Simulation > Creo Flow Analysis > Preprocessing > Defining Physics > Multicomponent Mixing > Physics
Physics
For a multicomponent flow, you solve the scalar transport equations for mixture velocity, pressure, temperature, turbulence, and other physical quantities. When multiple components are present, you must solve additional equations to determine how the components are transported within the fluid mixture.
Description of the Multiple Species (Component)
There are several different, but related variables to quantify the content of a component in component flow:
molar concentration of component
mass concentration of component
molar fraction of component
mass fraction of component
The four quantities are related as follows:
Equation 2.314
Equation 2.315
Equation 2.316
where,
molecular weight of component
mixture density
sum of the molar concentrations of all the components in a system:
and
Equation 2.317
where is the mixture molecular weight:
Equation 2.318
Equation 2.317 indicates that with mass-fraction weighted molecular weight for the mixture, equation 2.314 also applies to the mixture of the components.
Also, from the definitions in equation 2.315 and equation 2.316, the sum of both the molar and mass fractions must be unity:
Equation 2.319
In CFA solvers, you obtain the mass fraction of the arbitrary component , directly by solving partial differential transport equations. The other variables, , , and are auxiliary variables that you use for postprocessing.
Governing Equations
In a multicomponent flow, the bulk motion of the mixture is modeled using the single velocity, pressure, temperature, and turbulence fields. For the mixing and transport of the chemical species, each component has its own governing equation for the conservation of mass. The influence of multiple components on the bulk flow is felt through the variation of the mixture properties like density, viscosity, with the component properties and the local mass fractions.
• Mass Fraction Equations
For component mixture flow, if there are no chemical reactions, then transport of an arbitrary component is governed by the following equation:
Equation 2.320
where,
and
mixture density and velocity
any user-defined source
mass diffusion term
For laminar flows, the velocity vector and the mass fraction are instantaneous variables. For turbulent flows these velocity vectors are Favre-averaged quantities since multicomponent flows are considered as variable density or compressible flows.
In equation 2.320, the mixture quantities and the mass diffusion term are defined as follows:
◦ Mixture Density—Mass-averaged value of all the component densities:
Equation 2.321
For a mixture of gaseous species, the mixture density is computed using the ideal gas law based on the mixture molecular weight , that you calculate using equation 2.318:
Equation 2.322
where,
universal gas constant
mixture temperature
absolute pressure
If you use the operating pressure (constant), then equation 2.322 is reduced to the so-called incompressible ideal gas law. It is an appropriate assumption for the mixing and transport of species, where the gauge pressure is often negligible compared to the operating pressure.
◦ Mixture Velocity—Mass-averaged value of all the component velocities:
Equation 2.323
However, since only a single velocity is solved, you assume that the mixture velocity and all the component velocities have the same values.
◦ Mass Diffusion Flux—Mass diffusion flux of the component consists of two parts: the laminar and turbulent diffusion terms, which are expressed as:
Equation 2.324
In equation 2.324, is the laminar diffusion flux of the component which arises due to gradients of concentration and temperature. By default, Creo Flow Analysis uses the dilute approximation or Fick's law to model the mass diffusion due to concentration gradients. The laminar diffusion flux has the following formulation:
Equation 2.325
where is the mass diffusion coefficient for the component in the mixture; and is the thermal (Soret) diffusion coefficient.
For turbulent flows, the fluctuating term derived from Favre averaging the advection in equation 2.320, is modeled as turbulent diffusion:
Equation 2.326
where,
turbulent viscosity
turbulent Schmidt number
by default
Turbulent diffusion generally overwhelms the laminar diffusion. Specification of detailed laminar diffusion properties in turbulent flows is generally less important than the turbulent counterpart.
