CFD Models for Polydisperse Solids Based on the Direct Quadrature Method of Moments Rodney O. Fox H. L. Stiles Professor Department of Chemical Engineering Iowa State University Acknowledgements: US National Science Foundation, US Dept. of Energy, Numerous Collaborators CFD in CRE IV: Barga, Italy June 19-24, 2005 1
1. Introduction Population Balances Coupling with CFD Outline 2. Population Balances in CFD Population Balance Equation Direct Solvers Quadrature Methods 3. Implementation for Gas-Solid Flow Overview of MFIX Polydisperse Solids Model Application of DQMOM 4. Two Open Problems CFD in CRE IV: Barga, Italy June 19-24, 2005 2
Population Balances Number density function (NDF) particle surface area time particle volume (mass) spatial location CFD provides a description of the dependence of n(v,a) on x For multiphase flows, the NDF will include the phase velocities (as in kinetic theory) CFD in CRE IV: Barga, Italy June 19-24, 2005 3
Population Balances Moments of number density function Choice of k and l depends on what can be measured Solving for moments in CFD makes the problem tractable due to smaller number of scalars Multi-fluid model solves for moments from kinetic theory CFD in CRE IV: Barga, Italy June 19-24, 2005 4
Population Balances Physical processes leading to size changes Nucleation J(x,t) produces new particles, coupled to local solubility, and properties of continuous phase Growth G(x,t) mass transfer to surface of existing particles, coupled to local properties of continuous phase Restructuring particle surface/volume and fractal dimension changes due to shear and/or physio-chemical processes Aggregation/Agglomeration particle-particle interactions, coupled to local shear rate, fluid/particle properties Breakage system dependent, but usually coupled to local shear rate, fluid/particle properties CFD provides a description of the local conditions CFD in CRE IV: Barga, Italy June 19-24, 2005 5
Population Balances What can we compare to in-situ experiments? Sub-micron particles small-angle static light scattering Larger particles optical methods zero-angle intensity radius of gyration 1.8 < d f <3 length projected fractal dimension CFD model should predict measurable quantities accurately CFD in CRE IV: Barga, Italy June 19-24, 2005 6
Coupling with CFD Do particles follow the flow? Stokes number Particle diameter Kinematic viscosity If St > 0.14, particle velocities must be found from a separate momentum equation in the CFD simulation CFD in CRE IV: Barga, Italy June 19-24, 2005 7
Coupling with CFD Do PBE timescales overlap with flow timescales? Residence time Recirculation time Local mixing timescale Kolmogorov timescale CFD simulations w/o PBE can be used to determine timescales for a particular piece of equipment CFD in CRE IV: Barga, Italy June 19-24, 2005 8
Outline 1. Introduction Population balances Coupling with CFD 2. Population Balances in CFD Population Balance Equation Direct Solvers Quadrature Methods 3. Implementation for Gas-Solid Flow Overview of MFIX Polydisperse Solids Model Application of DQMOM 4. Two Open Problems CFD in CRE IV: Barga, Italy June 19-24, 2005 9
Population Balance Equation Typical NDF Transport Equation (small Stokes) Advection Diffusion Nucleation + Growth } Aggregation Breakage CFD in CRE IV: Barga, Italy June 19-24, 2005 10
Population Balance Equation Aggregation Kernel Shear-induced Brownian Sub-micron aggregates: Brownian >> Shear-induced Breakage and restructuring determine fractal dimension d f In granular flow, particle-particle collisions must be added CFD in CRE IV: Barga, Italy June 19-24, 2005 11
