Process design and optimization the case for detailed simulations Jos Derksen Chemical Engineering Delft University of Technology Netherlands & School of Engineering University of Aberdeen United Kingdom jderksen@abdn.ac.uk http://www.abdn.ac.uk/staffpages/jderksen/
(from my UG course in fluid mechanics) Fluid mechanics in chemical engineering my definition of chemical engineering transport gets a limiting factor flow of gas and/or liquid; motion of solids
Reactors from the outside from the inside bubbly flow sheared granular bed (Pickering) emulsion freeboard of a fluidized bed can we compute this?
The equations are known u= u ρ + ρ u u = p + µ u + f t cα + u c =Γ c +Ω α α cα, cβ, t T + u T = a T + q t the complexity is enormous turbulence multiphase (gas, liquid, solid) interfaces complex fluids chemical reactions ( ) already simplified incompressible Newtonian fluids Fick diffusion length scales vary wildly bubbles, drops, particles turbulent eddies (not to speak of molecules) versus the size of the reactor
What do you do (as chemical engineers)?
dimensional analysis reduce the number of variables & perform smart experiments Experiments ( Re, ε ) f = F D Re = F ( Eo, Mo) flow in pipes: Fanning friction factor cd = F ( Re,particle shape) rising bubbles and drops drag forces on objects
Computational Fluid Dynamics (in ChE?) CFD for aerospace and automotive design is well developed these are single-phase applications focus on dealing with & modeling of turbulence
CFD (in ChE?) turbulent flow in a mixing tank Re = ND ν = 1 5 for a particle or a drop or a bubble the average flow is an artefact average flow single realization
Resolve turbulence (as much as feasible) fully resolving turbulence would imply resolving the Kolmogorov length scaleη ηk L K Re if Re= 1 η 3 1 3 4 6 5 K L a three-dimensional simulation would require a grid spacing of η 5 K 3 1 L L 3 1 the number of grid cells would be of the order of which is unfeasible 4 ( ) 3 N 3 1 3 1 4 13
Large eddy simulation (LES) create a relatively coarse grid that does not resolve all turbulent scales devise a model that accounts for the effect of the unresolved eddies on the resolved eddies unresolved eddies resolved eddies such models are called subgrid scale models; they usually view the unresolved eddies as diffusion mechanisms ν ( ) 1 c S S with S u u i j eddy= S ij ij ij= + x j xi
LES in a mixing tank v tip Re = ND ν = 1 5 Smagorinsky SGS model (c S =.1) Lattice-Boltzmann discretization single realization: magnitude of GS and SGS velocity average flow single realization v /v tip.5.1..4.8.16.3.64 1.8
Experimental validation average flow experiment Rushton turbine Re=9, impeller blade k/v tip v tip experimental domain..8.16 > θ=1 o θ=31 o θ=49 o turbulent kinetic energy θ=1 o θ=31 o θ=49 o LES (interpolated to the experimental grid) LES experiment
a mixing tank should mix, i.e. homogenize Scalar mixing passive scalar active scalar red stuff is lighter than blue stuff
Some more mixing non-newtonian ( complex ) fluids ( N D N ) Y = τ ρ =.8 Y each movie lasts 1 impeller revolutions Y=. Y=.1
Some even more complex fluids thixotropy = time-dependent rheology network parameter λ - an active scalar λ λ + u 1 i = k 1γλ ɺ + k t x i ( λ) fiber suspensions form networks goes with the flow network breaks down due to shear the viscosity depends on λ (very simple, linear model) a ( 1 ) µ = µ + αλ builds under quiescent conditions; timescale 1/ k
α+ 1= 1 Db = N k What happens? liquid time scale Deborah= flow time scale at time zero: fully developed network: λ=1 everywhere Db= 1 < tn< 4 Db= 1 < tn< 5 Db= 1 < tn< 18 λ 1 λ contours - vertical cross sections, mid-baffle plane
Newtonian liquid - Add particles 1 liter vessel, d p =.3 mm, ρ part /ρ liq =.5, φ V =3.6%, n p =.4 1 7 St = ρ Re = ND ν ρ part liq d p 18ν 6N = 1 = O 5 ( 1) vertical cross section horizontal cross section
Solid phase dynamics Equations of motion for the spherical particles dx p = v added mass forces: p dt } gravity, drag, lift, from dv stress gradients p ( mp + ma ) = Fp dt dωp I = Tp dt Collisions: particle rotation: Magnus force rotational slip velocities hard-sphere particle-particle and particle-wall collisions parameters: restitution coefficient e (=1 mostly) friction coefficient µ f (= mostly) single-particle correlations d p <