To derive the mass continuity equation for the mixture flow, add all the component mass fraction equations and apply equation 2.319:
Equation 2.327
To satisfy the total mass conservation of the mixture flow, the sum of the diffusion terms for all components must be zero,
Equation 2.328
From equation 2.319 and equation 2.326, the turbulent diffusion term is always determined at zero. Therefore, for fully turbulent flows, you usually consider equation 2.328 automatically satisfied. However, for laminar flows, or when you cannot ignore the laminar mass diffusion in turbulent flows, equation 2.328 reduces to the following form:
Equation 2.329
Then to satisfy equation 2.329, you apply the two separate constraints:
Equation 2.330
Equation 2.331
The continuity equation of the multicomponent flows then has the final form:
Equation 2.332
• Diffusion Coefficients
To solve the transport equation 2.320 for multicomponent laminar flows, you require the mass diffusion coefficient and the thermal diffusion coefficient for each component in a mixture. The methods to determine and are the following:
◦ Mass Diffusion Coefficients—Formulation of the mass diffusion flux in laminar flows, equation 2.325, is strictly valid when the mixture composition is not changing, or when is independent of the composition. This is an acceptable approximation in dilute mixtures when is very small for all the components except for the carrier gas. For nondilute mixtures in multicomponent laminar flows, you calculate from the following formulation:
Equation 2.333
where is the binary mass diffusion coefficient of component in component , which you need to specify or calculate.
◦ Specified Value—Binary mass diffusion coefficient is a constant or function of temperature if heat transfer is accounted for. You can specify the value directly or obtain it from the specified Schmidt number:
Equation 2.334
where,
Schmidt number
Schmidt number is defined as the ratio of the viscous diffusion rate to the molecular (mass) diffusion rate.
If one value or one function of temperature applies for all the components, equation 2.333 is reduced to
Equation 2.335
Equation 2.335 is an appropriate approximation for modeling a dilute mixture, with the species present at low mass fractions in a carrier fluid that has high concentration. In such cases, you define directly as a constant or a function of temperature.
However, for non-dilute mixtures, with the specified , you use equation 2.333 to compute the individual mass diffusion coefficient in the mixture .
◦ Kinetic Theory—For an ideal gas, the binary mass diffusion coefficient can also be obtained using kinetic theory.
References: H. A. McGee, “Molecular Engineering”, McGraw-Hill, New York, 1991.
Equation 2.336
where is the absolute pressure, and is the diffusion collision integral, which is a measure of the interaction of the molecules in the system. is a function of the quantity , defined as:
Equation 2.337
is the Boltzmann constant, which is defined as the universal gas constant divided by the Avogadro number. for the mixture is the geometric average:
Equation 2.338
For a binary mixture, is calculated as the arithmetic average of the individual and :
Equation 2.339
and are the Lennard-Jones parameters for component in the mixture. Specifically, is the collision cross section of the sphere molecule with the diameter (note that a molecule sweeps out an area given by twice its diameter, as the molecules with which it collides also have diameter ); and =1.38064852(79) ×10-23(J/K) is the Boltzmann constant.
In Creo Flow Analysis, you specify the diameter and the energy to determine the two Lennard-Jones parameters.
◦ Thermal Diffusion Coefficients —Thermal Diffusion Coefficients can be defined as constants, polynomial functions of temperature, user-defined functions, or using the following empirically-based composition-dependent expression derived from:
References: K. K. Y. Kuo, “Principles of Combustion”, John Wiley and Sons, New York, 1986.
Equation 2.340
This form of the thermal diffusion coefficient causes heavy molecules to diffuse less rapidly, and light molecules to diffuse more rapidly, towards heated surfaces.
• Momentum Equations
With the mass-weighted properties and velocities, the momentum equations for the mixture of all the components have the same expression as those for single fluid flows:
Equation 2.341
where the mixture density and velocity are calculated using equation 2.321, equation 2.322, and equation 2.323. The turbulent viscosity is directly computed from the turbulence models based on the mixture flow so that its value is independent of the components. For the laminar viscosity, it is computed as follows:
◦ Mass-Averaged Laminar Viscosity—For non-ideal gas mixtures, the mixture viscosity is calculated based on a mass fraction average of the pure chemical species (components) viscosities:
Equation 2.342
◦ Kinetic Theory—For ideal gas mixtures, the mixture viscosity is computed based on the kinetic theory. For each component, the dynamic viscosity is based on the Boltzmann equation:
Equation 2.343
For the mass diffusivity, you require the Lennard-Jones parameters, and to calculate the viscosities of the gas components in a mixture.
Viscosity for the ideal gas mixture is then calculated as:
Equation 2.344
where,
Equation 2.345
• Energy Equation
As described in the Heat module, the energy equation for the mixture of all the components is expressed as:
Equation 2.346
where and are the total internal energy and total enthalpy of the component mixture. Along with the mixture specific heat and static enthalpy , they are obtained by mass-averaging the corresponding values of each component:
◦ Mass-Averaged Mixture Heat Capacity
Equation 2.347
◦ Mass-Averaged Mixture Energy and Enthalpy
Equation 2.348
Equation 2.349
Equation 2.350
Static enthalpy of a component consists of two parts: standard state reference enthalpy and sensible enthalpy. For multicomponent flows, you include both parts of the enthalpy (the absolute or total value) when you calculate .
