Cfdcre5 Petitti
-
Upload
dr-mohammed-azhar -
Category
Documents
-
view
217 -
download
0
Transcript of Cfdcre5 Petitti
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 1/33
1
Modelling the bubble size distribution ingas-liquid reactors
with QMOM implemented with a newcorrection algorithm
Miriam Petitti
A.Nasuti, D.L.Marchisio, M.Vanni, G. Baldi,N. Mancini, F. Podenzani, A. Bennardo
Dept. Materials Science and Chemical Engineering, Politecnico di Torino(Italy)ENI R&M, S.DonatoMilanese (Italy)
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 2/33
2
Outline
oGas-liquid systems modeling: challenges and issues
oSimulation of a gas-liquid stirred tank (air-water):oApproach
oDrag force evaluation
oPopulation Balance Model
oQMOM and correction algorithm
oResults:oGas volume fraction
oBubble Size Distribution (BSD)
oConclusion and possible next steps
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 3/33
3
Gas-liquid systems modeling: challenges and issues
o The simulation of poly-dispersed multiphase systems (i.e.,constituted by droplets, particles and bubbles) is a complexproblem
o Gas-liquid systems are the hardest to describe because fluidparticles have no fixed shape or size:
o Small bubblesà sphericalo Medium bubblesà ellipsoidal
o Big bubblesà spherical cup
o This has a tremendous impact on the interaction between theliquid (continuous phase) and the bubbles (disperse phase)
for:o Momentum phase transfer
o Mass phase transfer
o Enthalpy phase transfer
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 4/33
4
Phase coupling
Liquid Bubble – BubbleBubble
One – Way Coupling
Two – Way Coupling
Four – Way Coupling
Global gas hold-up
Drag forceVirtual mass force
Lift forceBubble breakup Collisions between bubbles:
Coalescence andmomentum exchange
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 5/33
5
BREAK-UP COALESCENCEThree conceptual steps
Bubbles come in close proximity because of
turbulent eddies
The controlling mechanism is deformationsinduced by turbulence
For gas bubbles binary breakageis the most common event
Approach: Four-way coupling
o The fluid exerts a drag force on the bubbles that is related tothe slip velocity, bubble size, turbulence intensity and bubbleconcentration (details in back-up slides)
o In addition there are the lift and virtual mass forces
LIQUID-BUBBLE FORCES
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 6/33
6
Approach
o The phases are simulated as interpenetrating continua thatobey transport equation and exchange momentum. Everyphase has its own mass and momentum conservationequation.
o Phase interaction occurs via interfacial terms (especiallydrag). Closeness of the model used for this terms to thereality determines accuracy of the model.
o Most of the interfacial terms depend on particle size andmost dispersed flows are not monodispersed, so
knowledge of representative bubble diameter is essential.
Eulerian Multifluid Model (Fluent)
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 7/33
7
Approach
o Expressions of the drag term tailored specifically forbubbles have been introduced as UDF
o A solver for the population balanceequation has beenincluded in the code to evaluate the bubble sizedistribution in every point of the system. The solver is
based on the QMOM method, which allows accurateestimation of the integral properties of the BSD withmoderate computational load (6 additional transportequations)
o Proper expressions for bubble breakage and coalescence
rates have been tested and fitted to experimental data sets
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 8/33
8
Drag force evaluation
Usually in stirred reactors BSD is included in the range of
ellipsoidal bubbles and is not so wide to cause a significantchange in the bubble terminal velocity, according toMendelsons’s law.
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 9/33
9
Drag force evaluation
The drag coefficient is evaluated from bubble terminal velocity: the d32
of the local BSD is considered and a unique value for bubble terminal
velocity is used in the calculation.
The drag force is evaluated on the basis of the local d32 and the local slipvelocity between the continuous and dispersed phases.
