Barba-Piña, J. L. A.

Burns, A.

Khatir, Zinedine

Mao, X.

Birmingham City University Open Access Repository

Numerical Study of Thermophoresis Mass Transport in Binary Fluid Mixtures Using OpenFOAM

BCU Open Access

Numerical Study of Thermophoresis Mass Transport in Binary Fluid Mixtures Using OpenFoamJose Lorenzo Alejandro Barba-Piña1, Alan Burns1, Xiaoan Mao1, Zinedine Khatir2,31School of Chemical and Process Engineering, University of Leeds, United Kingdom2School of Mechanical, University of Leeds, United Kingdom3School of Engineering and the EngineeringBuilt Environment, Birmingham City University, United KingdomINTRODUCTION• In fluid homogeneous mixtures, the mass transport process induced by the dependance of species concentration on temperature gradients is known as thermophoresis, thermodiffusion or “Soret effect”.• In absence of other external forces, the mass flux of the reference component “a” can be written as follows:𝐽𝑎 = 𝜌𝐷𝑎,𝑏∇𝑤𝑎 − 𝜌𝐷𝑇 𝑤𝑎0(1 − 𝑤𝑎0)∇𝑇∗ 𝑫𝒂,𝒃 is the mass diffusivity, 𝒘𝒂𝟎 is the initial mass fraction of component “a” and 𝑫𝑻 is the   thermodiffusion coefficient.• For binary gaseous mixtures: Heavy component usually moves to the colder region, while light component moves to the hotter region.• When steady state is reached, 𝐽𝑎 = 0 and concentration gradient can be written as:∇𝑤𝑎 =𝐷𝑇𝐷𝑎,𝑏𝑤𝑎0(1 − 𝑤𝑎0)∇𝑇JaJb• Thermodiffusion transport can be found in:• Natural processes:Thermohaline convection currents in oceans caused by the dependance of salinity gradients on temperature differences. Thermo-gravitational segregation in natural hydrocarbon  reservoirs.Particle deposition on hot/cold surfaces, etc.• Industrial processes: Isotope separation of liquid and gaseous mixtures with the Clusius-Dickel Column.Separation and characterization of polymers.Surface coating.Particle decontamination.*Lapeyra, E., Ghebardt M., Triller, T., Mialdun, A., Kolher, W., Shevtsova, V., Mounir-Bou Ali, M. Transport properties of the binary mixtures of the three organic liquids toluene, methanol, and cyclohexane. 2017. The Journal of Chemical Physics. 146(9):094507.-Sketch of a Clusius-Dickel column,Retrieved from Lapeyra et al. 2017Numerical Analysis Of Thermophoresis• Application of CFD on thermodiffusion processes: Compliments experimental analysis where visualization is limited. Quantification and parametrization of mass transfer dependency on thermophysical and geometrical variables.  Testing of different physical conditions (e.g., residual or zero gravity, ideal adiabatic systems, etc.) that are hard to replicate in experimental environments.Thermodiffusion Model in OpenFOAM• A basic thermodiffusion model is included into OpenFOAM’s thermophysical properties library.• Thermodiffusion coefficient is calculated with initial parameters: mass diffusivity 𝐷𝑎,𝑏, initial molar fraction 𝑥𝑎,0, and thermodiffusion factor 𝜶𝑻. • Then, thermodiffusion coefficient is corrected with the temperature variation of the simulation:𝐷𝑇 =𝛼𝑇𝑥𝑎,0 1 − 𝑥𝑎,0 𝐷𝑎,𝑏𝑇• This new implementation was added into two solvers for convective and non-convective thermodiffusion cases. • Laplacian solver for non-convective simulation • Species transport equation:𝜕𝑤𝑎𝜕𝑡+= 𝐷𝑎,𝑏∇2𝑤𝑎 + 𝐷𝑇𝑤𝑎,0 1 − 𝑤𝑎,0 ∇2𝑇• Energy equation:𝜕𝑇𝜕𝑡= 𝛼∇2𝑇• Compressible solver for convective simulation, • Navier-Stokes equation:𝜕(𝜌𝑣)𝜕𝑡+ 𝑣 ∙ ∇ 𝜌𝑣 = ∇𝑃 − ∇ ∙ 𝜏𝜏 = −𝜇 ∇𝑣 + ∇𝑣 𝑇 +23(∇ ∙ 𝑣)𝛿• Energy equation in terms of enthalpy:𝜕(𝜌ℎ)𝜕𝑡+ ∇ ∙ 𝜌𝑣ℎ +𝜕(𝜌𝐾)𝜕𝑡+ ∇ ∙ 𝜌𝑣𝐾 −𝜕𝑃𝜕𝑡= −∇ ∙ 𝑞 + ∇ ∙ 𝜏 ∙ 𝑣𝑞 = −𝜅∇𝑇; 𝐾 =|𝑣|22• species transport equation:𝜕(𝜌𝑤𝑎)𝜕𝑡+ 𝑣 ∙ 𝜌∇𝑤𝑎 = 𝐷𝑎,𝑏∇2 𝜌𝑤𝑎 + 𝐷𝑇𝑤𝑎,0 1 − 𝑤𝑎,0 ∇2(𝜌𝑇)OpenFOAM Thermodiffusion Solvers Test• Two cases from Kuwatani et. Al 2012 were selected for testing the new thermodiffusion solvers in OpenFOAM.