Multiphase Flow Modeling - University of Notre Damegtryggva/CFD-Course2010/2010... · ·...
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Computational Fluid Dynamics I Multiphase Flow Modeling Grétar Tryggvason Spring 2010 http://users.wpi.edu/~gretar/me612.html Computational Fluid Dynamics I Spray drying Pollution control Pneumatic transport Slurry transport Fluidized beds Spray forming Plasma spray coating Abrasive water jet cutting Pulverized coal fired furnaces Solid propellant rockets Fire suppression and control Examples: Computational Fluid Dynamics I Single component Multicomponent Single water flow air flow phase Nitrogen flow emulsions Multiphase Steam-water flow air-water flow Freon-Freon slurry flow vapor flow Computational Fluid Dynamics I Disperse flow Solid-liquid: Slurries, quicksand, sediment transport Solid-air: dust, fluidized bed, erosion Liquid-air: sprays, rain Air-liquid: bubbly flows Computational Fluid Dynamics I Mixed Stratified Dispersed Slugs Flow in pipes Computational Fluid Dynamics I This figure shows schematically one of several different configurations of a circulating fluidized bed loop used in engineering practice. The particles flow downward through the aerated “standpipe”, and enter the bottom of a fast fluidized bed “riser”. The particles are centrifugally separated from the gas in a train of “cyclones”. In this diagram, the particles separated in the primary cyclone are returned to the standpipe while the fate of the particles removed from the secondary cyclone is not shown. From: Computational Methods for Multiphase Flow, Edited by A.Prosperetti and G.Tryggvason
Transcript of Multiphase Flow Modeling - University of Notre Damegtryggva/CFD-Course2010/2010... · ·...
Computational Fluid Dynamics I
Multiphase Flow Modeling!
Grétar Tryggvason !Spring 2010!
http://users.wpi.edu/~gretar/me612.html!Computational Fluid Dynamics I
Spray drying!Pollution control!Pneumatic transport!Slurry transport!Fluidized beds!Spray forming!Plasma spray coating!Abrasive water jet cutting!Pulverized coal fired furnaces!Solid propellant rockets!Fire suppression and control!
Examples:!
Computational Fluid Dynamics I
Single component Multicomponent!
Single water flow air flow!phase Nitrogen flow emulsions!
Multiphase Steam-water flow air-water flow! Freon-Freon slurry flow! vapor flow!
Computational Fluid Dynamics I
Disperse flow!
Solid-liquid: Slurries, quicksand, sediment transport!
Solid-air: dust, fluidized bed, erosion!
Liquid-air: sprays, rain!
Air-liquid: bubbly flows!
Computational Fluid Dynamics I
Mixed!
Stratified!
Dispersed!
Slugs!
Flow in pipes!
Computational Fluid Dynamics I
This figure shows schematically one of several different configurations of a circulating fluidized bed loop used in engineering practice. The particles flow downward through the aerated “standpipe”, and enter the bottom of a fast fluidized bed “riser”. The particles are centrifugally separated from the gas in a train of “cyclones”. In this diagram, the particles separated in the primary cyclone are returned to the standpipe while the fate of the particles removed from the secondary cyclone is not shown. !
From: Computational Methods for Multiphase Flow, Edited by A.Prosperetti and G.Tryggvason!
Computational Fluid Dynamics I
Need model equations to predict flow rates, pressure drop, slip velocities, and void fraction!
Mixture models: one averaged phase!
Two-fluid models: two interpenetrating continuum!
Computational Fluid Dynamics I
Euler/Euler approach!All phases are treated as interpenetrating continuum!The dispersed phase is averaged over each control volume!Each phase is governed by similar conservation equations!Modeling is needed for!
!interaction between the phases!!turbulent dispersion of particles!!collision of particles with walls!
A size distribution requires the solution of several sets of conservation equations !Numerical diffusion at phase boundaries may result in errors!This approach is best suited for high volume fraction of the dispersed phase!