2
32
3
4
∞
−=
U
dC
l
gl
D ρ
ρ ρ
hold-up < 5%: about 13 cm/s (turbulence)
hold-up > 5% :about 8.5 cm/s (turbulence + effect near bubbles)
The damping effect of turbulence and the effect of high gas hold-upare considered for the terminal velocity evaluation:
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 10/33
10
Population BalanceEquation
Its solution describes the evolution of bubbles size distribution:
NumberDensityFunction
( )( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( )( ) ( ) ( ) ( ) ( )1 3 1 3
3 3 3 3
0 0
, ,, , , , , , ,
1, , , , , , , , , ,
2
d
d
d
d d d d d d d d d
L
L
d d d d d d d d d d d d
n L x tU n L x t g b L n x t d g L n L x tt
h L n L x t n x t d n L x t h L n x t d
λ λ λ λ
λ λ λ λ λ λ λ λ
∞
∞
∂ + ∇ ⋅ = − + ∂
+ − − −
∫
∫ ∫
Bb Db
Bc Dc
birth due to breakage death due to breakage
birth due to coalescence death due to coalescence
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 11/33
11
Kernels (or frequencies) quantifies the rate with which bubblesCOALESCENCE and BREAK.
Several expressions are available in the literature but in this work thefollowing have been used (Laakkonen et al., 2006):
Coalescence and breakup rates
( ) 1/3
1 2 32 /3 5 /3 1/ 3 4 /3erfc c
d
c d c d d
g L C C CL L
µ σ ε
ρ ε ρ ρ ε = +
BREAKUP KERNEL
( ) ( ) ( ) ( )1/ 221/ 3 2 / 3 2 / 3
7, ,d d d d d d d dh L C L L Lλ ε λ λ η λ = + +
( )
4
8 2, exp c c d d
d d
d d
LL C
L
µ ρ ε λ η λ
σ λ
= − +
C7 =0.88(from theory)
COALESCENCE KERNEL
Binary breakage, C1 from fitting
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 12/33
12
The PBE is solved in terms of the first k = 2N moments of the Number Density
Function (NDF) n(L):
0,1,2,3,4,5k=( ) ( )ˆkd k d d k d km u m S
t ρ ρ ρ ∂ +∇ =∂
The closure on the moments source terms Sk , concerning bubble coalescence and
breakup, is obtained by resorting to a quadratureapproximation :
QuadratureMethod of Moments
( )∫ +∞
≡0
dLLLnm kkwhere the moments mk refer to bubble size L:
L2
n(L)
LL1
w1
w2
L3
w3
N =3( ) ( ) ( )i
N
ii Lf wdLLf Ln ∑∫
=
+∞
≈10
weights nodes
the first 6 momentsare tracked
( ) ( ) ( ) ( ) ( )/3
( ) 3 3
1 1 1 1 1 1
1, ,
2
d d d d d dN N N N N Nk
k k kkk i i i i i i i j i j i j i i i j j
i i i j i j
dmS g L wb L g L w w w L L h L L L w h L L w
dt = = = = = =
= = − + + −∑ ∑ ∑ ∑ ∑ ∑
Bbk Bck DckDbk
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 13/33
13
QMOM and Moments corruption
The population balance is coupled with the multi-fluid model through the
moment transport equations. The moments can be considered as scalars that are
convected, mixed and diffused. However they are linked by mathematicalrelationships that assure the physical existence of the underlying distribution.
The independent transport of the moments does not preserve these relationships,
that are altered by the discretization schemes used by the CFD code, and can
therefore create a set of corrupted (and not valid) moments.
Non - valid moment set
Residuals
Instability problems
Negative moments, relationships altered
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 14/33
14
QMOM and Moments validity testThe validity of the moment set is checked by asserting the positivity of thedeterminants of Hankel-Hadamard (Shohat e Tamarkin, 1943):
1
1 2 1
,
1 2
...
