MassFluxValue

MassFluxValue[pred,vars,pars]

represents a mass flux boundary condition for PDEs with predicate pred indicating where it applies, with model variables vars and global parameters pars.

MassFluxValue[pred,vars,pars,lkey]

represents a mass flux boundary condition with local parameters specified in pars[lkey].

Details

  • MassFluxValue specifies a boundary condition for MassTransportPDEComponent and is used as part of the modeling equation:
  • MassFluxValue is typically used to model mass species flow through a boundary caused by a species source or sink outside of the domain.
  • A flow rate is the flow of a quantity like energy or mass per time. Flux is the flow rate through the boundary and is measured in the units of the quantity per area per time. A millimeter of rain per cross section of opening area per hour is a rain flux.
  • MassFluxValue models the rate of mass species flowing through some part of the boundary with dependent variable in [TemplateBox[{InterpretationBox[, 1], {"mol", , "/", , {"m", ^, 3}}, moles per meter cubed, {{(, "Moles", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]], independent variables in [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]] and time variable in [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]].
  • Stationary variables vars are vars={c[x1,…,xn],{x1,…,xn}}.
  • Time-dependent variables vars are vars={c[t,x1,…,xn],t,{x1,…,xn}}.
  • The non-conservative time-dependent mass transport model MassTransportPDEComponent is based on a convection-diffusion model with mass diffusivity , mass convection velocity vector , mass reaction rate and mass source term :
  • The conservative time-dependent mass transport model MassTransportPDEComponent is based on a conservative convection-diffusion model given by:
  • In the non-conservative form, MassFluxValue with mass flux in and boundary unit normal models:
  • In the conservative form, MassFluxValue models:
  • Model parameters pars as specified for MassTransportPDEComponent.
  • The following additional model parameters pars can be given:
  • parameterdefaultsymbol
    "BoundaryUnitNormal"Automatic
    "MassFlux"
  • 0
    		• , mass flux [TemplateBox[{InterpretationBox[, 1], {"mol", , "/(", , "m", , "s", , ")"}, moles per meter second, {{(, "Moles", )}, /, {(, {"Meters", , "Seconds"}, )}}}, QuantityTF]]
    "ModelForm""NonConservative"-
  • All model parameters may depend on any of , and , as well as other dependent variables.
  • To localize model parameters, a key lkey can be specified, and values from association pars[lkey] are used for model parameters.
  • MassFluxValue evaluates to a NeumannValue.
  • The boundary predicate pred can be specified as in NeumannValue.
  • If the MassFluxValue depends on parameters that are specified in the association pars as …,keypi…,pivi,…], the parameters are replaced with .

Examples

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Basic Examples  (2)

Set up a mass flux boundary condition in non-conservative form:

Set up a mass flux boundary condition in conservative form:

Scope  (10)

Define model variables vars for a transient species field with model parameters pars and a specific boundary condition parameter:

Define model variables vars for a transient species field with model parameters pars and multiple specific parameter boundary conditions:

1D  (1)

Model a 1D chemical species field in an incompressible fluid whose right side and left side are subjected to a mass concentration and inflow condition, respectively:

del .(-d del c(x))+v^->.del c(x)^(︷^( mass transport model )) =|_(Gamma_(x=0))q(x)^(︷^( mass flux value ))

Set up the stationary mass transport model variables vars:

Set up a region :

Specify the mass transport model parameters species diffusivity and fluid flow velocity :

Specify a species flux boundary condition:

Specify a mass concentration boundary condition:

Set up the equation:

Solve the PDE:

Visualize the solution:

2D  (1)

Model mass transport of a pollutant in a 2D rectangular region in an isotropic homogeneous medium. Initially, the pollutant concentration is zero throughout the region of interest. A concentration of 3000 is maintained at a strip with dimension 0.2 located at the center of the left boundary, while the right boundary is subject to a parallel species flow with a constant concentration of 1500 , allowing for mass transfer. A pollutant outflow of 100 is applied at both the top and bottom boundaries. A diffusion coefficient of 0.833 is distributed uniformly with a uniform horizontal velocity of 0.01 :

del .(-d del c(x,y))+v^->.del c(x,y)^(︷^( mass transport model )) =|_(Gamma_(y=0, y=10))q(x,y)^(︷^( mass flux value ))+|_(Gamma_(x=20))h (c_(ext)(x,y)-c(x,y))^(︷^( mass transfer value ))

