Process design and optimization the case for detailed simulations
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1 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
2 (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
3 Reactors from the outside from the inside bubbly flow sheared granular bed (Pickering) emulsion freeboard of a fluidized bed can we compute this?
4 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
5 What do you do (as chemical engineers)?
6 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
7 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
8 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
9 Resolve turbulence (as much as feasible) fully resolving turbulence would imply resolving the Kolmogorov length scaleη ηk L K Re if Re= 1 η 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
10 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
11 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
12 Experimental validation average flow experiment Rushton turbine Re=9, impeller blade k/v tip v tip experimental domain > 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
13 a mixing tank should mix, i.e. homogenize Scalar mixing passive scalar active scalar red stuff is lighter than blue stuff
14 Some more mixing non-newtonian ( complex ) fluids ( N D N ) Y = τ ρ =.8 Y each movie lasts 1 impeller revolutions Y=. Y=.1
15 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
16 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
17 Newtonian liquid - Add particles 1 liter vessel, d p =.3 mm, ρ part /ρ liq =.5, φ V =3.6%, n p = St = ρ Re = ND ν ρ part liq d p 18ν 6N = 1 = O 5 ( 1) vertical cross section horizontal cross section
18 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 <
19 averaged concentration midway between baffles Solids concentrations c/c av d p =.3 mm, φ V =.95% d p =.47 mm, φ V =3.6%
20 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 r coll /r coll,sm 1 9 r coll /r coll,sm v rel /v tip 55 o
21 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.) d p =.47 mm, φ V =3.6% d p =.47 mm d p =.3 mm v / rel v tip v rel / v tip
22 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 elastic, frictionless collision elastic, frictional collision ω / πn
23 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)
24 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) 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 = m /s (calcium ions) beads released in upper part (.9T-T) φ V = 1% (average 1%)
25 Particle distribution: < Nt 6 Nt= Nt= 5 Nt= 7 d p /d p Nt= 1 Nt= particles are 5 times enlarged
26 Scalar concentration distribution: < Nt Nt= Nt= 5 Nt= 7 c/c Nt= 1 Nt=
27 Snapshots spatial particle distributions Nt = 6.5 d p /d p.7.6 Nt = 6 d p /d p N p / N p d p /d p particles 1 times enlarged N p / N p d p /d p
28 N p / N p 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 = Sauter mean diameter Nt (-) Rep Sc d p d( d ) Sh d p dt p 1 d p
29 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
30 Velocity profiles simulations vs experiment u θ / Uin u ' U θ / in. u / U x in.5 u '/ U x.4 in. A B C A B C -1 r/ R 1 average tangential velocity -1 r/ R r/ R r/ R 1 RMS tangential velocity average axial velocity RMS axial velocity
31 .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 = 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
32 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 k U sgs in
33 Impressions One-way coupled simulation blue particles: Stk= red particles: Stk= side view (1: particles are on display) vertical cross section
34 One-way coupled simulation The separation process recirculations c/c in cross sections of the timeaveraged particle concentration Stk= high TKE region η grade efficiency ρp dpuin Stk Stk= : Stokes nmbr ρ 18νD g
35 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)
36 Response to switching on way coupling number of particles inside the cyclone as a function of time n p way way φ= way way φ=.5 way φ= t/t int t=: way coupling switched on way much less particles in the dustbin
37 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 r/d r/d effect of particles: loss of swirl turbulence damping
38 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 = Stk 5 = Stk 5 = φ.6.4. Stk 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)
39 Reactive flows: liquid mixing dye feed flow Flow geometry (Re=4,) 3D, time-resolved LIF experiment ν Sc = 1,9 Γ
40 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
41 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
42 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 =
43 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= (poor mixing) Φ Q /Φ P 1.5 Da= (good mixing) Da c B on a linear scale Red: c B =c B Blue: c B =
44 Where to go from here?
45 More details in solid-liquid suspensions 1 log(pdf) Experimental validation particle concentration profiles particle-impeller collisions no friction friction (µ f =.35) z/ D LES compared to experimental data of Michelettti et al. 3 c/ c av impact tests Kee & Rielly ω / πn
46 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
47 Macro-Meso-Micro macro meso micro equipment size assemblies of bubbles, drops, particles assemblies of molecules m cm mm µm nm
48 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
49 φ 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 collisional p streaming f streaming lubrication z/d p compaction void dilation p p~ p~ c 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
50 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
51 Liquid bridge (molecular) dynamics
52 Students Eelco van Vliet Acknowledgements Sponsors Bas Doelman Hugo Hartmann Andreas ten Cate Arjen Hoekstra
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