averaged concentration midway between baffles Solids concentrations c/c av.4 1...8 3.6 d p =.3 mm, φ V =.95% d p =.47 mm, φ V =3.6%
Particle-particle collisions refer collision rates to Von Smulochowski: d p =.3 mm, φ V =.95% r coll,sm 4 = γɺ d 3 3 p M γɺ = ε ν d p =.47 mm, φ V =3.6% collision intensities d p =.3 mm, φ V =.95% o proper collisions missed collisions proper collisions missed collisions 55 o d p =.47 mm, φ V =3.6% o 1 9..18 r coll /r coll,sm 1 9 r coll /r coll,sm..1.18 v rel /v tip 55 o
Particle-impeller collisions collision intensities d p =.3 mm, φ V =.95% front surface of impeller blade probability density function (pdf) of particle-impeller collision velocities pdf (a.u.) 1 1 1 1-1 d p =.47 mm, φ V =3.6% 1-1 -3 1-4 d p =.47 mm d p =.3 mm 1-5..4.6.8 1...1.18 v / rel v tip v rel / v tip
Collision mechanics t=t pdf of the particle s angular velocity (single realization, entire vessel) 1 log(pdf) -1 no friction friction (µ f =.35) t=t + t t=t + t - -3-4 elastic, frictionless collision elastic, frictional collision -5 1 3 4 ω / πn
Coupling particles and scalar: dissolution solid-to-liquid mass flux φ m c sat c φ m Sh = Γ = Sh d p ( c c) sat.5 +.6Re Sc p.33 * Re p based on local slip velocity Sh mass flux linear in concentration: no micro-mixing kd p = : Sherwood number Γ room for refinement c-v correlations do matter (but are neglected) fluid shear and particle rotation should be added to mass flux * Ranz and Marshall (195)
Flow system T =.3 m (1 liter vessel) working fluid water Re = 1 5 N = 16.5 rev/s (N js = 11.4 rev/s) 7 1 6 calcium-chloride beads d p =.3 mm; ρ p /ρ liq =.15 c sat = 6 kg/m 3 c =1 kg/m 3 Γ mol =.7 1-9 m /s (calcium ions) beads released in upper part (.9T-T) φ V = 1% (average 1%)
Particle distribution: < Nt 6 Nt= Nt= 5 Nt= 7 d p /d p 1.9.8.7.6.5.4.3..1 Nt= 1 Nt= particles are 5 times enlarged
Scalar concentration distribution: < Nt Nt= Nt= 5 Nt= 7 c/c 1.8 1.6 1.4 1. 1.8.6.4. Nt= 1 Nt=
Snapshots spatial particle distributions Nt = 6.5 d p /d p.7.6 Nt = 6 d p /d p.5.5.5 N p / N p.1.5.5 1 d p /d p particles 1 times enlarged N p / N p.1.5.5 1 d p /d p
N p / N p 1 1-3 1-6 Evolution particle size distribution Nt = Nt = 5 Nt = 7 Nt = 1 Nt = Nt = 4 Nt = 6 Nt = 8 Nt = 1 d 1 p /d p d 3 / d p (-) Sh φ m = 1.8.6.4. Sauter mean diameter 46 81 Nt (-).5.33 +.6Rep Sc d p d( d ) Sh d p dt p 1 d p
From mixing to separation gas-solid cyclones separators a challenge for CFD: prediction of the collection efficiency flow field predictions (average flow, turbulence quantities) particle transport modeling strongly swirling flow turbulence effects of solids loading
Velocity profiles simulations vs experiment u θ / Uin 1-1.4 u ' U θ / in. u / U x in.5 u '/ U x.4 in. A -.5 1-1 1-1.4..4..5 -.5.5.4..4. B C A B C -1 r/ R 1 average tangential velocity -1 r/ R 1 -.5 r/ R 1-1 -1 r/ R 1 RMS tangential velocity average axial velocity RMS axial velocity
.1R x z average vortex core position y Behavior of the vortex core vortex core precession in terms of power spectral density of a velocity signal psd (au) S =.7 1 3 S = 1.58 experiment S = fd/u 1 3 in fd U in fd/u in = 1.61 in terms of pressure field (horizontal cross section) -.6 (p-p )/p -.4 LES
Solid particle modeling dvp U = in 1 dt D Stk ( u v ) + g ρg 18νD local gas velocity: resolved part [u=f(time)] unresolved part isotropic random process with RMS u sgs = k sgs 3 k SGS p Stk = ρ = Ckcs SijSij Ck (in conjunction with the Smagorinsky model) p dpuin 5 : Stokes nmbr grid-scale turbulent kinetic energy subgrid-scale turbulent kinetic energy k U gs in.9.7.5.3.1 k U sgs in.9.7.5.3.1
Impressions One-way coupled simulation blue particles: Stk=1.8 1 - red particles: Stk=5. 1-4 side view (1: particles are on display) vertical cross section
One-way coupled simulation The separation process recirculations c/c in cross sections of the timeaveraged particle concentration.56.64.16.4.1 Stk=6.5 1-5 1.4 1-3.3 high TKE region η 1.8.6.4. grade efficiency 1-4 1-3 1 - ρp dpuin Stk Stk= : Stokes nmbr ρ 18νD g
