Mass Transport

This tutorial gives an introduction to modeling mass transport of diluted species. Equations and boundary conditions that are relevant for performing mass transport analysis are derived and explained.

Mass transport is a discipline of chemical engineering that is concerned with the movement of chemical species. The two mechanisms of mass transport are mass diffusion and mass convection. The driving force behind a mass diffusion is the difference in a species concentration at different locations. Mass convection, on the other hand, only occurs when species are transported in a moving fluid medium. The combination of these two mechanisms leads to changes in the species concentration field over time and is modeled with a mass balance equation.

The modeling process results in a partial differential equation (PDE) that can be solved with NDSolve. Furthermore, in this tutorial, different types of mass transport boundary conditions are introduced. For the boundary conditions given, the modeling of various real-world chemical species interactions is shown.

This tutorial provides an introduction overview of how to model mass transport. The focus is to present simple yet illustrative examples. Extended examples that show industrial applications of mass transport modeling can be found in the "Mass Transfer Model Collection".

Many of the animations of the simulation results shown in this tutorial are generated with a call to Rasterize. This is to reduce the disk space required. The downside is that the visual quality of the animations will not be as crisp as without it. To obtain high-quality graphics, remove or comment out the call to Rasterize.

To get high-fidelity visualizations, comment out the rasterization process:

The symbols and corresponding units used throughout this tutorial are summarized in the Nomenclature section.

Load the Finite Element package:

Mass Balance Equation

The following sections start with an introduction of the mass balance equation, followed by a derivation of the equation and examples of how it is mapped to the Wolfram Language.

Mass Balance Equation Introduction

There are two different formulations that can be used to model the transport of chemical species through mass diffusion and mass convection. These are the conservative and non-conservative forms of the mass balance equation:

The dependent variable in the mass balance equation is the species concentration , which varies with time and position . The partial differential equation (PDE) model describes how chemical species are transported over time in a solid or fluid medium.

Typically, the non-conservative formulation is appropriate for incompressible fluids where the concentration field is expected to be smooth. The derivation in the section Mass Balance Equation Derivation will help explain why this is the case.

Besides the time derivative part, both PDEs are made up of several components. First and foremost, there is a diffusive term: with a diffusion coefficient , which is also known as the species diffusivity. In some cases, the diffusion coefficient may depend on other properties, such as a temperature and the concentration itself making the equation nonlinear. The details of variable diffusion coefficients will be discussed later in the section Variable Mass Diffusion Coefficient.

The second component is a convective term. In the conservative formulation, the convective term is . In the non-conservative formulation, the convective term is . The difference between the conservative and the non-conservative forms is explained in the section Mass Balance Equation Derivation. In either case, this term is only present if mass transport occurs in a fluid medium with a fluid flow velocity . If the simulation medium is a solid, then this term is zero, regardless of which formulation is made use of.

Contrary to the convective term, the diffusive term may be present if the medium is solid or fluidic.

The term denotes a mass production or consumption rate of chemical species, typically due to a chemical reaction. This will be further explained in the Mass Transport with Chemical Reaction section.

Mass Balance Equation Derivation

To derive the mass balance equation, start from the law of mass conservation. In a system, the total mass change must be equal to net mass formation minus the net mass outflow flow. A mass balance can then be expressed as:

To formulate the mass balance equation, the mass of a chemical species is balanced within a unit volume domain . The mass balance is based on the concentration of the species, which denotes the amount of species per unit volume measured in moles. The molar concentration could be converted to a mass concentration by:

where is the amount of the species in moles, is the volume of the domain, is the mass of the species, and is the molar mass, which is the mass per unit mole of the species.

In the above graphic, is the concentration and is the area of the domain. The total mass within the control area is then equal to . The term denotes the net rate of mass formation due to chemical reactions inside the domain . The net mass flux represents the net rate of mass flow that exits through the boundaries.

The above consideration in 2D is valid in any dimension; for example, in a 3D domain, the concentration is given in and the volume in .

The unit of the mass flux through the boundary depends on the dimensionality of the boundary. In a 3D domain, the boundary is represented by 2D surfaces, and the mass flux has a unit of . In a 2D domain, however, the boundary is made up of 1D lines, which makes the unit of become . In 1D, the boundary is represented by a dimensionless point, and the heat flux has a unit of .

