis |

β_{i} | [-] | concentration analogue of the thermal expansion coefficient |

ν | [m^{2}/s] | kinetic viscosity |

ρ | [kg/m^{3}] | density |

τ | [-] | mesh relaxation time |

ω_{i} | [-] | weight coefficient depending on the number of discrete velocity |

Subscripts and Superscripts | ||

i, j, α | numbering of components | |

eq | equilibrium value | |

lbm | lattice Boltzmann equations method | |

phy | physical | |

^{/} | notion of the perturbed quantity |

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**Figure 1.**Simulation of mass transfer at the boundary of regime change: (

**a**) physical and calculated mixing areas; (

**b**) initial conditions for placing ternary mixtures in a diffusion channel.

**Figure 3.**Carbon dioxide isoconcentration lines for the 0.5 He + 0.5 CO

_{2}− N

_{2}system, p = 1.0 MPa, T = 298.0 K, Ra

_{1}= 4.34, Ra

_{2}= 8.31. Characteristic mixing times: (

**a**)—0.38 s; (

**b**)—0.57 s; (

**c**)—0.76 s; (

**d**)—0.95 s; (

**e**)—1.14 s; (

**f**)—1.71 s; (

**g**)—1.90 s.

**Figure 4.**Pressure distribution for the 0.5 He + 0.5 CO

_{2}− N

_{2}system, p = 1.0 MPa, T = 298.0 K, Ra

_{1}= 4.34, Ra

_{2}= 8.31. Characteristic mixing times: (

**a**)—0.38 s; (

**b**)—0.57 s; (

**c**)—0.76 s; (

**d**)—0.95 s; (

**e**)—1.14 s; (

**f**)—1.71 s; (

**g**)—1.90 s.

**Figure 5.**Average kinetic energy distribution for the 0.5 He + 0.5 CO

_{2}− N

_{2}system, p = 1.0 MPa, T = 298.0 K, Ra

_{1}= 4.34, Ra

_{2}= 8.31. Characteristic mixing times: (

**a**)—0.38 s; (

**b**)—0.57 s; (

**c**)—0.76 s; (

**d**)—0.95 s; (

**e**)—1.14 s; (

**f**)—1.71 s; (

**g**)—1.90 s.

Components | ρ, kg/m ^{3} | η, 10 ^{−5} Pa∙s | D_{12}, 10 ^{−4} m^{2}/s | D_{13}, 10 ^{−4} m^{2}/s | D_{23}, 10 ^{−4} m^{2}/s | Molar Mass, 10 ^{−3} kg/mole |
---|---|---|---|---|---|---|

He | 0.160 | 1.977 | 4.003 | |||

CO_{2} | 1.841 | 1.463 | 0.61 | 0.71 | 0.165 | 44.011 |

N_{2} | 1.146 | 1.775 | 28.016 |

**Table 2.**Characteristic mixing times for the different compositions of carbon dioxide at p = 1.01 MPa, T = 298.0 K.

Molar Composition of CO_{2}, mol. Fraction | Ra for CO_{2} | t_{1}, s | t_{2}, s | t_{3}, s | t_{4}, s | t_{5}, s | t_{6}, s | t_{7}, s |
---|---|---|---|---|---|---|---|---|

0.80 | 8.76 | 0.45 | 1.12 | 1.96 | 2.38 | 3.50 | 4.06 | 4.90 |

0.70 | 8.67 | 0.42 | 0.90 | 1.20 | 1.65 | 2.40 | 2.55 | 2.70 |

0.60 | 8.53 | 0.38 | 0.66 | 0.83 | 1.20 | 1.98 | 2.15 | 2.50 |

0.50 | 8.31 | 0.33 | 0.57 | 0.76 | 0.95 | 1.14 | 1.71 | 1.90 |

0.25 | 6.94 | diffusion |

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**MDPI and ACS Style**

Kossov, V.; Zhakebayev, D.; Fedorenko, O.; Zhumali, A.
Distinctions of the Emergence of Convective Flows at the “Diffusion–Convections” Boundary in Isothermal Ternary Gas Mixtures with Carbon Dioxide. *Fluids* **2024**, *9*, 47.
https://doi.org/10.3390/fluids9020047

**AMA Style**

Kossov V, Zhakebayev D, Fedorenko O, Zhumali A.
Distinctions of the Emergence of Convective Flows at the “Diffusion–Convections” Boundary in Isothermal Ternary Gas Mixtures with Carbon Dioxide. *Fluids*. 2024; 9(2):47.
https://doi.org/10.3390/fluids9020047

**Chicago/Turabian Style**

Kossov, Vladimir, Dauren Zhakebayev, Olga Fedorenko, and Ainur Zhumali.
2024. "Distinctions of the Emergence of Convective Flows at the “Diffusion–Convections” Boundary in Isothermal Ternary Gas Mixtures with Carbon Dioxide" *Fluids* 9, no. 2: 47.
https://doi.org/10.3390/fluids9020047