...0
...
n n n l
n n n l
n l
n l n l n l
m m mm m m
m m m
+ +
+ + + +
+ + + +
∆ = ≥M M M M
for n =0,1 and l ≥ 0
These conditions for the first few moments {m0 , m1 , m2 , m3 } are equivalent to the
convexity check of the curve ln(mk ) in function of k:
2
2 1 0k k kmm m− −− ≥
k
ln (mk)
k
ln (mk)
valid non-valid
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 15/33
15
QMOM and Moments correction
Adjust that moment whichafter adjustment maximizessmoothness through minimizationof a vector related to third orderdifferences
Initial moments set
Form second orderdifferences in ln(mk)
Satisfies convexity
Form third order differences inln(mk): verctor a
Determine k* for which
vector a is minimized
Adjust mk* to get newmoment sequence
check m4 and m5
Stop: momentsok
Minimum Square
Gradient Algorithm
(McGraw 2006)
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 16/33
16
QMOM and Moments correction1) In certain cases the iterative algorithm enters an infinite loop and
does not manage to correct the moment sequence.
In these cases the moment set is re-built through two log-normaldistributions whose parameters are obtained fromthe first 4 moments
of the original set that must be corrected (Wright 2007)
NDF(1) from m0 , m2 , m3 NDF(2) from m0 , m1 , m3
2 2
exp
2k T
km N k
σ µ
= +
( )
( )2
2
lnexp
2
2L T
L
n L N
L
µ
σ
σ π
− − = ⋅NDF: moments:
Momentsevaluated as averages of the momentsof the two distributions
2) Control on gas volume fraction, particularly useful at higher gassing
rates: if gas vof <10-3 sources =0; mk=0; d32=dbinlet
Without this change at higher gassing rates it is observed the formationof zones with m5 < 0, tending to spread in neighbouring zones
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 17/33
17
Effects of moments correction
Negativevalues! A
0
≅ -10 4
B103
0
103
0
m2 without correction m2 with correction
ResidualsNo instabilityproblemswhencorrectionapplied,alsowith highunder-relaxationfactors
Residuals
Instabilityproblems
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 18/33
18
Effects of moments correction
d32
withoutcorrection
d32
with correction
The correction algorithm does not alter the physical behaviour of the system,
the bubblesizedistribution
“Lucky” simulation with lownumerical diffusion for the moments
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 19/33
19
Standard Rushton turbine
Baffles and blades width: 1.03 cmShaft diameter: 3.3 cm
Disk diameter: 13.6 cm
Turbine diameter: 21cm
Disk width: 1.06 cm
Sparger position: z =-10.5 cm; d =3.3 cm
Sparger width d =15 mm
Reactor volume: 194 liter
Porous sparger
Reactor configuration - 1
rotatingfluid
baffles
shaft
impeller
spargerComputational domain on half reactorà Ncells: 226776
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 20/33
20
Ring sparger (12 holes 2 mm of diameter)
Computational domain on half reactorà Ncells: 226776
Standard Rushton turbine
Baffles and blades width: 1.03 cm
Shaft diameter: 3.3 cm
Disk diameter: 13.6 cm
Turbine diameter: 21cm
Disk width: 1.06 cm
Sparger position: z =-10.5 cm; d =3.3 cm
Sparger width d =15 mm
Reactor configuration - 2
Reactor volume: 194 liter
sparger
zona rotantebaffles
albero
impeller
sparger
zona rotantebafflesbaffles
albero
impeller
rotatingfluid
shaft
impeller
sparger
baffles
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 21/33
21
Results-fluid dynamics
The fluid dynamics predicted by the model was also tested for astandard geometry stirred reactor of 15.4 liter, agitated by a Rushtonturbine, in terms of:
-Velocity profiles
-Fluid dynamics regime transitions
-Gas cavity structure transitions
- Power dissipated
The agreement with experimental data and/orempirical correlations is good
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 22/33
22
Results – gas distribution
Reactorconfiguration 1
GAS VOLUME FRACTION
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 23/33
23
0.00
0.02
0.04
0.06
0.08
0 0.4 0.8 1.2gas flow rate, vvm
g l o b a l g a s h o l d - u p
Results – Gas hold-up
N = 390 rpmReactor configuration 2
T R A S I T I O N A L
R E G I M E Experimental
data
Simulation
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 24/33
24