Population Balance Equation Breakage Kernels exponential power law Breakage due to fluid shear only ==> additional term due to collisions in gas-solid flows Parameters determined empirically and depend on chemical/physical properties of aggregates CFD in CRE IV: Barga, Italy June 19-24, 2005 12
Population Balance Equation Daughter Distribution binary Equal sized: f = ½ Erosion: f << 1 10 14 Number density (# / m 3 ) 10 12 10 10 10 8 10 6 10 4 10 2 10 0 10 0 10 1 10 2 10 3 10 4 10 5 10 6 m(i) / m(1) f=0.5 f=0.1 f=0.01 f=0.001 CFD in CRE IV: Barga, Italy June 19-24, 2005 13
Direct Solvers Sectional or Class Methods 10 1 10 2 Finer grid N t=0 / N Finer grid R g / R g,p 10 1 10 0 0 10 20 30 Time / min 10 0 0 10 20 30 Time / min Accurate predictions for higher-order moments require finer grid (range: 25-120 bins) CFD in CRE IV: Barga, Italy June 19-24, 2005 14
Direct Solvers Difficulties encountered when coupled with CFD n(v; x, t) represented by N scalars n i (x,t) where 25<N<120 Depending on kernels, initial conditions, etc., source terms for these scalars can be stiff If particles are large (measured by Stokes number), multiphase models with N momentum equations required Extension to multi-variate distributions scales like N D accounting for morphology changes will be intractable Need methods that accurately predict experimentally observable moments, but at low computational cost CFD in CRE IV: Barga, Italy June 19-24, 2005 15
Quadrature Methods Quadrature Method of Moments (QMOM) weights abscissas k th moment of CSD: CFD in CRE IV: Barga, Italy June 19-24, 2005 16
Quadrature Methods Product-Difference algorithm (univariate CSD) Inverse problem solved on the fly in CFD simulation CFD in CRE IV: Barga, Italy June 19-24, 2005 17
Quadrature Methods Transport 2N moments in CFD simulation Advection Diffusion Nucleation + Growth Aggregation Breakage CFD in CRE IV: Barga, Italy June 19-24, 2005 18
Quadrature Methods Comparison with direct method R g / R g,p 12 9 6 3 Fixed pivot QMOM N = 2 QMOM N = 3 QMOM N = 4 QMOM N = 5 0 15 30 45 60 time, min Number density (# / m 3 ) 10 12 10 8 10 4 10 0 10 12 10 8 10 4 10 0 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 m(i) / m(1) Using 2N = 8 scalars, QMOM reproduces the grid-independent moments of the direct method CFD in CRE IV: Barga, Italy June 19-24, 2005 19
Quadrature Methods Multi-variate extension is straightforward weights abscissas (k,l) th moment of CSD: But inverse problem cannot be solved on the fly! CFD in CRE IV: Barga, Italy June 19-24, 2005 20
Quadrature Methods Direct Quadrature Method of Moments (DQMOM) Weights Volume Source terms found from linear system on the fly Area CFD in CRE IV: Barga, Italy June 19-24, 2005 21
Polydisperse Gas-Solid Flow DQMOM with size and momentum of solid phase Number Mass Momentum Source terms for mass and momentum can be found from kinetic theory for gas-solid flows Reduces to two-fluid model when α =1 CFD in CRE IV: Barga, Italy June 19-24, 2005 22
Outline 1. Introduction Population Balances Coupling with CFD 2. Population Balances in CFD Population Balance Equation Direct Solvers Quadrature Methods 3. Implementation for Gas-Solid Flow Overview of MFIX Polydisperse Solids Model Application of DQMOM 4. Two Open Problems CFD in CRE IV: Barga, Italy June 19-24, 2005 23