Computational Fluid Dynamics I
Euler/Lagrange approach!The fluid flow is found by solving the Reynolds-averaged Navier-Stokes equations with a turbulence model.!The dispersed phase is simulated by tracking a large number of representative particles.!A statistically reliable average behavior of the dispersed phase requires a large number of particles!The point particles must be much smaller than the grid spacing!Modeling is needed for!
!collision of particles with walls!!particle/particle collisions and agglomeration!!droplet/bubble coalescence and breakup!
A high particle concentration may cause convergence problems!
Computational Fluid Dynamics I
Although commercial codes will let you model relatively complex multiphase flows, it is really only in the limit of dispersed and dilute flows where we can expect reasonable accuracy!
To treat systems like this, the two-fluid model is usually used. The continuous phase is almost always used in an Eularian way where the continuity, momentum, and energy equations are solved on a fixed grid. !
Computational Fluid Dynamics I
The void fraction εp describes how much of the region is occupied by phase p.!
Obviously,!
Similarly, the effective density of phase p is!
�
εp =1∑
�
ˆ ρ = εpρp
Computational Fluid Dynamics I
While the averaging is similar to turbulent flows, here we must account for the different phases!
�
α p =1 inside phase p0 otherwise
⎧ ⎨ ⎩
�
εp = 1V
α p dvV∫
The void fraction is found by!
Computational Fluid Dynamics I
�
ˆ φ p = 1εpV
α pφ dvV∫Averages are found by!
Where the volume V goes to zero in some way!
The velocity is found by!
�
ˆ u p = 1εpV
upα p dvV∫
The averages can also be interpreted as time or ensemble averages!
Computational Fluid Dynamics I
�
α pρp dvV∫The total mass of phase p in a control volume is!
And the mass conservation equation can be averaged to yield!
�
∂∂tεpρp + ∇ ⋅ εpρpup( ) = ˙ m p
Here!
Since a mass that leaves one phase must add to another phase!�
˙ m p = 0∑
Computational Fluid Dynamics I
The conservation of momentum equation becomes!
�
∂∂t
εpρpup( ) + ∇ ⋅ εpρpupup( ) = −εp∇pp
+∇ ⋅ εpµpDp( ) + εpρpg + ∇ ⋅ εpρp < uu >( ) + Fint
In addition to the Reynolds stresses, it is now necessary to model the interfacial forces. The kinetic energy is often neglected, even though the fluctuations are non-zero in laminar flow!
interfacial forces!
Reynolds stresses!
Computational Fluid Dynamics I
If there is no mass transfer m=0 and F is the force that one phase exerts on the other!
�
Fp = 0∑
In principle the conservation equations can be solved for both the continuous and the dispersed phase (Euler/Euler approach).!
Computational Fluid Dynamics I
However, the dispersed phase is not all that continuous and an other approach is to explicitly tract (representative) particles by solving!
�
dudt
= Fp
If the particles have no influence on the fluid: one way coupling!If the particles exert a force on the fluid: Two way coupling!
Computational Fluid Dynamics I
Usually the force is written:!
�
Fp = kD u−up( ) + g ρD − ρρ
+ FotherOther forces due to added mass, pressure, lift, etc!
Gravity!buoyancy!
Drag force!
Computational Fluid Dynamics I
For the drag:!
�
Fp = k ur − up( )where!
�
k = 34CDεrρq
ur − up( )dr
�
CD = CD Re( )is obtained from experimental correlations, such as!
and!
�
CD = 24Re
1+ 0.15Re0.687( ) Re <103
For solid particles! Re based on slip velocity!
Computational Fluid Dynamics I
�
dupdt
= Fp
�
dx pdt
= up
Find particle trajectories by solving!
The force allows us to find the particle velocity by integrating:!
Computational Fluid Dynamics I
Turbulent flow!
Set particle velocity!
�
up + u'Random velocity fluctuations from !
This allows particles to cross streamlines as they do in turbulent flow!
