WCNSFV2PInterfaceAreaSourceSink

The interfacial area concentration is defined as the interface area between two phases per unit volume, i.e.,[ξp]=m2m3[\xi_p] = \frac{m^2}{m^3}. This parameter is important for predicting mass, momentum, and energy transfer at interfaces in two-phase flows.

The general equation for interfacial area concentration transport via the mixture model reads as follows:

(ρdξp)t+(ρduξp)(Dbξp)=13DρdDt+ST+ρd(SC+SB),\frac{\partial (\rho_d \xi_p)}{\partial t} + \nabla \cdot \left( \rho_d \vec{u} \xi_p \right) - \nabla \left( D_b \nabla \xi_p \right) = -\frac{1}{3} \frac{D \rho_d}{Dt} + S_T + \rho_d (S_C + S_B)\,,

where:

  • ρd\rho_d is the density of the dispersed phase dd, e.g., the density of the gas in bubbles,

  • tt is time,

  • u\vec{u} is the velocity vector,

  • DbD_b is a diffusion coefficient for the interface area concentration, which may be assumed to be 0 if unknown,

  • D()Dt=()t+u()\frac{D (\bullet)}{Dt} = \frac{\partial (\bullet)}{\partial t} + \vec{u} \cdot \nabla (\bullet) is the material derivative,

  • STS_T, SCS_C, and SBS_B are the interface area concentration sources due to mass transfer, coalescence, and breakage, respectively.

The terms on the left-hand side of this equation are modeled via standard kernels for the mixture model. For example, in an open flow case, the time derivative may be modeled using FVFunctorTimeKernel, the advection term using INSFVScalarFieldAdvection, and the diffusion term using FVDiffusion.

The terms on the right-hand side are modeled using a multidimensional version of Hibiki and Ishii's model (Hibiki and Ishii, 2000). In this one, the sources are approximated as follows:

Interface area concentration source due to mass transfer

The interface area concentration source due to mass transfer is modeled as follows:

ST={6αddp,if αd<αdco23hbd(1αd2.0),otherwise,S_T = \begin{cases} \frac{6 \alpha_d}{d_p}, & \text{if } \alpha_d < \alpha_d^{co} \\ \frac{2}{3} \cdot h^{b \rightarrow d} \left( \frac{1}{\alpha_d} - 2.0 \right), & \text{otherwise} \end{cases}\,,

where:

  • αd\alpha_d is the volumetric fraction of the dispersed phase, e.g., the void fraction if the dispersed phase is a gas,

  • αdco\alpha_d^{co} is a cutoff fraction for mass transfer model selection,

  • dpd_p is the best estimate for the dispersed phase particle diameter,

  • hbdh^{b \rightarrow d} is the mass exchange coefficient from the bulk to the dispersed phase,

The cutoff volumetric fraction αdco\alpha_d^{co} is included because the mass transfer term in Hibiki and Ishii's model is not physical for low volumetric fractions of the dispersed phase. Below the cutoff limit, the dispersed phase is modeled as spherical particles.

commentnote

The user should select the cutoff volumetric fraction αdco\alpha_d^{co} as the limit at which the modeled flows transition away from bubbly flow, i.e., below the cutoff limit there is an implicit assumption that the flow behaves as a bubbly flow.

Interface area concentration source due to coalescence

The reduction in interface area concentration due to coalescence of the dispersed phase is modeled as follows:

SC=(αdξp)2ΓCαd2uϵdp~11/3(αdmaxαd)exp(KCdp~5/3ρb1/2uϵ1/3σ1/2),S_C = -\left( \frac{\alpha_d}{\xi_p} \right)^2 \frac{\Gamma_C \alpha_d^2 u_{\epsilon}}{\tilde{d_p}^{11/3} (\alpha_d^{max}- \alpha_d)} \operatorname{exp} \left( -K_C \frac{\tilde{d_p}^{5/3} \rho_b^{1/2} u_{\epsilon}^{1/3}}{\sigma^{1/2}} \right)\,,

where:

  • αd\alpha_d is the volumetric fraction of the dispersed phase, e.g., the void fraction if the dispersed phase is a gas,

  • uϵ=(up/ρm)1/3u_{\epsilon} = \left( \| \vec{u} \| \ell \| \nabla p \| / \rho_m \right)^{1/3} is the friction velocity due to pressure gradients, with u\| \vec{u} \| being the norm of the velocity vector, \ell a characteristic mixing length, p\| \nabla p \| the norm of the pressure gradient, and ρm\rho_m the mixture density,

  • dp~=ψαdξp\tilde{d_p} = \psi \frac{\alpha_d}{\xi_p} is the model estimate for the dispersed phase particle diameter, with ψ\psi being a shape factor, which is, for example, ψ=6\psi=6 for spherical particles,

  • αdmax\alpha_d^{max} is the maximum volumetric fraction admitted by the model, in absence of data we recommend taking αdmax=1\alpha_d^{max}=1,

  • ρb\rho_b is the bulk phase density, e.g., for air bubbles in a water flow it would be the density of water,

  • σ\sigma is the surface tension between the two phases,

  • ΓC=0.188\Gamma_C = 0.188 and KC=0.129K_C = 0.129 are closure coefficients of the model.

commentnote

Many of the parameters in the coalescence model are provided as functors, which means that spatially dependent fields may be specified for these parameters. However, the validation of this model has only been performed using constant parameters.