Set up the mass transport model variables vars:

Set up a rectangular domain with a width of and a height of :

Specify model parameters species diffusivity and fluid flow velocity :

Set up a species concentration source of 0.2 length at the center of the left surface:

Set up a mass transfer boundary on the right surface:

Set up an outflow flux of on the top and bottom surfaces:

Set up the equation:

Solve the PDE:

Visualize the solution:

3D  (1)

Model a non-conservative chemical species field in a unit cubic domain, with two mass conditions at two lateral surfaces and a mass inflow through a circle with radius 0.2 at the center of the top surface, as well as an orthotropic mass diffusivity :

del .(-d del c(x,y,z))+v^->.del c(x,y,z)^(︷^( mass transport model )) =|_(Gamma_(z=1& (x-0.5)^2+(y-0.5)^2<=0.04))q(x,y,z)^(︷^( mass flux value ))

Set up the mass transport model variables vars:

Set up a region :

Specify a diffusivity and a flow velocity field :

Specify mass concentrations:

Specify a flux condition of through a regional circle on the top surface:

Set up the equation:

Solve the PDE:

Visualize the solution:

Material Regions  (1)

Model a 1D chemical species transport through different material with a reaction rate in one. The right side and left side are subjected to a mass concentration and inflow condition, respectively:

del .(-d del c(x))+a c(x)^(︷^( mass transport model )) =|_(Gamma_(x=0))q(x)^(︷^( mass flux value ))

Set up the stationary mass transport model variables vars:

Set up a region :

Specify the mass transport model parameters species diffusivity and a reaction rate active in the region :

Specify a species flux boundary condition:

Specify a mass concentration boundary condition:

Set up the equation:

Solve the PDE:

Visualize the solution:

Time Dependent  (1)

Model a 1D non-conservative chemical species field and a mass flux through part of the boundary with:

(partialc(t,x))/(partialt)+del .(-d del c(t,x))^(︷^( diffusion term )) +v^->.del c(t,x)^(︷^( convection term)) =|_(Gamma_(x=0))q(t,x)^(︷^( mass flux term))

Set up the time-dependent mass transport model variables vars:

Set up a region :

Specify the mass transport model parameters mass diffusivity and mass convection velocity :

Set up the equation with a mass flux of at the left end for the first 50 seconds:

Solve the PDE with an initial condition of a zero concentration:

Visualize the solution:

Nonlinear Time Dependent  (1)

Model a 1D non-conservative chemical species field with a nonlinear diffusivity coefficient and an outflow condition through part of the boundary, which is expressed as follows:

(partialc(t,x))/(partialt)+del .(-d del c(t,x))^(︷^( diffusion term )) +v^->.del c(t,x))^(︷^( convection term)) =|_(Gamma_(x=0.2))q(t,x)^(︷^( mass flux term ))

Set up the mass transport model variables vars:

Set up a region :

Specify a nonlinear species diffusivity and fluid flow velocity :

Specify an outflow flux of applied at the right end:

Specify a time-dependent mass concentration surface condition:

Set up an initial condition:

Set up the equation:

Solve the PDE:

Visualize the solution:

Coupled Time Dependent  (2)

Model a 1D coupled non-conservative dual chemical species field with corresponding mass flux through the left parts of the boundary:

(partialc_1(t,x))/(partialt)+del .(-d_(11) del c_1(t,x))^(︷^( mass transport model )) =|_(Gamma_(x=0))q_1(t,x)^(︷^( mass flux value ))

(partialc_2(t,x))/(partialt)+del .(-d_(22) del c_2(t,x))^(︷^( mass transport model )) =|_(Gamma_(x=0))q_2(t,x)^(︷^( mass flux value ))

