Macroscopic convective phenomena in non-uniformly heated liquid - PowerPoint PPT Presentation

VI International Workshop on Non-equilibrium Thermodynamics and III Lars Onsager Symposium Røros, Norway, August 19-24, 2012 Macroscopic convective phenomena in non-uniformly heated liquid mixtures Vitaly А . Demin 1 , Alexander F. Glukhov 2 1 – Theoretical Physics Department 2 – General Physics Department Perm State University, Russia Contact: demin@psu.ru

Chronicle of one PhD dissertation on thermal convection PhD thesis: Glukhov Alexander Experimental Investigation of Thermal Convection in Conditions of Gravity Stratification // Perm State University, Perm, 1995. – 140 p. Experimental investigation had three basic parts: � Thermal convection of liquid molecular mixtures in connected channels; � Thermal convection of ferrofluid in connected channels; � Evolution of particles distribution in vertical pipe filled by ferrofluid. Our present-day evaluation of dissertation results: ☺ Correct paradigm was formed for concerned phenomena in molecular mixtures; ☺ It was obvious that thermodiffusion and sedimentation must play definite role in these phenomena; � The contribution of each factor wasn’t clear; Alexander F. Glukhov, 2007 � Numerical model wasn’t built.

Experimental data z Binary mixtures 2 d 2 3 2 d One of the working fluids was a mixture of Carbon 4 h Tetrachloride CCl 4 and Decane C 10 H 22 1 2 1) Decane C 10 H 22 (Pr = 15), 2) CCl 4 (heavy admixture) 1 y Thermodiffusion properties of this mixture are not 2 investigated in details until now; x a b Schmidt number Sc = ν / D > 1000, ε − ? (1992); Thermodat COM PC The second liquid was a mixture of water and sulphate of natrium: COM Voltmeter 1) water H 2 O (Pr = 7), 2) natrium sulphate Fig. 1. Experimental setup ( a ): copper Na 2 SO 4 (heavy admixture in water) bar ( 1 ), heat exchangers ( 2 ), channels ( 3 ), Thermodiffusion properties of this mixture are thermocouples ( 4 ); coordinate system ( b ). well known : Schmidt number Sc = 2100, parameter of thermodiffusion ε = 0.36 (2005). Width and height of the channels: d = 3.2 mm; H = 50 mm

Harmonic oscillations and regime with regular redirection of fluid circulation Regime with regular redirection of fluid circulation 1992 1992 20 min 1992 Harmonic oscillations Fig. 2-4. First experimental data correspond to 5 - 15% solution of the denser CCl 4 in the less dense C 10 H 22 . The shape of the oscillations was transformed 20 min from near-sinusoidal to near-rectangular with the growth of supercriticality.

Principal explanation for molecular mixtures Mechanism responsible for the effects observed Cooler T 1 in experiments is mainly attributable to the θ u r ( x ) thermodiffusion separation of the mixture which is due to the horizontal temperature ∇ C l ( x ) T r ( x ) gradients θ / d = 3 K/cm rather than to the weak vertical gradients Θ /h = 0 . 3 K/cm with a characteristic component separation time h 2 /D ~ 103 hours; h – height, d – width of the channel. Horizontal gradients occur only in the T l ( x ) ∇ C r ( x ) circulating fluid. The separation time across the channel is d 2 /D ~ 1 hour, which coincides in u l ( x ) Θ = T 2 – T 1 Heater T 2 order of magnitude with the time of circulation of the fluid around the loop. Fig. 5. Schematic visualization of the admixture Liquid particle changes itself composition distribution. Left channel accumulates heavy during the motion in each channel. component, right one loses it.

Flow of ferrofluid with regular redirection of circulation 0.2 ϑ Solution of Na 2 SO 4 0.1 in water, 16%; 0 Θ = 10 ° С , -0.1 -0.2 t , h 0 0.5 1 1.5 2 2.5 3 0.3 Ferrofluid, 4%; ϑ 0.2 Θ = 2.0 ° С , ∇ φ = 0.58 ⋅ 10 -5 cm -1 0.1 f = 0.56 ⋅ 10 -2 c -1 0 -0.1 t , h 0 1 2 3 4 5 6 0.2 ϑ Ferrofluid, 12%; 0.1 Θ = 6.0 ° С , 0 ∇ φ = 2.62 ⋅ 10 -5 cm -1 f = 0.72 ⋅ 10 -2 c -1 -0.1 -0.2 t , h 0 1 2 3 4 5 6

Sedimentation in ferrofluid x , cm 10 4 Radius of particle: 10 nm; 2 1 Density of kerosene and magnetite: 2 ρ f = 0.82 g/cm 3 ; ρ m = 5.5 g/cm 3 L e 5 3 3 1 – 40 hours К L x C 2 – 6300 hours C e 0 3 – 10300 hours 0.045 0.055 0.065 Fig. 6. Scheme of experimental setup for the measurement of magnetic 1 particles concentration and concentration in dependence on height for different moment of time. 1 – metal support; 2 – test-tube with ferrofluid on the base of kerosene; 3 – inductive sensor of particles concentration; 4 – screw to move the test-tube Conclusion: The effect of particles sedimentation exists and can be estimated even in the beginning of thermal convection with the help of the formula for frequency of transitional oscillations : β 4 ∇ t d T f − − 2 4 5 − ∇ φ = = ÷ 1 f – frequency of transitional oscillations 10 10 cm β 2 2 π χ φ

Does convective behaviour of molecular mixtures and ferrofluids have common nature or only individual common features? = How can elephant be eaten? Answer: Only bit by bit.