Interface area concentration source due to breakage

The increase in interface area concentration due to breakage of the dispersed phase is modeled as follows:

SB=(αdξp)2ΓBαd(1αd)uϵdp~11/3(αdmaxαd)exp(KBσρbdp~5/3uϵ2/3),S_B = \left( \frac{\alpha_d}{\xi_p} \right)^2 \frac{\Gamma_B \alpha_d (1 - \alpha_d) u_{\epsilon}}{\tilde{d_p}^{11/3} (\alpha_d^{max}- \alpha_d)} \operatorname{exp} \left( -K_B \frac{\sigma}{\rho_b \tilde{d_p}^{5/3} u_{\epsilon}^{2/3}} \right)\,,

where:

  • αd\alpha_d is the volumetric fraction of the dispersed phase, e.g., the void fraction if the dispersed phase is a gas,

  • uϵ=(up/ρm)1/3u_{\epsilon} = \left( \| \vec{u} \| \ell \| \nabla p \| / \rho_m \right)^{1/3} is the friction velocity due to pressure gradients, with u\| \vec{u} \| being the norm of the velocity vector, \ell a characteristic mixing length, p\| \nabla p \| the norm of the pressure gradient, and ρm\rho_m the mixture density,

  • dp~=ψαdξp\tilde{d_p} = \psi \frac{\alpha_d}{\xi_p} is the model estimate for the dispersed phase particle diameter, with ψ\psi being a shape factor, which is, for example, ψ=6\psi=6 for spherical particles,

  • αdmax\alpha_d^{max} is the maximum volumetric fraction admitted by the model, in absence of data we recommend taking αdmax=1\alpha_d^{max}=1,

  • ρb\rho_b is the bulk phase density, e.g., for air bubbles in a water flow it would be the density of water,

  • σ\sigma is the surface tension between the two phases,

  • ΓB=0.264\Gamma_B = 0.264 and KB=1.370K_B = 1.370 are closure coefficients of the model.

commentnote

Many of the parameters in the breakage model are provided as functors, which means that spatially dependent fields may be specified for these parameters. However, the validation of this model has only been performed using constant parameters.

Input Parameters

  • pressureContinuous phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Continuous phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • rhoContinuous phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Continuous phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • rho_dDispersed phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Dispersed phase density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • uThe velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:The velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • variableThe name of the variable that this residual object operates on

    C++ Type:NonlinearVariableName

    Unit:(no unit assumed)

    Controllable:No

    Description:The name of the variable that this residual object operates on

Required Parameters

  • L1The characteristic dissipation length. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    Default:1

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:The characteristic dissipation length. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • blockThe list of blocks (ids or names) that this object will be applied

    C++ Type:std::vector<SubdomainName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of blocks (ids or names) that this object will be applied

  • cutoff_fraction0.1Void fraction at which the interface area density mass transfer model is activated. Below this fraction, spherical bubbles are assumed.

    Default:0.1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Void fraction at which the interface area density mass transfer model is activated. Below this fraction, spherical bubbles are assumed.

  • fd0Fraction dispersed phase. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    Default:0

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Fraction dispersed phase. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • fd_max1Maximum dispersed phase fraction.

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Maximum dispersed phase fraction.

  • k_c0Mass exchange coefficients from continous to dispersed phases. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    Default:0

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Mass exchange coefficients from continous to dispersed phases. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • particle_diameter1Maximum particle diameter. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    Default:1

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Maximum particle diameter. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

  • sigma1Surface tension between phases. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    Default:1

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:Surface tension between phases. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

  • vThe velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:The velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

  • wThe velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

    C++ Type:MooseFunctorName

    Unit:(no unit assumed)

    Controllable:No

    Description:The velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.

Optional Parameters

  • absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution

  • extra_matrix_tagsThe extra tags for the matrices this Kernel should fill

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The extra tags for the matrices this Kernel should fill

  • extra_vector_tagsThe extra tags for the vectors this Kernel should fill

    C++ Type:std::vector<TagName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The extra tags for the vectors this Kernel should fill

  • matrix_tagssystemThe tag for the matrices this Kernel should fill

    Default:system

    C++ Type:MultiMooseEnum

    Unit:(no unit assumed)

    Options:nontime, system

    Controllable:No

    Description:The tag for the matrices this Kernel should fill

  • vector_tagsnontimeThe tag for the vectors this Kernel should fill

    Default:nontime

    C++ Type:MultiMooseEnum

    Unit:(no unit assumed)

    Options:nontime, time

    Controllable:No

    Description:The tag for the vectors this Kernel should fill

Tagging Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:Yes

    Description:Set the enabled status of the MooseObject.

  • implicitTrueDetermines whether this object is calculated using an implicit or explicit form

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Determines whether this object is calculated using an implicit or explicit form

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    Unit:(no unit assumed)

    Controllable:No

    Description:The seed for the master random number generator

  • use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

  • ghost_layers2The number of layers of elements to ghost.

    Default:2

    C++ Type:unsigned short

    Unit:(no unit assumed)

    Controllable:No

    Description:The number of layers of elements to ghost.

  • use_point_neighborsFalseWhether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

Parallel Ghosting Parameters

References

  1. T Hibiki and M Ishii. One-group interfacial area transport of bubbly flows in vertical round tubes. International Journal of Heat and Mass Transfer, 43(15):2711–2726, 2000.[BibTeX]