The mass conservation within the domain can be described by the following equation:

That is, the rate of the mass change is equal to the net rate of mass formation inside the domain minus the net mass flux that exits the domain.

Here the mass flux can be divided into two parts: a diffusive mass flux and a convection mass flux . The diffusion flux describes the mass transport due to a concentration gradient in the mass species concentration and is proportional to its diffusivity :

Equation (1) is also known as Fick's law of diffusion [2]. The minus sign states that the mass diffusion is in the direction of decreasing concentration.

The convection flux denotes the mass transported by a flow of the medium and is proportional to the flow velocity :

If the mass transport occurs in a solid medium, then, because a solid can not have an internal velocity field by definition, the convection term is set to .

Inserting (3) and (4) into the mass conservation equation (5) yields the conservative mass transport model.

PDE for conservative mass transport model:

When expressing (6) in the coefficient form of PDEs: , one has to be careful about the sign of the convection term .

Conservative mass transport model expressed in the PDE coefficient form:

Equation (7) is known as the conservative mass balance equation since the convective mass flux term, , and with it the flow velocity is inside of the divergence operator.

To obtain the non-conservative formulation equation (8) is expanded using the chain rule:

This results in the terms and . For an incompressible fluid, the divergence of the flow velocity is zero () by definition, and thus the term equals zero, resulting in the non-conservative mass balance equation (9).

Non-conservative PDE for mass transport model:

If possible, it is recommended to use the non-conservative form for incompressible fluids when solving mass transport models. The two main reasons are:

The removal of the term ensures that the incompressible flow field will not induce any nonphysical mass production or consumption into the source term .

The mass convection term (10) is outside of the divergence operator, which prevents the growth of spurious oscillation of the solution field [11] and makes the PDE model more stable and efficient to solve.

However, the non-conservative formulation is only applicable for incompressible fluids where the concentration field is expected to be smooth. For mass transport in a compressible medium, the conservative formulation (12) should be used instead. Note that when the fluid flow velocity is zero, both models simplify to the same PDE.

This tutorial focuses on incompressible fluid media, and the non-conservative mass balance equation (13) will be used as the default PDE setup for the mass transport model. However, the later section Boundary Conditions in Mass Transport presents the boundary condition setup for both the conservative and the non-conservative formulations.

The mass balance equation may also be expressed in cylindrical and spherical coordinates. The details can be found in the appendix section Special Cases of the Mass Balance Equation.

The next section sets up the mass transport model function in the Wolfram Language.

Mass Transport Model Setup

The inputs needed for a mass transport model in Cartesian coordinates are

		the concentration variable
		the spatial independent variables
		a velocity field
		the species diffusivity
		the reaction rate (also known as the rate of mass formation)

Set up a 1D static mass transport model in the default non-conservative form:

Set up a 1D static mass transport model in conservative form:

Set up a 1D time-dependent heat transfer model in the default non-conservative form:

Set up a 1D time-dependent heat transfer model in the conservative form:

Note that this model definition uses inactive PDE operators. The tutorial "Numerical Solution of Partial Differential Equations" has several sections that explain the use of inactive operators.

In the next few sections, the mass transport model will be discussed in more detail. Various types of the usage of the mass transport PDE will be presented with examples.

Initial Mass Transport Example

The purpose of the 2D stationary example below is to demonstrate a typical workflow of mass transport modeling.

For this, consider a parallel-plate reactor with a species solved in water. As the water flow passes through the reactor, part of the species will be absorbed by the active surface located at the right side. The goal is to find the steady-state concentration field for the species within the reactor.

To make the example concrete, use a width of and a height of for the model domain , and set the absorption rate on the active surface at the right shown in light gray. The left-side wall shown in black is assumed to be impermeable by the species and thus has no mass flux across it.

Set up the model domain

Set up model variables and an empty parameter association:

This example uses an arbitrary mass diffusion coefficient.

Set up a mass diffusion coefficient:

In this example, the water is flowing through the reactor at an average velocity of . The flow velocity field within the reactor is determined by the laminar profile:

Specify the flow velocity field within the domain:

Next, the boundary conditions at each boundary are set up. The details of setting up mass transfer boundary conditions will be explained in detail in the section Boundary Conditions in Mass Transport.

At the flow inlet, the species concentration is held fixed at , as if there is an infinite supply of the species below the inlet.