Overview of MFIX Gas-solid multi-fluid model Momentum equations Mass & energy equations Chemical species equations Population balance equations ISAT DQMOM Kinetic theory Mass & heat transfer models Detailed chemistry Aggregation, breakage and growth CFD in CRE IV: Barga, Italy June 19-24, 2005 24
MFIX Governing Equations (I) Mass balances ( ε ρ ) + ( ε ρ u ) = t g g g g g gα n α = 1 n= 1 ( ε ρ ) + ( ε ρ u ) = t N N s sα sα sα sα sα gαn n= 1 N s M M Mass transfer from gas to solid phases Momentum balances N ( ερ g gug) + ( ερ g gugug) = σg + fgα + ερ g gg t N ( ε ρ u ) + ( ε ρ u u ) = σ -f + f + ε ρ g t α= 1 sα sα sα sα sα sα sα sα gα βα sα sα β= 1, β α g: Gas phase sα: Solid phases α=1, N Stress tensor Interaction with gas and other solid phases Body force CFD in CRE IV: Barga, Italy June 19-24, 2005 25
MFIX Governing Equations (II) Thermal energy balances ε ε ρ T ρ C Tsα + u T = q t + H H N g g gcpg + ug Tg = qg Hg α Hrg + Hwall Twall Tg t α = 1 sα sα psα sα sα sα gα rsα Conductive heat flux Chemical species balances Heat transfer between phases N ( ε ρ X ) + ( ε ρ X u ) = R M t ( ε ρ X ) + ( ε ρ X u ) = R + M t g g gn g g gn g gn gαn α = 1 sα sα sαn sα sα sαn sα sαn gαn Heat of reaction ( ) Heat lost to walls g: Gas phase sα: Solid phases α=1, N Reactions Mass transfer CFD in CRE IV: Barga, Italy June 19-24, 2005 26
Polydisperse Solids Model Population balance equation for solid phase L, u s Force acting to accelerate particles nl (, us; xt, ) t ( ) ( ) + u sn L,u s;x, t + u n L, (, ;, ) s F,u s;x t = S L us x t Joint size & velocity distribution function Aggregation, breakage and chemical reaction CFD in CRE IV: Barga, Italy June 19-24, 2005 27
Direct Quadrature Method of Moments nl (, us;,) xt + snl ( s, t) + nl (, ) (, ;, ) N s s t = L s xt t u,u;x u F,u;x S u nl (, us; xt,) = ωα(,) xtδ( L Lα(,)) xt δ( us us α(,)) xt α = 1 Integrate out solid velocity nlxt (;,) + s L nl (, t) = ( Lxt ;,) t u ;x S Use distribution function N N ( L L ) s ( L L ) ( L; x, t) t ωδ α α + u α ωδ α α = S α = 1 α = 1 N α = 1 [ δ δ ] ( L L ) a '( L L )( b L a ) = S( L; x, t) N α= 1 α α α α α α Simplify equation Moment transform k k 1 al α α + klα ( bα La α α) = Sk ( xt, ) Ax = w 1 w 2 L 1 L 2 L 3 u s1 u s2 u s3 N w3 ( ;x, ) ω ( x, ) δ - ( x, ) n L t = α t L Lα t b α = 1 { ωα + ( uαωα) = aα t ωαl α ( u ω L ) t + = b α α α α + N k k mk ( x, t) = n( L;x, t) LdL ω α L α 0 α= 1 L CFD in CRE IV: Barga, Italy June 19-24, 2005 28
Modifications to MFIX Relation between volume fractions and weights: ε = klω k v : volumetric shape factor 3 sα v α α Transport equations for volume fractions and lengths: ( ε sαρ sα) t ( ε L ρ ) sα α sα t + ( ε ρ u ) = 3k ρ L b 2k ρ L a 2 3 sα sα sα v sα α α v sα α α + ( ε L ρ u ) = 4k ρ L b 3k ρ L a 3 4 sα α sα sα v sα α α v sα α α CFD in CRE IV: Barga, Italy June 19-24, 2005 29
CFD in CRE IV: Barga, Italy June 19-24, 2005 30 DQMOM Source Terms 3 2 1 0 S S S S = Ax b 2 2 2 1 3 2 3 1 2 1 2 2 2 1 3 3 2 2 2 2 1 1 0 0 0 0 1 1 L L L L L L L L 2 1 2 1 b b a a Matrix A relates moments to weights and lengths Source term x is obtained by forcing moments to be exact
Aggregation and Breakage Death due to Birth due to a aggregation aggregation Bk (,) x t D a k (,) xt a a b b Sk(,) x t = Bk(,) x t Dk(,) x t + Bk(,) x t Dk(,) x t Birth due to breakage B b k (,) x t L Death due to breakage b Dk (,) xt a 1 + + 3 3 k /3 Bk (,) x t = n(;,) λ x t β(, u λ)( u λ ) n(;,) u x t dudλ 2 + 0 0 a k Dk (,) xt = LnLxt (;,) β(, Lλ)(;,) nλ xtdλdl b 0 0 k Bk (,) x t = L a()( λ b L λ)(;,) n λ x t dλdl a + + + + 0 0 + k Dk (,) x t = La() L n( LxtdL ;, ) 0 Apply DQMOM 1 S x t L L L ab L a N N N N N N 3 3 k/3 k ( k ) k k (, ) = ωω i j ( i + j ) βij ωω i j i βij + ωi i i ωi i i 2 i= 1 j= 1 i= 1 j= 1 i= 1 i= 1 CFD in CRE IV: Barga, Italy June 19-24, 2005 31
