MassTransportPDEComponent

MassTransportPDEComponent[vars,pars]

yields a mass transport PDE term with variables vars and parameters pars.

Details

  • MassTransportPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
  • MassTransportPDEComponent models the generation and propagation of diluted mass species in physical systems such as mixtures, solutions and solids by mechanisms of diffusion or convection.
  • MassTransportPDEComponent models is applicable when the concentration of the diluted species is at least one order of magnitude less than the concentration of the solvent.
  • MassTransportPDEComponent models mass transport phenomena with dependent variable in , independent variables in and time variable in .
  • Stationary variables vars are vars={c[x1,,xn],{x1,,xn}}.
  • Time-dependent variables vars are vars={c[t,x1,,xn],t,{x1,,xn}}.
  • MassTransportPDEComponent provides both a conservative model to be used with compressible fluids and a non-conservative model to be used with incompressible fluids.
  • 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:
  • The non-conservative stationary mass transport PDE term is given by:
  • The implicit default boundary condition for the non-conservative model is a MassOutflowValue.
  • The conservative stationary mass transport PDE term is given by:
  • The implicit default boundary condition for the conservative model is a MassImpermeableBoundaryValue.
  • The difference between the non-conservative and the conservative models is the treatment of a convection velocity .
  • The non-conservative model is the default model. The conservative model should be used when the divergence of convection velocity is nonzero.
  • The units of the mass transport PDE terms are in .
  • The following model parameters pars can be given:
  • parameterdefaultsymbol
    "MassConvectionVelocity", flow velocity in
    "DiffusionCoefficient"IdentityMatrix, mass diffusivity in
    "MassReactionRate"0, mass reaction rate in
    "MassSource"0, mass source in
    "ModelForm""NonConservative"
  • All parameters may depend on any of , and , as well as other dependent variables.
  • The number of independent variables determines the dimensions of and the length of .
  • The mass convection velocity specifies the velocity with which a fluid transports mass.
  • A mass reaction term models mass chemical reactions of masses.
  • A mass source models mass that is generated (positive) or absorbed (negative).
  • Possible choices for the parameter "ModelForm" are "Conservative" and "NonConservative".
  • Coupled equations can be generated with the same input specification as with the corresponding operator terms.
  • If no parameters are specified, the default mass transport PDE is:
  • (partialc(t,x))/(partialt)+ del .(-del c(t,x))^(︷^(          diffusive term        ))

  • If the MassTransportPDEComponent depends on parameters that are specified in the association pars as ,keypi,pivi,], the parameters are replaced with .

Examples

open allclose all

Basic Examples  (3)

Define a time-dependent mass transport model:

Set up a time-dependent mass transport model with particular material parameters:

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:

Scope  (18)

Basic Usage  (9)

Set up a time-dependent mass transport model with numeric material parameters:

Define a 2D stationary mass transport model:

Set up a 2D stationary mass transport model with an orthotropic mass diffusion:

Set up a 2D stationary mass transport model with a diffusivity matrix:

Set up a 2D stationary mass transport model with an anisotropic diffusivity matrix:

Define a 2D stationary mass transport model with a material change:

Define a 2D stationary mass transport model with a material change and an anisotropic material:

Set up a coupled 2D stationary mass transport model:

Set up a coupled 2D transient mass transport model with cross-coupled nonlinear reaction terms:

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  (2)

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 in 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:

Symmetry boundaries can be used to reduce the size of the geometry of the model. Set up a mass transport equation:

Set up and visualize a region:

Solve and visualize the equation:

Set up a region about the symmetry axis at :

Solve and visualize the equation with a symmetry boundary at :

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 center of 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 the 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 4 and 8 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 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 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  (3)

Single Equations  (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 the 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  (2)

Solve a coupled heat and mass transport model:

(partialT(t, x))/(partialt)+del .(-k del Theta(t,x))-Q^(︷^(                  heat transfer model                  ))  = 0; (partialc(t,x))/(partialt)+del .(-d del c(t,x))-R^(︷^(                mass transport model                ))  = 0

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 :

Set up the model and initial conditions:

Set up initial conditions:

Solve the model:

Visualize the solution:

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 Theta(t,x))-Q^(︷^(                      heat transfer model                         ))  = |_(Gamma_(x=1))h (Theta_(ext)(t,x)-Theta(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:

Properties & Relations  (1)

Model a 1D chemical species field once with a conservative model and once with a non-conservative model. For a constant velocity flow field, both models return the same result. The right side and left side are subjected to a mass concentration and inflow conditions, respectively:

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

 del .(-d del c(x))+v^->.del c(x)^(︷^(   non-conservative 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 mass concentration boundary condition:

Specify species flux boundary condition:

Set up the equation:

Solve the PDE:

Visualize that the difference in the solutions is minimal:

Possible Issues  (2)

The implicit default boundary condition changes depending on the model form. For a conservative model, an implicit Neumann 0 boundary condition is equivalent to specifying an impermeable boundary condition. For a non-conservative model, an implicit Neumann 0 boundary condition is equivalent to specifying an outflow boundary condition.

Considering this, for a constant velocity field, both the conservative and non-conservative models return the same result. A comparison between the conservative and non-conservative fields are conducted based on the following models:

 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     ))

 del .(-d del c(x,y))+del v^->.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:

Specify the 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:

Since the default boundary condition for a conservative model is an impermeable boundary, an impermeable boundary condition is added to the non-conservative mode:

Set up the equation:

Solve the PDEs:

Visualize the difference in the solutions:

The scale of the differences in the solutions is expected and comes from numerical differences in how the operators are computed.

When a discretized region is given and the mesh does not meet the quality criteria for a large convection-to-diffusion ratio, a message is generated. Model a 1D non-conservative chemical species field with a high convection velocity to diffusivity ratio, 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 mass concentration surface conditions:

Set up an initial condition:

Set up the equation:

Solve the PDE with a refined mesh:

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

Text

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

BibTeX

@misc{reference.wolfram_2020_masstransportpdecomponent, author="Wolfram Research", title="{MassTransportPDEComponent}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/MassTransportPDEComponent.html}", note=[Accessed: 18-April-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_masstransportpdecomponent, organization={Wolfram Research}, title={MassTransportPDEComponent}, year={2020}, url={https://reference.wolfram.com/language/ref/MassTransportPDEComponent.html}, note=[Accessed: 18-April-2021 ]}

CMS

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

APA

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