Set up a concentration boundary condition at the flow inlet:

On the active surface, the species is continuously absorbed at a given rate of . This is modeled by a mass flux boundary condition.

Set up a mass flux boundary condition on the active surface

Since the left-side wall is impermeable by the species , an impermeable boundary condition is used to model the boundary where there is no mass flux across it.

Set up an impermeable boundary condition at the flow outlet

A default outflow boundary condition is implicitly applied at the top boundary to model the flow outlet where the species is transported out of the domain.

Next, solve the mass transport PDE model with a prescribed flow field and an arbitrarily chosen species diffusivity of .

Solve the 2D static mass transport PDE:

Note that the argument "NoReaction" is used to denote that there is no internal reaction within the domain.

Visualize the steady-state concentration field

In this example, the species is exclusively absorbed on the active surface at the right boundary at a rate of , resulting in a layer-like concentration field.

The Catalytic Converter application model is a time-dependent mass transport mode.

Mass Transport with a Chemical Reaction

In the previous section, the chemical species did not undergo a chemical reaction. In the following section, the modeling of chemical reactions is explained.

The term in the mass balance equations (14) and (15) is used to model a mass formation () or consumption rate () within the domain. Typical phenomena that lead to a mass formation or consumption are chemical reactions. As an example, consider the following 2D transient model:

The model domain has a internal rectangular region where an unimolecular reaction occurs. A unimolecular reaction happens when a chemical species undergoes a self-transformation into a single product :

This example assumes the reaction to be a first-order reaction [16]; that is, the mass consumption rate of the reactant will be linearly proportional to the concentration of the reactant itself:

Here, is the reaction rate constant and has a unit of . For most chemical reactions, the rate constant is temperature dependent and can be computed by the Arrhenius equation [17]. In this example, however, a constant is chosen for the sake of simplicity.

Typically, applications where the reaction rate is temperature dependent are modeled as coupled equations where a heat equation is linked with a mass transport equation. An example of the setup of and modeling with a temperature-dependent reaction rate can be found in the multiphysics application model Thermal Decomposition.

Define the model domain and a rectangular reacting region:

Set up model variables and a diffusion coefficient parameter:

Specify the rate constant

and the mass consumption rate

with a RegionMember function:

Note that by using exact numbers for the region, the region membership test is simplified to an If statement.

To highlight the effect of the chemical reaction, assume that there is no moving flow within the domain, which means that the mass transport is by diffusion only.

Define the mass transport PDE with the mass consumption rate

, an initial concentration field

and a species diffusivity

Solve the concentration field of the species

for

seconds:

Visualize the time-dependent concentration field

of the species

See this note about improving the visual quality of the animation.

The species has an initial concentration field of and is gradually consumed by the reaction within the region . The excess species outside the region then fills in by diffusion due to the concentration gradient, which brings down the overall concentration within the model domain.

An application example that shows a more evolved reaction model is given in the Microscale Simulation of Catalyst Deactivation model. A further model is the Gas Absorption at Liquid Surface model.

Anisotropic and Orthotropic Mass Diffusion

In previous sections, the assumption was that the species is transported in an isotropic medium; that is, the rate of the diffusive mass transfer is independent of its direction and given by the same concentration gradient . In reality, however, a medium may be anisotropic. Anisotropic diffusion means that the species diffuses in different directions with a different rate. Orthotropic diffusion is a special case of anisotropic mass diffusion that will be described below. Anisotropic behavior can be found all the way from natural products to sophisticated composite materials and as such, is an important phenomenon to be able to model.

Since the species diffusivity varies in different directions, the diffusion flux term (18) is then rewritten as:

or more concisely as:

with the symmetric matrix:

is the diffusivity tensor. and are called the principle diffusion coefficients and off-diagonal diffusion coefficients, respectively.

With orthotropic diffusion, the species diffusivity of a material is symmetric but distinct along the principal directions , and , such as shown in the graphics below. All the off-diagonal diffusion coefficients are zero. This behavior can be seen, for example, in fiber composite materials. Then the diffusion tensor becomes:

As an example, consider a 2D composite material with a layered-like structure. The initial concentration of the species can be described by a Gaussian bell shape centered at . The species then gradually diffuses throughout the entire domain.