Aggregation and Breakage Kernels Aggregation and breakage kernels are obtained from kinetic theory Number of collisions: 1 2 3 4 θ mi + m s j 2 Nij = πωiω jσ ijg ij ( s ) u σ ij π 2mm i j 3 Aggregation kernel: Breakage kernel: β ij Nij = ψ a ωω i j a i Nij = ψ b ω i i Efficiencies (ψ a and ψ b ) depend on temperature, particle size, etc. CFD in CRE IV: Barga, Italy June 19-24, 2005 32
PSD Effect on Fluidization No aggregation and breakage Breakage dominant average size decreases, FB expands CFD in CRE IV: Barga, Italy June 19-24, 2005 Aggregation dominant average size Increases, FB defluidizes 33
Volume-Average Mean Diameter Case 1 Case 2 β = 0; a = 0 β = a = 5 3 1 10 m /s; 0.1 s 1 Case 3 N = 2 filled symbols N = 3 empty symbols N = 4 lines β = a = 1s 5 3 1 10 m /s; 1 CFD in CRE IV: Barga, Italy June 19-24, 2005 34
Volume-Average Normalized Moments N = 2 filled symbols N = 3 empty symbols N = 4 lines + N k k mk ( x, t) = n( L;x, t) L dl ω α L α 0 α = 1 CFD in CRE IV: Barga, Italy June 19-24, 2005 35
Extension to Energy/Species Balances ε ρ Thermal energy balance T C + u T = q + H H t sα sα sα psα sα sα sα gα rsα + k ρ L C c k ρ L C T a 3 3 v s α ps T, α v s α ps sα α Changes due to aggregation and breakage Multi-variate DQMOM H = ε R H rsα sα sα, C r R k c P * m sα, C = p[ ] C 2 2 CFD in CRE IV: Barga, Italy June 19-24, 2005 36 2
1. Introduction Population Balances Coupling with CFD Outline 2. Population Balances in CFD Population Balance Equation Direct Solvers Quadrature Methods 3. Implementation for Gas-Solid Flow Overview of MFIX Polydisperse Solids Model Application of DQMOM 4. Two Open Problems CFD in CRE IV: Barga, Italy June 19-24, 2005 37
Two Open Problems 1. How to extend DQMOM to systems with unknown fluxes at boundaries in phase space? Model problem: pure evaporation How can we estimate it? Estimate flux in DQMOM variables, test with exact solutions: Define vectors: Define cross product : Linear constraint: CFD in CRE IV: Barga, Italy June 19-24, 2005 38
Simple case with monotone flux (N = 2): CFD in CRE IV: Barga, Italy June 19-24, 2005 39
Harder case with multimode flux (N = 2): CFD in CRE IV: Barga, Italy June 19-24, 2005 40
Harder case with N = 3: CFD in CRE IV: Barga, Italy June 19-24, 2005 41
Two Open Problems 2. What is best choice of moments for multivariate DQMOM? Model problem: homogeneous aggregation Aggregation Aggregation + sintering Moments (mass k=1, l=0; area k=0, l=1) CFD in CRE IV: Barga, Italy June 19-24, 2005 42
Choice of moments affects the condition of matrix C B F E D A All choices yield nearly same weights and abscissas Choose moments with lowest condition number? CFD in CRE IV: Barga, Italy June 19-24, 2005 43
Another example: Williams Spray Equation CFD in CRE IV: Barga, Italy June 19-24, 2005 44
Coefficients depend on choice of 5N moments: Condition number of A depends on choice of k, l, m, p In general, A matrix will become singular if 1 < l + m + p Choose l, m, p = (0,1) and vary k to yield 5N distinct moments Number: (k, l, m, p)= 0 Mass: k =1, (l, m, p) = 0 X-Mom: k =1, l = 1 Y-Mom: k = 1, m = 1 Z-Mom: k = 1, p = 1 Is there a general method for choosing moments? CFD in CRE IV: Barga, Italy June 19-24, 2005 45