Particles can accumulate here!�
kp = u'u'∑
Computational Fluid Dynamics I
Usually a large number of particles is used to get a well converged particle distribution!
Notice that almost all the interactions (particles/flow) particle/particle, particle/wall) are highly empirical!
Computational Fluid Dynamics I
Similar approach can be taken for the temperature and the size of a particle (heat and mass transfer)!
�
mpcpdTpdt
= hAp (Tf −Tp ) + εpApσ (T∞4 −Tp
4 )
�
dmp
dt= ˙ m p
For dilute flows this does work reasonably well — if the initial or inlet conditions are knows!
Mass transfer due to evaporation, for example!
Computational Fluid Dynamics I
Turbulent in the continuous phase!
�
DkDt
=+ <U ⋅Fp >
�
<U ⋅Fp >= τρ
< uf (uf − up ) >= τρ(< uf uf > − < uf up >)
This term can lead to both reduction and increase in the turbulence in the liquid!
Either ignore the contributions of the dispersed phase when computing the flow, or use a k-ε model!Solve for k and ε in the liquid and kp. Called k ε kp models.! The k equation is!
Computational Fluid Dynamics I
The full two-fluid model suffers from several problems, in addition to uncertainties about the various closure assumptions:!
The major one is that the full equations are ill-posed and one cannot expect a fully converged solution under grid refinement!
One possible way around this is to use the “drift flux approximation” where the particle velocity is assumed to be a given function of the local conditions.!
Computational Fluid Dynamics I
Modeling of Laminar Flow in a Vertical Channel!
Computational Fluid Dynamics I
Flow! Gravity!
Bubbly flow in a vertical channel!
Need to know!• The bubble distribution!• The velocity profile and the flow rate!
x
y
�
∂∂y
= 0
Assume that the flow is independent of y, so!
�
∂pl∂y is given!
but!
S.P. Antal, R.T. Lahey and J.E. Flaherty. Intʼl. J. Multiphase Flow 17 (1991), 635-652.!
Computational Fluid Dynamics I
�
ε ∂ε∂x
Ur2
51−ε( ) = −εCLUr
∂ul∂x
−ε Cw1 + Cw2Rb
s⎛ ⎝ ⎜
⎞ ⎠ ⎟ Ur
2
Rb�
(1−ε) dpldy
+ 1−ε( )ρlgy = (1−ε)µl∂2vl∂x 2
+ 38
εRb
CDρlUr Ur
�
ε x( ) = 1L
εdx0
L
∫ , ul 0( ) = ul H( ) = 0
Simple two-fluid model for laminar multiphase flow
Comparison with a two-fluid model!
�
ε dpdy
+ ερggy = − 38
εRb
CDρlUr Ur
�
dpgdy
= dpldy
= dpdy
Bubble vertical momentum!Liquid vertical momentum!Bubble horizontal momentum!
Lift! Wall repulsion!(away from wall or zero)!
�
CD = 24Re1+ 0.1Re0.75( )
�
Re=2Rbρl Ur
µm
�
µm = µ l
1−ε
Computational Fluid Dynamics I Comparison with a two-fluid model!
Comparison with experimental results. Graph from: S.P. Antal, R.T. Lahey and J.E. Flaherty. Intʼl. J. Multiphase Flow 17 (1991), 635-652.!
Computational Fluid Dynamics I
For more information about computing multiphase flow, see:!
Modeling of multiphase flows is still a very immature area. Interpret the results with care!!
Computational Fluid Dynamics I
Combustion Modeling!
http://users.wpi.edu/~gretar/me612.html!
Grétar Tryggvason !Spring 2010!
Computational Fluid Dynamics I
Gas combustion!!Gasoline engines, !!gas burners!
Spray combustion!!Jet engines, !!diesel engines!
Combustion of solids!!coal, !!wood, !!polymers!
Examples!
Computational Fluid Dynamics I
Diffusion flames!Most burners, candle!
Flame stays at the boundary between the fuel and the oxidizer!
Premixed flames!Some burners!Hazards!IC engines!