8 different operating conditions for reactor configuration 1 and 7different conditions for reactor configuration 2 were considered
resulting in the following best fit values:C1=6 (from fitting) and C7 =0.88 (from theory)
Results Bubble size distribution
G a s f l o w r a t e , v v m
Stirring rate, RPM
Reactorconfiguration 1
Reactorconfiguration 2
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 25/33
25
0
1
2
3
4
0 2 4 6 8
R2
2.90
2.77R4
3.25
3.19 R8
2.43
2.90R9
2.23
2.71 R12
3.34
3.30
Simulation resultsExperimental data
Reactor configuration 1N =250 RPM; 0.093 vvm
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
B S D
Mean bubblesize, mm
d, mm
R2 R4
R8
R9
R12
Results – Bubble size distribution
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 26/33
26
R2_A
2.24
2.86
R2_B
2.90
2.77
R2_C
2.252.74
R2_D
2.03
2.47
R2_E
2.92
3.37
Simulation results
Experimental data
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
0
1
2
3
4
0 2 4 6 8
R2_E R2_D
R2_C
R2_B
R2_A
B u b b l e v o l u m
e
d e n s i t y
d, mm
d32, mm
Reactor configuration 1 – N =250 RPM; 0.093 vvm
Results – Bubble size distribution
l bbl i di ib i
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 27/33
27
Results – Bubble size distribution
N =250 RPM; 0.052 vvmN =157 RPM; 0.052 vvm
R2
R8R9
R4
R12
R2
R8R9
R4
R12
Reactor configuration 1
N =250 RPM; 0.072 vvm
ExperimentalSimulation (45° )
Simulation (135°)
Simulation (45° )
Simulation (135°)
R lt B bbl i di t ib ti
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 28/33
28
Results – Bubble size distributionReactor configuration 1
Experimental
Simulation
N =157 RPM; 0.052 vvm N =250 RPM; 0.052 vvm
N =250 RPM; 0.072 vvm
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 29/33
29
Results – Bubble size distribution
experimental
simulation
experimental
simulation
Reactor configuration 2 – N =390 RPM; 0.7 vvm
measurement points
d 3 2 , m
m
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 30/33
30
Reactor configuration 2 – N =390 RPM
Results – Bubble size distribution
0.000
0.002
0.004
0.006
0.008
0 0.2 0.4 0.6 0.8 1 1.2
gas flow rate, vvm
m e a n b u b b l e
s i z e , m
Experimental data (derivedthrough emp. correlation)
Simulation
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 31/33
31
• Some solid guidelines for the simulation of gas-liquid stirrer tanks inrealistic conditions have been identified and validated.
•Great attention must be paid to the calculation of the drag force (thiscan be calculated via the bubble terminal velocity taking into account theeffect of turbulence and of the other bubbles for very high hold-ups)
• The algorithm for the calculation of the bubble size distribution (QMOM+correction algorithm) was found to be stable under very different
operating conditions and reactor configurations.
•The coalescence and breakage kernels and the drag law are able todescribe with satisfactory accuracy both the global hold up, the bubblesize distribution and the mean bubble size.
•The simulation settingscan be used to describe the behavior of the
stirrer tank under different operating conditions ranging from very lowgas hold-up (0.5 %) to high gas hold-up (7 % for the air-water system)
Conclusions
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 32/33
32
n Application to mass transfer in gas-liquid systems
Mass transfer in gas-liquid systems is very sensitive to the bubble sizedistribution, because of the effect of interfacial area and because thetransfer coefficient depends on gas-liquid slip velocity (which in turndepends on bubble size). An important point is that there may be adifference between the size range of bubbles that control fluid dynamics(usually the larger ones) and those that determine interfacial area (the
smaller ones). This point could be captured well by proper use of thepopulation balance method.
n Gas-liquid chemical reactions
Introduction of chemical reaction paths and interfacial mass transfermodels for process reactions in multicomponent two-phase systems.Probable need to consider heat generation/transfer effects as well.
Possible next steps
7/30/2019 Cfdcre5 Petitti
http://slidepdf.com/reader/full/cfdcre5-petitti 33/33
33
Thankyouverymuch for your
kindattention