• Case 1 closed system:• H2-CO2 mixture• Geometry: Insulated region, with fixed temperature at left and right boundaries.• Hexahedral mesh of 3200 elements.• Zero flux condition at solid boundaries:∇𝑤𝑎 =𝐷𝑇𝐷𝑎,𝑏𝑤𝑎0(1 − 𝑤𝑎0)∇𝑇*Kuwatani, S., Watanabe, S., Ono, N. Study and Development of a Mini-Tube Gas Separator Utilizing the Soret Effect. 2012. Journal of Thermal Science and Technology. 7(1), 31-43.- Sketch of thermophoresis separation open system, (retrieved from Kuwatani et Al. 2012)Simulation 𝑇𝑐 [𝐊] 𝑇𝑯[𝐊] 𝑥𝑎,0 𝛼𝑇 𝑫𝒂,𝒃 [𝒎/𝒔]1 273.15 473.15 0.48 0.3602 6.34𝑒−52 ---------- 523.15 ---------- ---------- ----------3 ---------- 573.15 ---------- ---------- ----------4 ---------- 593.15 ---------- ---------- ----------5 ---------- 693.15 ---------- ---------- ----------6 ---------- 723.5 ---------- ---------- ----------Experiment 313.15 670.15 ---------- ---------- ----------Notice: Molar fraction is converted to mass fraction for species transport equation consistency for both cases.• Case 2 Open system :• H2-CO2 mixture.• Forced convection, absolute pressure, P = 1𝑒5• 6000 hexahedral elements.• Zero flux condition at solid boundaries and outlet boundary:∇𝑤𝑎 =𝐷𝑇𝐷𝑎,𝑏𝑤𝑎0(1 − 𝑤𝑎0)∇𝑇- Sketch of thermophoresis separation open system, (retrieved from Kuwatani et Al. 2012)Volume flow rate (mL/min)𝑇𝑖𝑛𝑙𝑒𝑡 [𝐊] 𝑇𝑐 [𝐊] 𝑇𝑯[𝐊] 𝑥𝑎,0150 300.15 273.15 4 0.5𝑫𝒂,𝒃 [𝒎/𝒔] 𝛼𝑇 𝝁 [𝑷𝒂 ∙ 𝒔]𝑪𝒑𝑱𝑲𝒈 ∙ 𝑲𝑷𝒓 =𝝁𝑪𝒑𝝆𝜿6.34𝑒−5 0.3596 1.02𝑒−5 1427 0.217CASE 1: CONCENTRATION DIFFERENCE IN THE CLOSED SYSTEM• Results of the six different steady-state simulations are compared with theoretical calculations and simulations from Kuwatani et al. 2012. Theoretical calculations are computed by:∆𝑥 = −𝐾𝑇𝑙𝑛𝑇ℎ𝑇𝑐𝐾𝑇 =𝐷𝑇𝐷𝑎,𝑏𝑇 = 𝛼𝑇𝑥𝑎,0 1 − 𝑥𝑎,0• Comparison with experimental case data (Tc=313.15k, Th=670.15):∆𝑥𝑂𝐹= 7.3%∆𝑥𝑡ℎ𝑒𝑜𝑟𝑦= 6.8%∆𝑥𝑒𝑥𝑝= 6.6%• Greater differences between OpenFOAM results and theoretical calculations are appreciated for higher temperature differences.234567891011150 170 190 210 230 250 270 290 310 330 350 370 390 410 430 450 470 490H2-Concentration Difference [%]Temperature Difference [K]TheoryCFD-Phoenics(Kuwatani et al. 2012)CFD-OpenFOAMCase 2: Concentration Difference in the Closed SystemCase 2: Concentration Difference in the Open System• Qualitative agreement can be observed although the numerical results differ greatly from Kuwataniet al. 2012 reported data.• Computations were carried out by considering convective terms for transport equations, while theoretical calculations, other CFD results and Series solution does not consider those terms. 0123456789100 2 4 6 8 10 12 14 16 18 20H2-Concentration difference [%]Distance [mm]TheoryCFD-OpenFOAMFourier Series Expansion(Kuwatani et al. 2012)CFD-Phoenics (Kuwataniet al. 2012)CASE 2: CONCENTRATION DISTRIBUTION, OPENFOAM SIMULATION VS. KUWATANI ET AL., 2012 -Distribution of H2 Concentration (OpenFOAM). -Distribution of H2 Concentration (Retrieved from: Kuwatani et al. 2012) 

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Numerical Study of Thermophoresis Mass Transport in Binary Fluid Mixtures Using OpenFOAM

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