The flame separates unburned and burned mixture of fuel and oxidizer!
Computational Fluid Dynamics I
Diffusion Flames!
Computational Fluid Dynamics I
Diffusion flames!
Fuel, CH4 (methane), for example!
Oxidizer, O2!
The thickness of the flame depends on the ratio of the reaction rates to the diffusion times (Dahmköler number)!
Computational Fluid Dynamics I
Diffusion flames!
CH4! O2!
Interim species!
The thickness of the flame depends on the ratio of the reaction rates to the diffusion times (Dahmköler number)!
CH4! O2!
Interim species!Fast Reaction!
Slow Reaction!
Computational Fluid Dynamics I
In turbulent flows the flame sheet usually folds in complex ways!
P = 4, inj =
0.3
P = 4, inj =
0.1
P = 4, inj =
0.5
Computational Fluid Dynamics I
In general, the combustion is a very complex process involving O(100) species and reaction rates. These are reasonably well known for a number of reaction, but still an open research field in general.!
For combustion of natural gas, GRIMech, for example!
Computational Fluid Dynamics I
For the full problem it is necessary to track the mass fraction of every species along with the momentum, mass, and energy conservation equations!
�
∂∂t
ρmi + ∇ ⋅uρmi = ∇ ⋅ J + R
�
∂∂tci + ∇ ⋅uci = R
�
R = TnAE−Ek /RT Πcii
Arrenius reaction rates!
Only a handful of computations of the full problem have been done so far!
Computational Fluid Dynamics I
For realistic situations, the problem must be simplified!
Diffusion Flames!• Use one-step (overall) reaction rates ! (if Da -> ∞, Burke Shuman limits)!• Use a reduced set of chemical reactions!
Can work very well for laminar flames. For turbulent flows the reaction rates have to be modified to account for stretching and folding of flame sheets.!
Computational Fluid Dynamics I
This is a conserved variable that is simply advected with the flow!
Given f, we can find each species fraction by!
For a simple one step reaction it can be shown that it is sufficient to follow one variable, called the mixture fraction!
�
φi = φi( f )�
∂∂t(ρf ) + ∇ ⋅ (ρfu) = ∇ ⋅D∇f�
f =mf
mf + mo
Computational Fluid Dynamics I
The Φ function can be constructed either assuming infinitely fast reactions (flame sheets) or equilibrium. The library is constructed once only. If the system is non-adiabatic, Φ is a function of the enthalpy also!
• The chemical system must be a diffusion flame and consists of a fuel and an oxidizer!• The Lewis number must be unity (all diffusion coefficients equal)!• Only one fuel type (can be a mixture)!• Only one type of oxidizer (can be a mixture)!• Incompressible turbulent flow!
Computational Fluid Dynamics I
For turbulent flows we solve for both f and the fluctuations of f and use those to determine the species fraction!
Determines how f is distributed and therefore how the species are distributed!
In the actual code a 2D look up table is first constructed, given the shape of the pdf!
�
f '( )2
smaller!
pdf!
f!
�
φ = φ f , f '( )2⎛ ⎝ ⎜ ⎞
⎠ ⎟
Computational Fluid Dynamics I
pdf!
f!
f!
t!
Constructing the pdf from measurements!
Computational Fluid Dynamics I
Premixed Flames!
Computational Fluid Dynamics I
Premixed flames!
�
∂∂t
ρG + u ⋅ ∇ρG =Uf ∇ρGMotion due to fluid flow!
Motion due to burning!
For turbulent flow the flame speed is different from laminar flow due to wrinkling!
The flame speed is found experimentally or by detailed computations !
�
Uf
Flame is marked by G=0!
G<0!
G<0!
Computational Fluid Dynamics I
Other combustion models!
Droplets! Solid particles!
Evaporation!
Burning (usually)!
Gasification!
Burning!
Computational Fluid Dynamics I
As for multiphase flows, many issues are still unresolved in modeling of combustion and these models should be applied with care!