In equation 2.336, the first term on the right side represents the diffusion of the energy. It consists of three parts: heat conduction, energy transport due to diffusion of the species, and viscous heating. For the mixture heat conduction, it is modeled in the same way as in the single fluid flow. In Creo Flow Analysis the mixture heat conductivity is calculated as follows:
◦ Mass-Averaged Heat Conductivity—For non-ideal gas mixtures, the mixture heat conductivity is computed based on a simple mass fraction average of the pure species or components heat conductivities:
Equation 2.351
This is the default method in Creo Flow Analysis.
◦ Kinetic Theory—For ideal gas mixtures, the mixture heat conductivity can be computed based on kinetic theory. For each component, the heat conductivity has the form:
Equation 2.352
where,
universal gas constant
molecular weight
specified or computed viscosity of the component
specified or computed specific heat capacity of the component
Note that as the laminar viscosity, , the specific heat can also be obtained using kinetic theory:
Equation 2.353
where is the number of modes of energy storage (degrees of freedom) for the gas component .
The heat conductivity for the ideal gas mixture is then calculated as:
Equation 2.354
where is expressed in equation 2.335.
The second diffusion term,
Equation 2.355
represents the transport of enthalpy due to the diffusion of the chemical species in the component flow. This term can have a significant effect on the enthalpy field and should not be neglected. When the Lewis number, the ratio of thermal diffusivity to mass diffusivity :
Equation 2.356
for any species is not unity, neglecting this term can result in significant errors.
The third diffusion term is the viscous heating contribution . Though it is treated in the same way as in the single fluid flow, you calculate the shear using the mixture laminar and turbulent viscosities. The general source term is the total external or user heat source on all the components.
• Turbulence Models—With the mixture density , molecular viscosity , and velocity , the turbulence modeling equations in both the standard k-ε and RNG k-ε models have the same general forms as in the single fluid turbulence models. These are described in the Turbulence module. The turbulent viscosity for the mixture is computed directly from the expression:
Equation 2.357
Also, the production of turbulent kinetic energy is calculated based on mixture turbulent viscosity and velocity gradients.
Modeling of Multicomponent Boundaries
In a multicomponent flow, boundary conditions for the flow, energy, and turbulence modeling equations are the same as those in the single phase flows, described in the Flow, Heat, and Turbulence modules. For the mass fractions of a component, the boundary conditions consist of specified value, specified volumetric flux, and/or gradient.
• n-Component Inlet Boundary
At an inlet boundary, the net transport of a component can consist of both convection and diffusion contributions. The convection is determined by the specified inlet species mass fraction. The diffusion depends on the gradient of the computed mass fraction field. At very small convective inlet velocities, substantial mass can be gained or lost through the inlet due to diffusion. For this reason, the inlet diffusion is not included by default, but can be enabled as an option.
◦ Specified Value—For component flow, the inlet mass fractions are predetermined for components, while the mass fraction of the component is obtained using the physical constraint equation 2.319:
Equation 2.358
Equation 2.359
Also, the mass fraction for each component must be nonnegative.
◦ Specified Volumetric Flux—Assuming that is the predescribed inlet volumetric flux for component , you have the mass flux of each component and the total mass flux at the inlet as:
Equation 2.360
where is the inlet density of the component .
By definition, the mass fraction is computed as:
Equation 2.361
◦ Outlet, Symmetry, Wall Boundary—For components, zero gradient conditions apply for all the outlet, symmetry, and wall boundaries, while the phase is obtained using the physical constraint:
Equation 2.362
Equation 2.363
where is the boundary value obtained from equation 2.347.
Numerical Considerations
The above governing equations, turbulence models, and boundary conditions form the foundation of the multicomponent mixing model. Without external or user source terms and chemical reactions, they are a closed system of equations that you solve numerically using a pressure-based finite volume solver.
The mass-fraction transport equations are solved for all the components. To satisfy the physical constraint, the actual mass fractions are scaled by the sum of the solved values for all the components:
Equation 2.364
where is the value obtained from solving equation 2.320. The actual mass fraction is:
Equation 2.365