The illustration above depicts the orthotropic case where the species diffuses slower in the vertical direction, where it passes across the internal boundaries of the layered structure shown as gray lines. The diffusivity tensor can be described by:

Set the diffusion tensor

of the orthotropic material:

Set up the initial concentration field

Solve the mass transport PDE model:

Visualize the time-dependent concentration field

of the species

See this note about improving the visual quality of the animation.

The species is transported faster in the horizontal direction, resulting in a higher concentration zone within the band between .

Variable Mass Diffusion Coefficient

In the preceding sections, the assumption was that the mass diffusion coefficient and hence the species diffusivity are either a constant scalar for an isotropic medium or a constant tensor for an anisotropic medium. In reality, however, the diffusivity might change significantly with temperature and the species concentration . Both phenomena will be examined in turn.

Temperature dependence of the diffusion coefficient

When modeling mass transport in a medium with a nonuniform temperature field, it is better to account for the temperature dependence of the diffusion coefficient . As a starting model, the diffusion coefficient can be predicted by Arrhenius's equation:

is the species diffusivity ,
is the maximum species diffusivity at infinite temperature ,
is the absolute temperature ,
is the activation energy for diffusion ,
is the gas constant .

The exponential form of this relation means that the species diffusivity can grow quickly with temperature .

As an example, consider a 1D domain with a prescribed temperature field increase from left to right as .

Set the maximum diffusivity

, the activation energy

and the gas constant

Set the temperature-dependent species diffusivity with a prescribed temperature field

At time , the species is confined around the center and can be described by a Gaussian distribution.

Set up the initial concentration field

by a Gaussian distribution centered at

Solve the time-dependent mass transport PDE model for

seconds:

Visualize the time-dependent concentration field

of the species

At time , the species is confined around the center and then gradually diffused throughout the domain. Note that the species diffused faster to the right side due to the higher temperature in that region.

In this example, the temperature field is prescribed. However, to solve a mass transport model with an unknown temperature field, a multiphysics model has to be constructed that couples a heat transfer model with a mass transport model. Such a coupled heat transfer mass transport multiphysics application model is shown in the model collection in the Thermal Decomposition example.

Concentration dependence of the diffusion coefficient

For some mass transport applications such as drug delivery in blood and pollutants emission in air, the transported species may have unique interaction with the molecules of the medium [19, 20]. In such cases, the diffusion coefficient will be affected by the concentration field of the transported species, leading to a nonlinear mass balance equation. The non-conservative nonlinear mass balance equation is given as:

The diffusion coefficient in the PDE model is now a function of the concentration itself. The expression of the nonlinear diffusion coefficient is usually determined experimentally and depends on the type of transported species and the medium.

As an example, consider a 1D mass transport model with an initial concentration field of , a reference diffusion coefficient and a concentration-dependent diffusion coefficient . In this case, the species diffusivity will increase with higher concentration .

To model the supply of species into the domain, a constant mass flux is applied at the left-hand boundary.

Set up model variables and a diffusion coefficient parameter:

Set up a mass flux boundary condition at the left end with NeumannValue:

Solve the nonlinear mass transport PDE with a concentration-dependent diffusion coefficient

, an initial concentration field

and a reference diffusivity

To better understand the effects of the nonlinearity, compare to a linear mass transport PDE.

Set up and solve a linear mass transport PDE with a constant diffusion coefficient

Make an animation of the solutions using Plot and ListAnimate:

In both the linear and nonlinear model, the species entered the domain from the left with a constant mass flux . However, for the nonlinear model, the diffusion coefficient will increase with the concentration , which further speeds up the mass transport and results in a flatter concentration field.

Interphase Mass Transfer

In the preceding sections, the mass transfer by mass convection and mass diffusion in a single phase has been considered. However, there are many situations in nature where two different phases are in contact, and mass may be transported from one phase to the other. This process is known as interphase mass transfer.

Unlike the single-phase mass transfer where the concentration field is always continuous, the species concentration can jump at the interface between two states as follows:

Here and are the concentration of species dissolved in phase and phase . The subscription denotes the species concentration at the interface between two phases.

A separate mass balance equation should be used to describe the species concentration in phase and in the phase :

At the interface of two phases, the concentration may be discontinuous between and . The ratio is known as the equilibrium distribution coefficient :

The coefficient depends on pressure, temperature and the chemical properties of the transported species, as well as the media of both phases. The value can be determined by experimental measurement [21].