Particle-to-gas coupling particle-source in cell (PSIC) method* ΛF p g F g p extrapolation of the force with the same coefficients that were used for interpolating velocities Λ: number of particles in a parcel (or: the trick to get appreciable mass-loading) our system: Λ=8 1 5 for mass-loading φ=.1 *Crow et al. Ann. Rev Fl. Mech. 8 (1996)
Response to switching on way coupling 1.5 1 7 number of particles inside the cyclone as a function of time n p 1 1 7 1way way φ=.1 5 1 6 1way way φ=.5 way φ=.1 5 1 15 t/t int t=: way coupling switched on way much less particles in the dustbin
way coupled gas flow u tan /U in tangential velocity turbulent kinetic energy.1 k/u in.5 A 1way way φ=.5 way φ=.1 u tan /U in - A.1 k/u in B A B B - -.5..5 r/d.5 -.5..5 r/d effect of particles: loss of swirl turbulence damping
Separation performance / pressure drop p 1 ρgu in 5. pressure drop η 1.8 grade efficiency 1way way φ=.5 way φ=.1 Stk 5 =.3 1-3 Stk 5 =.1 1-3 Stk 5 =.86 1-3.5.5.1.15. φ.6.4. Stk 5 1-4 1-3 1 - Stk η f 1.95 fractional efficiency η f = η ( dp ) ψ( dp ) 3 ψ( d ) d p p d 3 p dd dd p p.9.1. φ (efficiency by weight)
Reactive flows: liquid mixing dye feed flow Flow geometry (Re=4,) 3D, time-resolved LIF experiment ν Sc = 1,9 Γ
Tubular reactor: passive scalar transport Vertical cross-section time-averaged concentration fields: experimental validation D-LIF experiment LES Vertical cross-sections log(c/c ) -6
φ + v φ = J + t Turbulent reactive flows convection diffusion reaction equation for scalar vector ϕ filter φ + t ( φ) ωφ ( ) ω ωφ ( ) ( φ v) = Γ φ + ωφ ( ) σ k A + B C r = kc A c B B A A B real life LES Lagrangian model the motivation for pdf methods ( ) = kψaψbpl ( ψa, ψb ) dψadψb ωφ solve transport equations for the pdf s joint pdf of A and B
Lagrangian (Monte-Carlo) methods B A A B Lagrangian model define particles and move them around in physical and composition space real life Lagrangian model Evolution in physical space ( ( t ), t ) dt + E( x( t ) t ) dw( t ) dx = D x, drift (convection) ( ) D = v + Γ + Γ e diffusion (random process) ( Γ + ) E = Γ e Evolution in composition space ( ( t ) t )dt d φ = Bφ, ( φ φ) ωφ ( ) B Ω + = m micro-mixing reaction IEM model C ( Γ + Γ ) Ω e Ωm =
A sample application k A + B 1 P k A + C Q k 1 =1 3 k Damköhler number: turbulent Da = = tchem A: in the jet; B and C in the bulk flow t ( D / U )( k c c ) bulk A C Da=.5 1 3 (poor mixing) Φ Q /Φ P 1.5 Da=.5 1-5 (good mixing) 1-1 1 1 4 1 6 Da c B on a linear scale Red: c B =c B Blue: c B =
Where to go from here?
More details in solid-liquid suspensions 1 log(pdf) Experimental validation particle concentration profiles particle-impeller collisions... -1 - -3 no friction friction (µ f =.35) z/ D 1-1 1.. 3. LES compared to experimental data of Michelettti et al. 3 c/ c av impact tests Kee & Rielly 4-4 -5 1 3 4 ω / πn
Unresolved vs resolved particles a p = d < particle size < fluid grid spacing particle dynamics based on empirical force correlations up to 1 8 particles a p = d > particle size > grid spacing no need for empiricism* up to 1 4 particles multi-scale
Macro-Meso-Micro macro meso micro equipment size assemblies of bubbles, drops, particles assemblies of molecules m cm mm µm nm
Mesoscale example: liquid-solid fluidization typically: 1 mm glass beads in water z Experimental result*: narrow liquid-fluidized beds show a planar wave instability Computational approach: fully periodic 3D box d p x6d p x6d p g z liquid flow body force on fluid * Duru et al., JFM 45 () gravity
φ.6.5.4 σ~ u ch φ av =.55 σ = ρ u 1 z/d p f ν = d ch p g Momentum transfer ( stress ) relative magnitude of zz stresses ɶσ 4 1 3 3 1 3 1 3 1 1 3 collisional p streaming f streaming lubrication 1 1 3 1 z/d p compaction void dilation p p~ p~ c 4 1 3 3 1 3 1 3 1 1 3 c c 1 = 3 p = ρ u ( σ + σ + σ ) f c,xx c ch c,yy negative φ s -slope positive φ s -slope φ s p c not a unique function of φ s c,zz negative φ s -slope: compaction positive φ s -slope: dilation
Molecular scale example: aggregation of nanoparticles & liquid bridges Molecular Dynamics TiO nanoparticles 8 nm spheres in a classical Lennard- Jones fluid (red=vapor; yellow=liquid; green=solid) f 1 R R 1 = σπr (+ is attractive) 1 3 V molar NB: 3nm N Av
Liquid bridge (molecular) dynamics
Students Eelco van Vliet Acknowledgements Sponsors Bas Doelman Hugo Hartmann Andreas ten Cate Arjen Hoekstra