Absorption of gas by a falling liquid film
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1 Absorption of gas by a falling liquid film Christoph Albert Dieter Bothe Mathematical Modeling and Analysis Center of Smart Interfaces/ IRTG 1529 Darmstadt University of Technology 4th Japanese-German International Workshop on Mathematical Fluid Dynamics, Tokyo Mathematical Modeling and Analysis Christoph Albert
2 Wavy Falling Films Advantages Good heat transfer due to small thickness Large Interface Applications Evaporation Cooling Absorption Tokyo Mathematical Modeling and Analysis Christoph Albert 2
3 Hydrodynamic Model Assumptions Incompressible, Newtonian two-phase flow No phase transition Constant surface tension t (ρu)+ (ρu u)+ p= S+ρ g, Ω Σ u=0, Ω Σ p I S n Σ =σ κ n Σ, Σ u =0, Σ S=η( u+( u) T ) Tokyo Mathematical Modeling and Analysis Christoph Albert 3
4 The Volume of Fluid Method One Fluid Formulation: t (ρu)+ (ρu u)+ p= S+ρ g+δ f Σ u=0 t f +u f = Density and viscosity depend on volume fraction Volume fraction has to be transported Additional surface tension force term f Σ Tokyo Mathematical Modeling and Analysis Christoph Albert 4
5 Parasitic Currents VOF simulations suffer from unphysical oscillation of velocity, so-called parasitic currents Stem from numerical treatment of interfacial jump condition for stress: p I S n Σ =σ κ n Σ Especially serious in stagnant flow situations Parasitic Currents in a VOF simulation of a static droplet Also problematic in simulations of falling films, which are convection dominated Tokyo Mathematical Modeling and Analysis Christoph Albert 5
6 Numerical Setup u(t, y) x=0 =(1+ϵsin(2 π ω t )) ( ρ l μ L g δ 0 2 [ y δ ( y δ 0 ) 2 ], 0 ) Tokyo Mathematical Modeling and Analysis Christoph Albert 6
7 Continuum Surface Stress (CSS) ( f σ(i n Σ n Σ )) Body Force, Approximate n Σ by differentiating a smoothed f-field Momentum conservative Standard Surface Tension model in FS3D; delivers good results in many twophase flow situations Tokyo Mathematical Modeling and Analysis Christoph Albert 7
8 Falling Films with CSS Water/Air, Re cells per mean film thickness (0.265mm) A = 30%, f = 20 Hz Tokyo Mathematical Modeling and Analysis Christoph Albert 8
9 Continuum Surface Force (CSF) Body Force σ κ f In the case of a sphere and constant curvature, an exact balance between surface tension and pressure can be achieved: Balanced Force Renardy, J. Comput. Phys. 183 (2002) h i Greatly reduces parasitic currents Relies on good curvature information Easy for a falling film: κ= h xx 3 2 (1+h x 2 ) i 1 i i Tokyo Mathematical Modeling and Analysis Christoph Albert 9
10 Falling Films with CSF Water/Air, Re cells per mean film thickness (0.265mm) A = 30%, f = 20 Hz Tokyo Mathematical Modeling and Analysis Christoph Albert 10
11 Comparison to experiment: Dietze Experimental data from DMSO/Air = m 2 Re = 8.6 f = 16 Hz s G.F. Dietze, Flow Separation in Falling Liquid Films, 2010, = kg m 3, = N m A = 40% Resolution Tokyo Mathematical Modeling and Analysis Christoph Albert 11
12 Comparison to experiment: Film thickness Film thickness at x = 56 mm over a time span of 0.3 s CSS overshoots Tokyo Mathematical Modeling and Analysis Christoph Albert 12
13 Species Transport in Falling Films Data from Yoshimura et.al., 1996 Transport of Oxygen into a water film at 18 C Sc= ν L D =570 Sh= k Lδ 0 D k L = Γ L ln ( C S C in C S C out ) Tokyo Mathematical Modeling and Analysis Christoph Albert 13