Binary molecular mixtures z Cooler u T g x H − 2 2 C d Physical requirement: 1 2 H >> d Heater 1) Straight-trajectories approximation is applied 2) Boundaries of channels have high heat conductivity 3) Antisymmetric solutions for fields of temperature, velocity and concentration are valid.

Basic assumptions The diffusion and heat fluxes are related with the concentration and temperature gradients in general case by the formulas: � � ( ) ( ) = − ρ ∇ + ∇ α = − λ + α Λ ∇ − Λ∇ j D C T q D T D C The effects associated with the presence of an admixture are characterized by the coefficients of diffusion D and thermodiffusion α . ∂ ρ ⎛ ⎞ 1 The concentration density coefficient β c β = ⎜ ⎟ c Expansion of density: describes the dependence of the density on the ρ ∂ ⎝ C ⎠ , o T p concentration: ( ) ρ = ρ − β + β The equations for an incompressible fluid in the Boussinesq 1 T C o t c approximation had been used to simulate the convective flows of a binary mixture: � ∂ ∂ � � � � С � v 1 ( ) + ∇ = − ∇ + Δ ν + β − β γ + ∇ = Δ + α Δ v v p v T C v С D C D T ( ) g ( ) t c ∂ ρ t ∂ t o ( ) ∂ T � C – mass concentration 2 + ∇ = χ + α Λ Δ + α ΛΔ v T D T D C ( ) ∂ t of heavy admixture Shaposhnikov I.G. Theory of Convective Phenomena in a Binary Mixture // Prikl. Matem. Mekh., 1953.

Equations in non-dimensional form and control parameters • Pressure p – [ ρ ο ν 2 / d 2 ], • Length L – [2 d ], • Velocity v – [ ν / d ], • Concentration C – [ Θβ t / β c ], Units: • Time t – [ d 2 / ν ], • Temperature T – [ Θ ]. Here Θ is the temperature difference between heat exchangers. � 4 β t d Θ ∂ v RaH = g 1 � � � � � ( ) Ra + ∇ = −∇ + Δ + − γ = ( v ) v p v T C , div v 0, νχ ∂ h t Pr Pr αβ ∂ ∂ T � 1 С � 1 ( ) c + ∇ = Δ ε = + ∇ = Δ + Δ ε ( v ) T T , ( v ) С C T . β ∂ ∂ t Pr t Sc t ν Nondimensional parameters: thermal Rayleigh number, = Pr thermodiffusive parameter, Prandtl and Schmidt numbers: χ ∂ ∂ С T ν Boundary conditions on vertical + ε = . = 0 Sc � � ∂ ∂ walls for field of concentration: n n D Γ Γ

Mechanical equilibrium state Boundary condition Conditions of mechanical equilibrium: ∂ on concentration: � = = = = = v p p T T C C 0, , , , 0, o o o ∂ t ∂ ∂ С T + ε = Equations system: . 0 � � ∂ ∂ n n Γ Γ � ( ) Δ = Δ + Δ ε = , T C T ∇ − ∇ × γ = 0, 0 T C 0 o o o o o T o ( z ) = – z / H , Equilibrium distributions of temperature and concentration: C o ( z ) = ε z / H . � It’s possible to use the straight - ( ) v U x y t 0, 0, ( , , ) Stationary flow: trajectory approximation. Fields distributions in cross-section: = F = C + ε T F x y z f z x y ( , , ) ( )s ( )c ( ) 13 13 π π x y ⎛ ⎞ ⎛ ⎞ = ⋅ U x y u ( , ) sin cos ⎜ ⎟ ⎜ ⎟ π π x x ⎛ ⎞ 1 ⎛ 3 ⎞ ⎝ 2 ⎠ ⎝ 2 ⎠ = − s ( ) x sin sin ⎜ ⎟ ⎜ ⎟ 13 ⎝ 2 ⎠ 3 ⎝ 2 ⎠ π π ⎛ x ⎞ ⎛ y ⎞ π π y y = θ ⎛ ⎞ 1 ⎛ 3 ⎞ T x y z z ( , , ) ( )sin cos ⎜ ⎟ ⎜ ⎟ = + c ( ) y cos cos ⎜ ⎟ ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ 13 ⎝ 2 ⎠ 3 ⎝ 2 ⎠

Stationary solution 10 u Fig. 7. Amplitude curves for different values of thermodiffusion parameter: 1 – ε = – 0.01; 2 – ε = 0.02; 3 – ε = 0. There is formula in limit u → 0 , ε = 0: 5 4 π = Ra c = π z H 2 2 ⎛ ⎞ 1 1 − z 4 1 tanh ⎜ ⎟ 1 z ⎝ ⎠ 1 In limiting case H → ∞ formula gives well- 2 known value of critical Rayleigh number 1 Ra Ra = π 4 /4. 0 3 0 40 80 Solution for arbitrary values of thermodiffusion parameter, Prandtl and Schmidt numbers: − 1 4 ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ π ε π ⎪ Sc ⎪ H 1 1 3 10 ( ) = + ε − + − Ra z z = 1 1 tanh 0.45 tanh z ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ c 1 2 2 z Pr z 4 20 ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ 1 2