Assuming the molecules of species can penetrate quickly through the interface of two phases, the equilibrium condition (22) at the interface will be reached instantaneously and is satisfied at all times.

This assumption was first proposed by Whitman [23] as the two-resistance theory and is valid for most cases of the interphase mass transfer. However, in case a chemical reaction occurs at the interface, additional repulsive or attractive forces may act on the transported species, and then this assumption will no longer be valid.

Consider a 1D model where sulfur dioxide, , is transferred from air into water . To model the interphase mass transfer, a thin interphase region will be defined between the two different phases, which will allow us to enforce the equilibrium condition (24) via a coupled fictitious mass sources and handle the discontinuity of the concentration at .

Here the thickness of the interphase region is set to be of the length of the domain. Note that if is set too small, a numerical instability may occur around the interface.

Create a boundary ElementMesh with a thin interphase region between

Next, a full element mesh with internal markers is defined to distinguish the gas, liquid and interphase regions. For clearer assignment of markers, an association is used.

Create a full ElementMesh with internal markers of the gas, liquid and interphase regions:

Set up coupled model variables and an empty parameter association:

The diffusion coefficients of in the gas and liquid phases are given by and , respectively. Note that and are only valid within the interphase region as well as their own phase.

Specify the

diffusivity in the gas and liquid phases:

To model the transfer of between two phases, add the coupling mass source terms and in the governing mass balance equation (25), leading to:

Here and are set at , so that the equilibrium condition (26) can be enforced at the interface. In the equation, is the interphase mass transfer coefficient, and is a switch that turns on within the interphase region and off otherwise.

Set a factor

to switch on within the interphase region:

Based on the two-resistance theory mentioned above, the equilibrium at the interface is considered to be reached instantaneously and maintained at all times. This condition can be modeled by setting the mass transfer coefficient to be infinitely large.

In practice, can be chosen to be greater than the species diffusivity and ) by several orders of magnitude.

Set a large value of

to model an instantaneous mass transfer across the interphase region:

Set up the coupling source terms

and

Note that due to the mass conservation, the coupling source terms and have the same magnitude but opposite sign.

To model the supply of into the domain, a constant mass flux is applied at the left-hand boundary of the air region.

Set up a mass flux boundary condition at the left end with NeumannValue:

Define the interphase mass transport PDE with an initial concentration field

Define two mass transport models to describe

concentration

in the gas phase and

in the liquid phase:

To study the effect of the equilibrium coefficient on the interphase mass transfer, the PDE model is solved repetitively with ParametricNDSolveValue at , and :

Solve the interphase mass transport model with an equilibrium coefficient

and

Visualize the

concentration in the air and water regions:

Note that as the equilibrium coefficient deviates from , the jump between and increases.

The strategy presented in this section can also be applied in modeling mass transfer across a membrane. In that case, however, an additional repulsive force (i.e. surface resistance) may play a role, and the time for the membrane to reach the equilibrium condition (27) should also be considered. This can be done by selecting a value in (28) based on the surface resistance of the membrane.

Mass Source Types

Mass sources like volumetric, layer and point sources can be modeled in a similar fashion, as explained in the corresponding Heat Transfer model section.

Multiphysics Mass Transport

Mass transport can be combined with other fields of physics. What follows is a list of multiphysics application examples that make use of mass transport:

Fluid Flow

Water Condensation of Passive Dew Condensers

Heat Transfer

Radial Effects in a Tubular Reactor

Thermal Decomposition

Water Condensation of Passive Dew Condensers

Solid Mechanics

Hygroscopic Swelling

Boundary Conditions in Mass Transport

The most common boundary conditions in mass transport modeling can be modeled with DirichletCondition, NeumannValue and PeriodicBoundaryCondition and can be categorized in the following four types:

Dirichlet-type boundary conditions. This type of boundary condition specifies the species concentration at the boundary and can be modeled with DirichletCondition.

Neumann-type boundary conditions. This type of boundary condition specifies the mass flux at the boundary and can be modeled with NeumannValue.

Robin-type boundary conditions. This type of boundary condition specifies the relation between the species concentration and its normal derivatives at the boundary and can modeled with a NeumannValue, since Robin-type boundary conditions are technically generalized Neumann boundary conditions.

