CFD Models for Polydisperse Solids Based on the Direct

PowerPoint PresentationCFD in CRE IV: Barga, Italy June 19-24, 2005 1
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 2
Outline 1. Introduction
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
Population Balances
• Number density function (NDF)
particle surface area time
spatial locationparticle volume (mass)
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)
Population Balances
• Moments of number density function
Solving for moments in CFD makes the problem tractable due to smaller number of scalars
Multi-fluid model solves for moments from kinetic theory
Choice of k and l depends on what can be measured
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
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
length
Coupling with CFD
Particle diameter
Kinematic viscosity
If St > 0.14, particle velocities must be found from a separate momentum equation in the CFD simulation
Coupling with CFD
Residence time
Recirculation time
Kolmogorov timescale
CFD simulations w/o PBE can be used to determine timescales for a particular piece of equipment
Outline
Population Balance Equation
Advection
Diffusion
In granular flow, particle-particle collisions must be added
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
102
104
106
108
1010
1012
1014
Direct Solvers
Accurate predictions for higher-order moments require finer grid (range: 25-120 bins)
Direct Solvers
• Difficulties encountered when coupled with CFD – n(v; x, t) represented by N scalars ni(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 ND –
accounting for “morphology” changes will be intractable
Need methods that accurately predict experimentally observable moments, but at low
computational cost
Quadrature Methods
weights abscissas
Quadrature Methods
Quadrature Methods
Advection
Diffusion
Quadrature Methods • Comparison with direct method
0 15 30 45 60
3
6
9
12
Fixed pivot QMOM N = 2 QMOM N = 3 QMOM N = 4 QMOM N = 5
time, min
⟨R g⟩
g, P
Using 2N = 8 scalars, QMOM reproduces the grid-independent moments of the direct method
100
104
108
1012
Quadrature Methods • Multi-variate extension is straightforward
weights abscissas
Quadrature Methods • Direct Quadrature Method of Moments (DQMOM)
Weights
Volume
Area
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
Outline
Overview of MFIX
Gas-solid multi-fluid model
MFIX Governing Equations (I) • Mass balances
• Momentum balances
1 1
M t α
s s s s s g n n
=
∂ +∇ ⋅ =
∂ ∑u
1 ( ) ( )
N
g g g g g g g g g g gt α α
ε ρ ε ρ ε ρ =
∂ +∇⋅ =∇⋅ + +
1, ( ) ( )
N
s s s s s s s s g s st α α α α α α α α α βα α α β β α
ε ρ ε ρ ε ρ = ≠
∂ +∇⋅ =∇⋅ + +
∂ ∑u u u -f f gσ
Stress tensor Body force Interaction with gas and other solid phases
g: Gas phase sα: Solid phases α=1, N
Mass transfer from gas to solid phases
MFIX Governing Equations (II) • Thermal energy balances
• Chemical species balances
1
( ) N
g g g pg g g g g rg wall wall g
T C T H H H T T
t α α
TC T H H t α
α α α α α α α αε ρ ∂ + ⋅∇ = −∇⋅ + − ∂ u q
Conductive heat flux
1
( ) ( ) N
g g gn g g gn g gn g nX X R M t α
α
∂ ∑u
+∇ ⋅ = + ∂
Polydisperse Solids Model
L, us Force acting to accelerate particles
( ) ( )( , ; , ) , , ( , ; , ) s
s s s s s
n L x t n L t n L t L x t t
∂ +∇⋅ +∇ ⋅ = ∂ u
Aggregation, breakage and chemical reaction
Joint size & velocity distribution function
Direct Quadrature Method of Moments
1
( , ; , ) ( , ) ( ( , )) ( ( , )) N
s s sn L x t x t L L x t x tα α α α
ω δ δ =
s s s s s
n L x t n L t n L t L x t t
∂ +∇⋅ +∇ ⋅ = ∂ u
Integrate out solid velocityIntegrate out solid velocity
( )( ; , ) , ( ;| , )s n L x t n L t L x t
t L∂ +∇⋅ = ∂
ω δ =
1 1
( ) ( ) ( ; , ) N N
sL L L L L x t t α α α α α
α α
1 1
( ) ( ) ( ; , ) N N
sL L L L L x t t α α α α α
α α
1
1 ( ) ( , )
N k k
−
=
+ − = ∑
( ) ( ) 10
, , N
k k km t n L t L dL Lα α
α
ω +∞
α α α α ω ω∂
+∇⋅ = ∂
u [ ]
1
( ) '( )( ) ( ; , ) N
L L a L L b L a S L x tα α α α α α α
δ δ =
Modifications to MFIX
• Relation between volume fractions and weights:
3 s vk Lα α αε ω= kv: volumetric shape factor
• Transport equations for volume fractions and lengths:
2 3( ) ( ) 3 2s s s s s v s v sk L b k L a
t α α
+ ⋅ = − ∂
u∇
3 4( ) ( ) 4 3s s s s s v s v s
L L k L b k L a t
α α α α α α α α α α α α α
ε ρ ε ρ ρ ρ∂ +∇ ⋅ = −
∂ u
DQMOM Source Terms

−− −−
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
3 3 /3
a k k
a k k
b k k
a k k
S x t B x t D x t B x t D x t
B x t n x t u u n u x t dud
D x t L n L x t L n x t d dL
B x t L a b L n x t d dL
D x t L a L n
λ β λ λ λ
β λ λ λ
λ λ λ λ
+∞
∫
( , ) b kD x t
Death due to breakage
Birth due to breakage
( , ) a kB x t ( , )
a kD x t
( , ) b kB x t
)
N N N N N N k k k
k i j i j i j i ij i i i i i j i j i i
= = = = = =
= + − + −∑∑ ∑∑ ∑ ∑
Aggregation and Breakage Kernels • Aggregation and breakage kernels are obtained from kinetic
theory Number of collisions:
Efficiencies (ψa and ψb) depend on temperature, particle size, etc.
1 2
i js ij i j ij ij s
ij i j
Aggregation dominant average size Increases, FB defluidizes
Breakage dominant average size decreases, FB expands
PSD Effect on Fluidization
No aggregation and breakage
Volume-Average Mean Diameter Case 1
0; 0aβ = =
−
= ×
= NN = 2 filled symbols= 2 filled symbols NN = 3 empty symbols= 3 empty symbols NN = 4 lines= 4 lines
Volume-Average Normalized Moments
( ) ( ) 10
, , N
k k km t n L t L dL Lα α
α
ω +∞
=
= ≈∑∫x ;xNN = 2 filled symbols= 2 filled symbols NN = 3 empty symbols= 3 empty symbols NN = 4 lines= 4 lines
Extension to Energy/Species Balances
v s ps T v s ps s
TC T H H t
k L C c k L C T a
α α α α α α α α α
α α α α
Multi-variate DQMOM
H R H
α
ε =
Outline
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:
Simple case with monotone flux (N = 2):
Harder case with multimode flux (N = 2):
Harder case with N = 3:
Two Open Problems 2. What is “best” choice of moments for multivariate
DQMOM?
Aggregation + sintering
Moments (mass k=1, l=0; area k=0, l=1)
• Choice of moments affects the condition of matrix
A
B
F
C
All choices yield nearly same weights and abscissas Choose moments with lowest condition number?
Another example: Williams’ Spray Equation
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 Models for Polydisperse Solids Based on the Direct Quadrature Method of Moments
Outline
Modifications to MFIX
DQMOM Source Terms
Coefficients depend on choice of 5N moments:
Coupling with CFD
Coupling with CFD

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CFD Models for Polydisperse Solids Based on the Direct