Bubble impingement with particle

the soft matter engineering group @unina
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Università degli Studi di Napoli ”Federico II”
Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale
Tesi di Laurea Magistrale in Ingegneria Chimica
Impingement of gas bubbles in liquid matrices in the
presence of a solid particle
Candidato:
Alessandro Esposito
Matricola: M55/418
Relatore
Ch.mo Prof.Ing. Gaetano D’avino
Correlatore
Dott.Ing. Massimiliano Maria Villone

25/12/2016 - 2/20
Foams
Foams are dispersions of gas in a liquid or a solid matrix
liquid foam polyurethane foam metallic foam

25/12/2016 - 3/20
Foam structure
Foams may have different structure, in relation to their production process
Foam realization steps [1]:
Nucleation &
Single bubble growth
Impingement Rupture Retraction
Key steps to determine the final
structure are the rupture and the
consequent film retraction
[1] Tammaro et al., Chem. Eng. J., 2016
Open and closed cell foam

25/12/2016 - 4/20
How the presence
of a solid particle
can influence the
foam structure?
Investigation
through 3D
numerical
simulations
Basic problem: two
gas bubbles in a
liquid matrix in the
presence of a solid
particle
Objective of the work
Rupture may be induced by different factors:
• Crystallization
• Presence of solid objects (filled polymeric foams)
OBJECTIVE

25/12/2016 - 5/20
L
L
d
R0 R0
Rp
L
B1 B2
P
Problem formulation
Geometrical domain: Assumptions:
• Newtonian liquid matrix
• Incompressible flow
• Single, rigid, spherical particle
• Creeping flow regime (Re << 1)
– force-and torque-free particle
• Constant pressure inside the
bubbles
• Isothermal problem
• L >> R(t) such that bubbles are not
affected by domain boundaries
x
z
y

25/12/2016 - 6/20
Mathematical model
Fluid motion equations
Mass balance:
𝜵 ∙ 𝒗 = 0
Momentum balance:
𝜵 ∙ 𝑻 = 𝟎
𝑻 = −𝑝𝑰 + 2𝜇 𝑫
𝑫 = (𝜵𝒗 + 𝜵𝒗 T
)/2
Particle motion equations
Kinematic equation for translation:
𝑑𝒙p
𝑑𝑡
= 𝒗p
Kinematic equation for rotation:
𝑑𝜽
𝑑𝑡
= 𝝎
NB: since we impose that bubble growth is driven by a constant pressure
difference between gas and ambient fluid, no mass balance equation is
solved for the gas phase[2]
[2] Favelukis et Al. Bubble growth in viscous newtonian and non-newtonian liquids

25/12/2016 - 7/20
Boundary conditions
BCs on the fluid domain
Outflow on the external domain boundaries (Ω):
𝑻 ∙ 𝒏 = −𝑝out 𝒏
Young – Laplace condition on the bubbles (G2 e G3):
𝑻 ∙ 𝒏 = Γ 𝐧𝛁 ∙ 𝒏 − 𝑝g 𝒏
No-slip on solid particle (G1):
𝒗 = 𝒗p + 𝝎 × (𝐱 − 𝐱p)
BCs on the solid particle
Force-free condition:
𝑭 =
G1
𝑻 ∙ 𝒏 𝑑𝑆 = 𝟎
G3
Ω
Ω
Ω
G2
G1
Torque-free condition:
𝚿 =
G1
𝐱 − 𝐱p × 𝑻 ∙ 𝒏 𝑑𝑆 = 𝟎

25/12/2016 - 8/20
Parameters
System parameters
• 𝑅0: initial bubble radius
• 𝑅p: particle radius
• 𝒙p: initial position of particle center
• 𝑑 = |𝒙1 − 𝒙2|: distance between bubbles
• L: cube side
• 𝛤: surface tension
• 𝑝0: pressure inside bubbles
• 𝑝out: ambient pressure
Dimensionless parameters
𝑑∗
=
𝑑
𝑅0
𝑅 𝑝
∗ =
𝑅p
𝑅0
𝒙 𝑝
∗ =
𝒙p
𝑅0
𝐶𝑎 =
𝑅0(𝑝0−𝑝out)
2Γ

25/12/2016 - 9/20
Results
Two bubbles growth without solid particle:
• bubble dynamics
• mesh visualization (remeshing)
𝐶𝑎 = 0.5
𝑑∗ = 5

25/12/2016 -
Strain rate
Two bubbles growth without solid particle:
• strain rate ( 𝜀 = 2𝑫: 𝑫) on the xy-plane
𝐶𝑎 = 0.5
𝑑∗ = 5

25/12/2016 -
Results
𝐶𝑎 = 0.5
𝑑∗
= 5
𝑅p
∗
= 0.1
𝑥p
∗
= (0,2.4,0)
𝐶𝑎 = 0.5
𝑑∗
= 5
𝑅p
∗
= 0.1
𝑥p
∗
= (0,0.4,0)
Two bubbles growth with solid particle: two possible scenarios

25/12/2016 -
Two bubbles growth with solid particle:
Strain rate
𝐶𝑎 = 0.5
𝑑∗ = 5
𝑅p
∗
= 0.1
𝑥p
∗
= (0,2.4,0)

25/12/2016 -
Two bubbles growth with solid particle:
Strain rate
𝐶𝑎 = 0.5
𝑑∗ = 5
𝑅p
∗
= 0.1
𝑥p
∗
= (0,0.4,0)

25/12/2016 -
Symmetry
x
z
y
On the y-z plane each
radial position of the
particle is
symmetrical.

25/12/2016 -
Attraction basins
𝑅p = 0.5
𝐶𝑎 = 0.5
𝑑∗ = 5

25/12/2016 -
Attraction basins
𝐶𝑎 = 0.5
𝑑∗ = 5
𝑅p = 0.05
𝑅p = 0.1
𝑅p = 0.5

25/12/2016 -
Attraction basins
𝑑 = 3
𝑑 = 5
𝑑 = 7
𝐶𝑎 = 0.5
𝑅 𝑝
∗
= 0.1

25/12/2016 -
Conclusions
The results found shows that the presence of the
solid particle can induce:
- Higher values of strain rate.
- Bubble edge deformation nearby the particle.
The influence of the solid particle is relevant if it is actually «trapped» between the
bubbles. Parameters like particle radius and the bubble distance, (concentration)
can influence the attraction basins of the particle, driving the system to different
structures

25/12/2016 -
Future developements
- Investigate the effect of 𝐶𝑎 on the
attraction basins of the particle
𝐶𝑎 =
2Γ
𝑅0(𝑝0−𝑝 𝑜𝑢𝑡)
- Remove the assumptions of Newtonian
fluid and approach different constituive
equation (UCM, Oldroyd – B)
𝑻 + λ 𝑻 = 2𝜇D
- Consider multiple bubble interactions

25/12/2016 -
Thank you for your attention

Bubble impingement with particle

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