Periodic boundary conditions. This type of boundary condition specifies the species concentration at one part of the boundary to be the same at another part and can be modeled with PeriodicBoundaryCondition.

Under these four types, the following boundary conditions are introduced:

Dirichlet type

Concentration Boundary Conditions

Neumann type

Mass Flux Boundary Conditions

Impermeable Boundary Conditions

Symmetric Boundary Conditions

Outflow Boundary Conditions

Robin type

Surrounding Flux Boundary Conditions

Periodic type

Periodic Boundary Conditions

The following section describes several physical boundary conditions common in modeling mass transport and how they can be represented with the use of DirichletCondition, NeumannValue and PeriodicBoundaryCondition. For this purpose, the boundary condition currently discussed is always at the left-hand side of the example simulation domain. In some examples, additional boundary conditions are at the right-hand side to better demonstrate the behavior of the boundary condition at the left-hand side.

Note that the setup of Neumann-type boundary conditions will be slightly different between the conservative model and non-conservative model. The details of this difference are presented in the following section.

Neumann Values for Conservative and Non-conservative Models

In a previous section, the derivation and setup of the conservative and non-conservative mass transport model was presented. The difference of the model formulation has an impact on how Neumann-type boundary conditions are to be set up and what they mean.

With the conservative formulation, the NeumannValue specifies the boundary values for . With NeumannValue[val,X∈Γ_b], this means:

With the non-conservative formulation, the NeumannValue specifies the boundary values for . With NeumannValue[val,X∈Γ_b], this means:

Note that for the conservative formulation, the term: , which is the mass transported by a flow of the medium across the boundary, is also included in the Neumann value.

In some of the boundary conditions presented, the form of the Neumann value needs to changed. This means that, for example, a Neumann value in the conservative form like (29) will have to be transformed into:

Transformations of this sort pose a problem, as usually the normal in a Neumann formulation like the left side of (30) is never actually computed but replaced with the value of the right-hand side of (31). Once a normal appears on the right-hand side of the equation, that value of the normal will have to be computed, and this is done automatically. If the boundary unit normal is simple, it can be specified with the model parameter "BoundaryUnitNormal". This may increase computational efficiency both in speed and memory usage.

Before the mass transport boundary conditions for both the conservative model and non-conservative model are presented, some example model parameters are set up.

Model Parameter Setup

The following model parameters are made use of in the examples of mass transport boundary conditions. These parameters define the simulation domain and the simulation end time .

Set up model parameters for the domain

and the simulation end time

In some examples, a smoothed step function will be used to prescribe a time profile for a transient parameter, such as, for example, the mass flux or the concentration on the boundary. The smoothed step function is defined as follows:

Here, the minimum value and the maximum value of the function can reach are denoted by and . The location of the step is controlled by , and the smoothed steepness is controlled by .

Define and visualize the smoothed step function

Concentration Boundary Condition

Purpose

The purpose of a concentration boundary condition is to set a specific species concentration on some part of the boundary.

Formulation

With a specified concentration on the boundary , the concentration boundary condition is given for both conservative and non-conservative models by:

Since the concentration boundary condition is a Dirichlet-type condition and thus is not associated with the model equation, its formulation is the same for both the conservative model and non-conservative model.

A concentration boundary for the dependent variable

modeled with DirichletCondition:

Derivation

A given species concentration is prescribed on a boundary. This prescribed concentration can either be a constant or time-dependent value and is set with a DirichletCondition in the mass transport PDE model.

Example

To model, for example, the supply of a given species on a boundary, a transient species concentration can be set up at the left end. Note that a Neumann zero condition is implicitly applied at the right end as an outflow boundary.

Here a smoothed step function is used to describe the profile of the species concentration on the boundary from to . The parameters and are arbitrarily chosen to simulate the supply of species from the boundary.

Specify the species concentration

on the boundary using a smoothed step function:

Set up the concentration boundary condition with DirichletCondition:

Next, set up the mass transport model with a flow field , an initial concentration field and a diffusion coefficient .

Set up the transient mass transport PDE:

Solve the PDE with NDSolveValue:

Visualization

Make an animation of the solution using Plot and ListAnimate:

The simulation begins with an undisturbed domain where . As the concentration increases at the left boundary, the excess species are then transported to the right and bring up the overall species concentration within the domain. The speed of the mass transport depends on the species diffusivity and the flow field.