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

Inheritance diagram for PrinceBlanch:

Collaboration diagram for PrinceBlanch:

Public Member Functions
	TypeName ("PrinceBlanch")
	Runtime type information. More...

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

virtual	~PrinceBlanch ()
	Destructor. More...

virtual void	precompute ()
	Precompute diameter independent expressions. More...

virtual void	addToCoalescenceRate (volScalarField &coalescenceRate, const label i, const label j)
	Add to coalescenceRate. 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)

autoPtr< coalescenceModel >	clone () const

virtual	~coalescenceModel ()
	Destructor. More...

Additional Inherited Members
Static Public Member Functions inherited from coalescenceModel
static autoPtr< coalescenceModel >	New (const word &type, const populationBalanceModel &popBal, const dictionary &dict)

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]
	=	Initial film thickness [m]
	=	Critical film thickness [m]
$\epsilon_c$	=	Continuous phase turbulent dissipation rate [m2/s3]
	=	Diameter of bubble i [m]
	=	Diameter of bubble j [m]
$u_{ri}$	=	Rise velocity of bubble i [m/s]
$S_{ij}$	=	Collision cross sectional area [m2]
	=	Gravitational constant [m/s2]
$\dot{\gamma}_c$	=	Continuous phase shear strain rate [1/s]
	=	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