Public Member Functions | List of all members
PrinceBlanch Class Reference

Model of Prince and Blanch (1990). The coalescence rate is calculated by. More...

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Public Member Functions

 TypeName ("PrinceBlanch")
 Runtime type information. More...
 
 PrinceBlanch (const populationBalanceModel &popBal, const dictionary &dict)
 
virtual ~PrinceBlanch ()
 Destructor. More...
 
virtual void precompute ()
 Pre-compute diameter independent expressions. More...
 
virtual tmp< volScalarField::Internal > rate (const label i, const label j) const
 Return the coalescence rate between two groups. More...
 
- Public Member Functions inherited from coalescenceModel
 TypeName ("coalescenceModel")
 Runtime type information. More...
 
 declareRunTimeSelectionTable (autoPtr, coalescenceModel, dictionary,(const populationBalanceModel &popBal, const dictionary &dict),(popBal, dict))
 
 coalescenceModel (const populationBalanceModel &popBal, const dictionary &dict)
 Construct from a population balance and its dictionary. More...
 
virtual ~coalescenceModel ()
 Destructor. More...
 
virtual bool coalesces () const
 Does this model coalesce? More...
 

Additional Inherited Members

- Static Public Member Functions inherited from coalescenceModel
static autoPtr< coalescenceModel > New (const populationBalanceModel &popBal, const dictionary &dict)
 Selector. More...
 
- Protected Attributes inherited from coalescenceModel
const populationBalanceModel & popBal_
 Reference to the populationBalanceModel. More...
 

Detailed Description

Model of Prince and Blanch (1990). The coalescence rate is calculated by.

\[ \left( \theta_{ij}^{T} + \theta_{ij}^{B} + \theta_{ij}^{LS} \right) \lambda_{ij}\;, \]

with the coalescence efficiency

\[ \lambda_{ij} = \mathrm{exp} \left( - \sqrt{\frac{r_{ij}^3 \rho_c}{16 \sigma}} \mathrm{ln} \left(\frac{h_0}{h_f}\right) \epsilon_c^{1/3}/r_{ij}^{2/3} \right)\;, \]

the turbulent collision rate

\[ \theta_{ij}^{T} = C_1 \pi (d_i + d_j)^{2} \epsilon_c^{1/3} \sqrt{d_{i}^{2/3} + d_{j}^{2/3}}\;, \]

the buoyancy-driven collision rate

\[ \theta_{ij}^{B} = S_{ij} \left| u_{ri} - u_{rj} \right|\;, \]

and the laminar shear collision rate

\[ \theta_{ij}^{LS} = \frac{1}{6} \left(d_i + d_j\right)^{3} \dot{\gamma}_c\;. \]

The rise velocity of bubble i is calculated by

\[ u_{ri} = \sqrt{2.14 \sigma / \left(\rho_c d_i \right) + 0.505 g d_i}\;, \]

the equivalent radius by

\[ r_{ij} = \left( \frac{1}{d_i} + \frac{1}{d_j} \right)^{-1}\;, \]

the collision cross sectional area by

\[ S_{ij} = \frac{\pi}{4} \left(d_i + d_j\right)^{2}\;, \]

and the shear strain rate by

\[ \dot{\gamma}_c = \mathrm{mag}(\mathrm{symm}(\mathrm{grad}(U_c)))\;. \]

Note that in equation 2, the bubble radius has been substituted by the bubble diameter, which leads to a different coefficient in equation 8. Also the expression for the equivalent radius, equation 19 was corrected.

$ \theta_{ij}^{T} $ = Turbulent collision rate [m3/s]
$ \theta_{ij}^{B} $ = Buoyancy-driven collision rate [m3/s]
$ \theta_{ij}^{LS} $ = Laminar shear collision rate [m3/s]
$ \lambda_{ij} $ = Coalescence efficiency [-]
$ r_{ij} $ = Equivalent radius [m]
$ \rho_c $ = Density of continuous phase [kg/m3]
$ \sigma $ = Surface tension [N/m]
$ h_0 $ = Initial film thickness [m]
$ h_f $ = Critical film thickness [m]
$ \epsilon_c $ = Continuous phase turbulent dissipation rate [m2/s3]
$ d_i $ = Diameter of bubble i [m]
$ d_j $ = Diameter of bubble j [m]
$ u_{ri} $ = Rise velocity of bubble i [m/s]
$ S_{ij} $ = Collision cross sectional area [m2]
$ g $ = Gravitational constant [m/s2]
$ \dot{\gamma}_c $ = Continuous phase shear strain rate [1/s]
$ U_c $ = Continuous phase velocity field [m/s]

Reference: 

        Prince, M. J., & Blanch, H. W. (1990).
        Bubble coalescence and break‐up in air‐sparged bubble columns.
        AIChE journal, 36(10), 1485-1499.
Usage
Property Description Required Default value
C1 coefficient C1 no 0.356
h0 initial film thickness no 1e-4m
hf critical film thickness no 1e-8m
turbulence Switch for collisions due to turbulence yes none
buoyancy Switch for collisions due to buoyancy yes none
laminarShear Switch for collisions due to laminar shear yes none
Source files

Definition at line 259 of file PrinceBlanch.H.

Constructor & Destructor Documentation

◆ PrinceBlanch()

PrinceBlanch ( const populationBalanceModel &  popBal,
const dictionary &  dict 
)

Definition at line 54 of file PrinceBlanch.C.

References Foam::dimLength, Foam::dimVelocity, populationBalanceModel::mesh(), dimensioned< Type >::name(), coalescenceModel::popBal_, IOobject::time(), and Foam::Zero.

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◆ ~PrinceBlanch()

virtual ~PrinceBlanch ( )
inlinevirtual

Destructor.

Definition at line 302 of file PrinceBlanch.H.

Member Function Documentation

◆ TypeName()

TypeName ( "PrinceBlanch"  )

Runtime type information.

◆ precompute()

void precompute ( )
virtual

Pre-compute diameter independent expressions.

Reimplemented from coalescenceModel.

Definition at line 95 of file PrinceBlanch.C.

References Foam::fvc::grad(), Foam::mag(), Foam::sqrt(), and Foam::symm().

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◆ rate()

Foam::tmp< Foam::volScalarField::Internal > rate ( const label  i,
const label  j 
) const
virtual

Return the coalescence rate between two groups.

Implements coalescenceModel.

Definition at line 106 of file PrinceBlanch.C.

References Foam::cbrt(), Foam::dimTime, Foam::dimVolume, Foam::exp(), Foam::log(), Foam::mag(), DimensionedField< Type, GeoMesh, PrimitiveField >::New(), Foam::constant::mathematical::pi(), Foam::pow(), Foam::pow3(), tmp< T >::ref(), Foam::constant::physicoChemical::sigma, Foam::sqr(), and Foam::sqrt().

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The documentation for this class was generated from the following files: