gas absorption into a spherical liquid droplet: numerical ... filegas absorption into a spherical...
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Gas absorption into a spherical liquid droplet: numerical
and theoretical study C. Wylock, P. Colinet, B. Haut
22th International Symposium on Chemical Reaction Engineering
Maastricht, The Netherlands
September, the 3rd, 2012 ISCRE 22
Framework
Spray absorbers
• Absorption of gas materials by reactive liquid droplets
• Important mass transfer devices for several industrial applications, such as air pollution control (e.g. scrubbing of SO2 by limestone slurry)
• Nowadays, efficient design and optimization require deep understanding of phenomena taking place during the gas-droplet mass transfer BUT complex problem
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Framework
Multiscale problem
Several coupled phenomena in an absorbing droplet
• Interactions gaseous flow – liquid flow
• Interactions flow – mass transfer
• If enhanced interactions reactions - mass transfer
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Spray absorber Droplet cloud Droplet Gas-liquid interface
This work
Framework
Development of mass transfer model
• Detailed modelling of phenomena and their interactions
fundamental interest for spray absorber modelling
• Looking for «simplicity» (simple but not simplistic) of the mass transfer model in order to be applicable in an absorber model (e.g. for the scaling)
• Approach :
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"Complex" model
Analysis "Simplified"
model Comparison
Studied case
Liquid water droplet in free fall in gaseous air
• Non-deformable sphere
• Non-oscillating interface
• Incompressible laminar flows
• At terminal velocity (stationary flow)
A gaseous component A is transferred from the gas into
the liquid droplet
Possibly irreversible chemical reaction with a
component B in the liquid phase following:
A + B C
Problem studied by Direct Numerical Simulation (DNS)
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dd ≤ 1mm ; Re ≤ 250
Outline
Mathematical modelling
• Domain and equations
Physical absorption
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Outline
Mathematical modelling
• Domain and equations
Physical absorption
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Mathematical modelling
Computational domain
• 2-D axisymmetric domain
• Meshing with boundary layer meshes
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Mathematical modelling
Equations in the gaseous phase
• Navier-Stokes and continuity
• Concentration transport
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with
Mathematical modelling
Equations in the liquid phase
• Navier-Stokes and continuity
• Concentration transport
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Physical absorption : Ha=0
with
Mathematical modelling
Conditions at the gas-liquid interface
• Navier-Stokes and continuity
• Concentration transport
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Mathematical modelling
Numerical resolution
• Finite element resolution with COMSOL Multiphysics 3.4
Simulations realized for:
• Physical absorption : various flow regimes <-> various Re
• Absorption coupled with a reaction : various flow and chemical regimes <-> various Re and Ha
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Mathematical modelling
Post-processing: study of time evolution of
• Saturation
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Physical absorption
Absorption coupled with a chemical reaction
* see our paper doi/10.1016/j.cej.2012.07.085
*
*
Mathematical modelling
Post-processing: study of time evolution of
• Saturation
• Time-averaged Sherwood number
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* see our paper doi/10.1016/j.cej.2012.07.085
*
If physical absorption:
Outline
Mathematical modelling
• Domain and equations
Physical absorption (Ha=0)
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Physical absorption: simulation results
Time evolution of
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Saturation Time-averaged Sherwood number
Analysis of time evolution of concentration fields
• Re = 0.1
Similar to a pure diffusive process in a sphere
Physical absorption: discussion
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Analysis of time evolution of concentration fields
• Re = 200
1. Fast saturation of the vortex periphery
2. Mainly ‘diffusive’ process through toroidal vortex
Physical absorption: discussion
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Analysis of time evolution of concentration fields
• Re=5
Combination of diffusive and convective transport
Physical absorption: discussion
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Outline
Mathematical modelling
• Domain and equations
Physical absorption (Ha=0)
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Physical absorption: analytical solutions
1. Limit case Re 0 : diffusion in a sphere
Model equation:
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Physical absorption: analytical solutions
2. Limit case Re ∞
• ‘Instantaneous’ saturation of a vortex periphery
• ‘Diffusion’ through a toroidal vortex
• Torus cut and unfold gives a cylinder
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Physical absorption: analytical solutions
2. Limit case Re ∞ : diffusion in a cylinder representing
a toroidal vortex
– Model equation:
with and
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Adjustable parameters Fitted from DNS results
0.97 2.4
Physical absorption: analytical solutions
3. Intermediate Re
i. Diffusion in a boundary layer (thiner if Re larger)
similar to penetration-film model with thickness df
Model equation:
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df
x
Physical absorption: analytical solutions
3. Intermediate Re
i. Diffusion in a boundary layer (thiner if Re larger)
similar to penetration-film model with thickness df
with
ii. Diffusion through the toroidal vortex
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Adjustable parameter Fitted from DNS results
𝛿𝑓 = 0.1825 Re−0.587
Physical absorption: analytical solutions
Comparison of the analytical expressions for the mass
transfer rate (continuous lines) with the DNS results (dashed
lines)
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Outline
Mathematical modelling
• Domain and equations
Physical absorption (Ha=0)
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Physical absorption: summary
3 steps can be observed:
1. Diffusion at interface vicinity directed towards the droplet center.
2. Convection enhancing the transfer by depleting the interface boundary layer
3. Saturation of the vortex periphery. The transfer rate becomes limited by diffusion towards the vortex inside.
Following the flow regime:
• Re 0 : only the 1st step is observed.
• Re ∞ : the 3rd step is directly observed.
• Intermediate Re : mass absorption rate controlled successively by these 3 limiting steps, transition time decreases as Re increases.
Enables to propose simplified mechanisms to describe the mass transfer rate evolution for any flow regime
Analytical solutions for
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Outline
Mathematical modelling
• Domain and equations
Physical absorption
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction (Ha>0)
• Simulation results and discussion
• Summary
Final conclusion
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Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 0.1
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Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 0.1
Ha = 0.1
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CA CB
Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 0.1
Ha = 10
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CA CB
Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 100
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Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 100
Ha = 0.1
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CA CB
Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 100
Ha = 10
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CA CB
Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 1
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Absorption coupled with chemical reaction:
simulation results and discussion
Time evolution of Sh and f and concentration fields
analysis
• Re = 10
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Outline
Mathematical modelling
• Domain and equations
Physical absorption
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction (Ha>0)
• Simulation results and discussion
• Summary
Final conclusion
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Absorption with reaction : summary
Prevalent mass transport mechanism: depends on Re
Slow reaction (Ha<<1)
• Saturation is delayed by B as it increases absorption capability
• Transfer rate remains mostly controlled by transport of A
Fast reaction (Ha>>1)
• Saturation not delayed
• Transfer rate is first controlled by the transport of B and becomes controlled by transport of A when B is depleted
Moderate reaction (Ha~1) : intermediate situation
• Saturation possibly delayed
• Transfer rate gets influenced by the transport of A and B
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Outline
Mathematical modelling
• Domain and equations
Physical absorption
• Simulation results and discussion
• Development of analytical solutions
• Summary
Absorption coupled with a chemical reaction
• Simulation results and discussion
• Summary
Final conclusion
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Final conclusion
Results show the importance of taking into account all the
phenomena simultaneously in gas-droplet mass transfer
modeling
Detailed analysis enables looking “smartly” for simplified
approaches:
• Approximate analytical expressions for physical absorption: OK
• Expressions when coupled with an instantaneous reaction: in development
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