14 Species Transport in Falling Films: Model Assumptions: Dilute system => species bears no mass or momentum, and does not affect viscosity No adsorption at the interface => surface tension stays constant No chemical reaction Local thermodynamic equilibrium at the interface => Henry's law holds Constant Henry coefficient t c+u c=d Δ c, D c n Σ =0, c L =H c G, Ω G (t ) Ω L (t) Σ Σ Tokyo Mathematical Modeling and Analysis Christoph Albert 14
15 Numerical Approach Two scalar approach with one-sided concentration gradient transfer flux Concentration is advected in the same way as volume fraction Two concentrations stored in each interfacial cell Both concentrations and Henry's law yield interfacial concentration Concentration gradient in liquid phase computed between interface and some value in bulk by subgrid model Diffuse flux from gas to liquid according to Fick's Law ϕ G =c χ ΩG ϕ L =c χ ΩL Tokyo Mathematical Modeling and Analysis Christoph Albert 15
16 Calculations Water film at 18 C in Oxygen atmosphere δ 0 Resolution 16 Re=31 Sc=50 k H,cc = c G =4.19e-5 c L,0 =2.37e-7 D=2.14e Tokyo Mathematical Modeling and Analysis Christoph Albert 16
17 Validation: Method Solve stationary advection diffusion problem on domain with boundary conditions. u( y) x c=d 2 y c [0, x max ] [0,δ 0 ] c x=0 =c L,0 y c y=0 =0 x c x=10cm =0 Solved with Matlab ODE Solver, by defining x as pseudotime. y c y=δ0 =k H,cc c G Tokyo Mathematical Modeling and Analysis Christoph Albert 17
18 Validation: Result Tokyo Mathematical Modeling and Analysis Christoph Albert 18
19 0 Hz 0.46mm 0mm 5.90cm 8.53cm Planar Interface Pure Diffusion Tokyo Mathematical Modeling and Analysis Christoph Albert 19
20 20 Hz 0.53mm mm 5.25cm 9.19cm Time-periodic wave structures appear Large Wave humps, preceeded by several smaller capillary waves Tokyo Mathematical Modeling and Analysis Christoph Albert 20
21 20 Hz 0.53mm mm 6.33cm 6.70cm Filaments of high concentration in the large wave humps Develop along the streamlines of the large vortex Touch the interface at a hyperbolic point Tokyo Mathematical Modeling and Analysis Christoph Albert 21
22 20Hz Highly nonmonotonous concentration profiles Tokyo Mathematical Modeling and Analysis Christoph Albert 22
23 20 Hz Strong contribution from the capillary wave region Tokyo Mathematical Modeling and Analysis Christoph Albert 23
24 20 Hz 0.53mm 0mm 6.59cm 6.77cm Flow up the wall in the reference frame of the wall Tokyo Mathematical Modeling and Analysis Christoph Albert 24
25 20 Hz 0.33mm 0mm 6.55cm Pressure in film according to Young- Laplace: Δ p σ κ 6.79cm Increase in pressure large enough to drive water up the wall Compare Dietze et al., J. Fluid Mech., 637, Tokyo Mathematical Modeling and Analysis Christoph Albert 25
26 30 Hz 0.53mm 0mm 7.22cm 9.84cm Less capillary waves Wave length, peak height, and wave velocity decrease Tokyo Mathematical Modeling and Analysis Christoph Albert 26
27 40 Hz 0.53mm 0mm 7.39cm 9.65cm No capillary waves; waves become sinusoidal Wave length, peak height, and wave velocity decrease further Tokyo Mathematical Modeling and Analysis Christoph Albert 27
28 40 Hz 0.39mm 0mm 7.91cm 8.03cm At this Reynolds Number, there exists backflow even when Capillary Waves are absent Tokyo Mathematical Modeling and Analysis Christoph Albert 28
29 Re 15 / 15Hz 0.43mm 0mm 5.84cm 7.87cm At this Reynolds Number, concentration profiles are monotonous Tokyo Mathematical Modeling and Analysis Christoph Albert 29
30 Re 15 / 15Hz Tokyo Mathematical Modeling and Analysis Christoph Albert 30
31 Re 15 / 15Hz 0.43mm mm 6.98cm 7.12cm No Vortex relative to wave velocity Tokyo Mathematical Modeling and Analysis Christoph Albert 31
32 Thank you for your attention! Tokyo Mathematical Modeling and Analysis Christoph Albert 32
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