Aggregation and Breakage of Nanoparticle Dispersions in Heterogeneous Turbulent Flows

Transcription

1 ggregation and reakage of Nanoparticle Dispersions in Heterogeneous Turbulent Flows M. Soos,, D. Marchisio*, J. Sefcik, M. Morbidelli Swiss Federal Institute of Technology Zürich, Z CH *Politecnico di Torino, IT CFD in Chemical Reaction Engineering arga,, Italy, June 25

2 Problem definition - coagulation process Dispersion of stable primary particles ggregates / Granules destabilization Coagulator types: Stirred tank Couette device Pipe [m]

3 Problem definition - coagulation process To have good product quality: appropriate morphology effective mixing V = 2.5 L Turbulent flow V =.3 L Stirred tank Taylor-Couette device

4 Particle/aggregate characterization technique Focused eam Reflectance Method (LSENTEC) Stirred vessel 2 Light scattering instrument 3 Syringe pump Light Scattering (Malvern Mastersizer) 3 2 Probe of FRM

5 ggregation and reakage kinetics ggregation j i k=i+j K N N ij, i j ggregation Rate k reakage k-j j K N k k reakage Rate ssumption about daughter distribution function

6 Population alance Equation Reynolds-averaged mass balance ( CFD & PE ): (PE mass based) ( ξ x ) ( ξξ ) ( ξ ; x, ) d ξ ξ ( ξ; x, ) ( ξ x ) n ;, t n ;, t + ui n( ;, t) D t t t x ξ x = i xi xi ξ K, n( ;, t) n( ;, t) d n( ;, t) K, n( ;, t) d 2 ξ ξ ξ ξ ξ x ξ x ξ ξ x ξ ξ ξ x ξ Kξ b n t K n t ξ + ggregation source term reakage source term Only unknown are the values of aggregation and breakage kernels There are several numerical approaches to solve this equation: Classes method Method of moments* Monte Carlo method b(ξ ξ ) - Daughter distribution function of produced fragments

7 ggregation and reakage rate expressions (kernels) ggregation kernel: reakage kernel: K = K + K S ξξ, ξξ, ξξ, K ξ = P ( )( / d ) f ξ 3 P G 2kT Kξξ, = ξ + ξ ξ + ξ 3µ W K GR p ( / d / ) 3 f df + S 3 ξξ, = α ξ ξ Fractal scaling: R R ξ p ξ / d f ( d )( ) f df df df Occupied volume fraction: ( ξ) φocc = n V ξ φocc.5 gelation occurs α, P, P 3, d f from experiment or assume G either from experiment (difficult) or from CFD ε G = ν Connection to light scattering R R ( + ) N 2 d f g w iξ i= i = N 2 p w i = iξi w i weights of quadrature app. ξ i abscissas of quadrature app. 2

8 Solution of PE - QMOM vs. fixed pivot method Fixed pivot QMOM N = 2 QMOM N = 3 QMOM N = 4 QMOM N = τ = t G φ Quadrature Method Of Moments (QMOM) mk ( x, t) mk ( x, t) + u m ( x, t) D = 2 t x x x i k t i i i N N k N N ( k ), ( + ) + K ξ ξ ξ ξ ww K b w K ξ w k k k i j i j i j i j i i i i i i i= j= i= i= quadrature approximation N k k k = ( ξ ) ξ dξ iξi i= m n w Weights w i and abscissas ξ i calculated by PD algorithm Number density (# / m 3 ) Cluster mass distribution (CMD) 7.5 min 5 min min 45 min 2 3 m(i) / m() Steady state is accurately modeled with two nodes (N = 2) For lower fractal dimensions (d f 2) larger number of nodes (N = 4-5) needed to be used

9 Estimation of model parameters from experiment Dimensionless form of PE dxk( τ ) K = ( τ ) ( τ) ( τ) ( τ) dτ 2 V G V G ij Kik xi xj xk xi i+ j= k i= Γ K K + ( ) xk( τ) G mk m k xm τ m= k+ φg φ Ni ( τ ) V xi ( τ ) = φ τ = tgφ ggregation is linearly dependent on G K reakage is nonlinearly dependent on G K ξ = P ( )( / d ) f ξ 3 P G ( / d / ) 3 f df + ξξ, = αg ξ ξ 2 P (G), P 3 / R g,p α, d f 2 rpm 4 rpm 6 rpm 8 rpm 5 5 τ = t G φ R P =.85 µm α =.2 φ = 5-5 d f = 2. P =.55 2 G 2.82 P 3 = 4

10 Dynamics of the system full CFD (TC) Taylor-Couette apparatus, 2D simulation, RSM of turbulence Lumped model (often used in literature): particle distribution homogeneous shear rate is everywhere equal to the G G G = G 2 = G i = G G 2 CFD model: particle distribution heterogeneous shear rate distribution heterogeneous Outer wall Homogeneous model: particle distribution homogeneous shear rate distribution heterogeneous breakage kernels properly averaged over volume G P G G P 2 2 Is the effect of spatial shear rate heterogeneity significant? G 2 G G 2 G i G G ε = ν Inner wall 2

11 Dynamics of the system full CFD (TC) 2 φ = 5-5 φ = Lumped model 3 CFD Homogeneous model τ ssumption about particle distribution homogeneity is valid

12 Dynamics of the system full CFD (TC) 2 9 φ = 5-5 φ = 5-5 φ = -3 6 Lumped model 3 CFD Homogeneous model τ 3 φ = τ ssumption about particle distribution homogeneity is NOT valid

13 Fluid element trajectories Outer wall Spatial shear rate distribution in the gap of Taylor-Couette Inner wall G, s Time, s G = s - Two examples of fluid element trajectories corresponding to the starting point from low and high shear rate value Shear rate [/s]

14 Fluid element tracking Process timescales Macromixing: τ M = k ε τ, τ >> τ M k, ε CFD G φ S = 5 x -5 ε = ν τ = 2 ggregation / breakage at steady state: ( ; g, g ) K G R R N G, s τ = ( ; ) K G R g Homogeneous model: particle distribution homogeneous aggregation / breakage kernels properly averaged over volume Kξξ, = Kξξ, ( G) dv V Kξ = Kξ ( G) dv V V Time, s V Time, s Shear rate history Calculation starts from the steady state distribution corresponding to the starting point of shear rate

15 Fluid element tracking Process timescales Macromixing: τ M = k ε k, ε CFD G ε = ν τ = 2 ggregation / breakage at steady state: ( ; g, g ) K G R R N τ = ( ; ) K G R g G, s Time, s Shear rate history Calculation starts from the steady state distribution corresponding to the starting point of shear rate τ, τ << τ M Time, s φ S = x -2 Equilibrium model: particle distribution relaxes immediately to steady state according to local shear rate

16 nalysis of timescales Conclusion Lumped model Homogeneous model CFD model Equilibrium model R R g g max min. (τ < τ for this system) φocc. ased on timescales analysis it is possible to decide which model is appropriate for certain conditions t low volume fraction (< 4-4 ) CFD model can by efficiently replaced by homogeneous model timescales, s τ, τ M Homogeneous model Coupled CFD & PE model appropriate Need full CFD & PE w/ micromixing model & multiphase flow Kinetic parameters obtained from lumped model are not applicable for different vessel geometry significant effect of shear rate heterogeneity CFD + PE need to be used already for rather mild solid volume fractions φ S Note: region of validity of homogeneous model in stirred tank is shifted to left compare to TC Note: occupied volume fraction can be used to check significance of viscosity on flow field