Set up the time dependent mass transport model variables vars for the and species, respectively:

Set up a uniform region :

Specify the mass transport model parameters mass diffusivity and for the and species:

Set up the boundary conditions with a mass flux of and for and at the left end for the first 50 seconds:

Set up the equation:

Set up initial conditions:

Solve the PDEs:

Visualize the solution:

Model a 1D coupled chemical species field with a convection velocity and a mass flux through the left boundary:

(partialc_1(t,x))/(partialt)+del .(-d_(11) del c_1(t,x))+v^->.del c_1(t,x)^(︷^( mass transport model )) =|_(Gamma_(x=0))q_1(t,x)^(︷^( mass flux value ))

(partialc_2(t,x))/(partialt)+del .(-d_(22) del c_2(t,x))+v^->.del c_2(t,x)^(︷^( mass transport model )) =|_(Gamma_(x=0))q_2(t,x)^(︷^( mass flux value ))

Set up the time-dependent mass transport model variables vars for and species, respectively:

Set up a uniform region :

Specify the mass transport model parameters mass diffusivity and for the and species:

Set up the equation with a mass flux of 6 and 12 for and at the left end for the first 50 seconds:

Set up the equation:

Set up initial conditions:

Solve the PDEs:

Visualize the solution:

Applications  (2)

Single Equation  (1)

Model mass transport of a pollutant in a 2D rectangular region in an isotropic homogeneous medium. Initially, the pollutant concentration is zero throughout the region of interest. A concentration of 3000 is maintained at a strip with dimension 0.2 located at the center of the left boundary, while a pollutant outflow of 100 is applied at both the top and bottom boundaries. A diffusion coefficient of 0.833 is distributed uniformly, but both horizontal and vertical velocity are spatial dependent:

del .(-d del c(x,y))+v^->.del c(x,y)^(︷^( mass transport model )) =|_(Gamma_(y=0, y=10))q(x,y)^(︷^( mass flux value ))

Set up the mass transport model variables vars:

Set up a rectangular domain with a width of and a height of :

Specify model parameters species diffusivity and fluid flow velocity :

Set up a species concentration source of 0.2 length at the center of the left surface:

Set up an outflow flux of on the top and bottom surfaces:

Set up the equation:

Solve the PDE:

Visualize the solution:

Coupled Equations  (1)

Solve a coupled heat transfer and mass transport model with a thermal transfer value and a mass flux value on the boundary:

(partialT(t, x))/(partialt)+del .(-k del T(t,x))-Q^(︷^( heat transfer model )) = |_(Gamma_(x=1))h (T_(ext)(t,x)-T(t,x))^(︷^( heat transfer boundary )); (partialc(t,x))/(partialt)+del .(-d del c(t,x))-R^(︷^( mass transport model )) = |_(Gamma_(x=0||x=1))q (t,x)^(︷^( mass flux boundary ))

Set up the heat transfer mass transport model variables vars:

Set up a region :

Specify heat transfer and mass transport model parameters, heat source , thermal conductivity , mass diffusivity and mass source :

Specify boundary condition parameters for a thermal convection value with an external flow temperature of 1000 K and a heat transfer coefficient of :

Specify the equation:

Set up initial conditions:

Solve the model:

Visualize the solution:

Wolfram Research (2020), MassFluxValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MassFluxValue.html.

Text

Wolfram Research (2020), MassFluxValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MassFluxValue.html.

CMS

Wolfram Language. 2020. "MassFluxValue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MassFluxValue.html.

APA

Wolfram Language. (2020). MassFluxValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MassFluxValue.html

BibTeX

@misc{reference.wolfram_2024_massfluxvalue, author="Wolfram Research", title="{MassFluxValue}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/MassFluxValue.html}", note=[Accessed: 26-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_massfluxvalue, organization={Wolfram Research}, title={MassFluxValue}, year={2020}, url={https://reference.wolfram.com/language/ref/MassFluxValue.html}, note=[Accessed: 26-April-2024 ]}