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DUNE for Multi-{Phase, Component, Scale, Physics, ...} flow and transport in porous media
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Dumux::DimensionlessNumbers< Scalar > Class Template Reference

Collection of functions which calculate dimensionless numbers. Each number has it's own function. All the parameters for the calculation have to be handed over. Rendering this collection generic in the sense that it can be used by any model. More...

#include <dumux/common/dimensionlessnumbers.hh>

Description

template<class Scalar>
class Dumux::DimensionlessNumbers< Scalar >

Collection of functions which calculate dimensionless numbers. Each number has it's own function. All the parameters for the calculation have to be handed over. Rendering this collection generic in the sense that it can be used by any model.

Static Public Member Functions

static Scalar reynoldsNumber (const Scalar darcyMagVelocity, const Scalar charcteristicLength, const Scalar kinematicViscosity)
 Calculate the Reynolds Number [-] (Re). More...
 
static Scalar prandtlNumber (const Scalar dynamicViscosity, const Scalar heatCapacity, const Scalar thermalConductivity)
 Calculate the Prandtl Number [-] (Pr). More...
 
static Scalar nusseltNumberForced (const Scalar reynoldsNumber, const Scalar prandtlNumber, const Scalar porosity, NusseltFormulation formulation)
 Calculate the Nusselt Number [-] (Nu). More...
 
static Scalar schmidtNumber (const Scalar dynamicViscosity, const Scalar massDensity, const Scalar diffusionCoefficient)
 Calculate the Schmidt Number [-] (Sc). More...
 
static Scalar sherwoodNumber (const Scalar reynoldsNumber, const Scalar schmidtNumber, SherwoodFormulation formulation)
 Calculate the Sherwood Number [-] (Sh). More...
 
static Scalar thermalDiffusivity (const Scalar &thermalConductivity, const Scalar &phaseDensity, const Scalar &heatCapacity)
 Calculate the thermal diffusivity alpha [m^2/s]. More...
 

Member Function Documentation

◆ nusseltNumberForced()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::nusseltNumberForced ( const Scalar  reynoldsNumber,
const Scalar  prandtlNumber,
const Scalar  porosity,
NusseltFormulation  formulation 
)
inlinestatic

Calculate the Nusselt Number [-] (Nu).

     The Nusselt Number is a measure for the relation of convective- to conductive heat exchange.

     The Nusselt number is defined as Nu = h d / k,
     with h= heat transfer coefficient, d=characteristic length, k=heat conductivity(stagnant).
     However, the heat transfer coefficient from one phase to another is typically not known.
     Therefore, Nusselt numbers are usually given as *empirical* Nu(Reynolds, Prandtl) for a given flow
     field --forced convection-- and *empirical* Nu(Rayleigh, Prandtl) for flow caused by temperature
     differences --free convection--. The fluid characteristics enter via the Prandtl number.

     This function implements an *empirical* correlation for the case of porous media flow
     (packed bed flow as the chemical engineers call it).

     source for Nusselt number definition: http://en.wikipedia.org/wiki/Nusselt_number
     source for further empirical correlations for Nusselt Numbers:
     VDI-Gesellschaft, VDI-Waermeatlas, VDI-Verlag Duesseldorf, 2006
Parameters
reynoldsNumberDimensionless number relating inertial and viscous forces [-].
prandtlNumberDimensionless number relating viscosity and thermal diffusivity (temperaturleitfaehigkeit) [-].
porosityThe fraction of the porous medium which is void space.
formulationSwitch for deciding which parametrization of the Nusselt number is to be used. Set via the property NusseltFormulation.
Returns
The Nusselt number as calculated from the input parameters [-].

◆ prandtlNumber()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::prandtlNumber ( const Scalar  dynamicViscosity,
const Scalar  heatCapacity,
const Scalar  thermalConductivity 
)
inlinestatic

Calculate the Prandtl Number [-] (Pr).

     The Prandtl Number is a measure for the relation of viscosity and thermal diffusivity (temperaturleitfaehigkeit).

     It is defined as
     \f[
     \textnormal{Pr}= \frac{\nu}{\alpha} = \frac{c_p \mu}{\lambda}\, ,
     \f]
     with kinematic viscosity\f$\nu\f$, thermal diffusivity \f$\alpha\f$, heat capacity \f$c_p\f$,
     dynamic viscosity \f$\mu\f$ and thermal conductivity \f$\lambda\f$.
     Therefore, Pr is a material specific property (i.e.: not a function of flow directly
     but only of temperature, pressure and fluid).

     source for Prandtl number definition: http://en.wikipedia.org/wiki/Prandtl_number
Parameters
dynamicViscosityDynamic (absolute) viscosity over density. http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s]
heatCapacityHeat capacity at constant pressure. Specifies the energy change for a given temperature change [J / (kg K)]
thermalConductivityConductivity to heat. Specifies how well matter transfers energy without moving. [W/(m K)]
Returns
The Prandtl Number as calculated from the input parameters.

◆ reynoldsNumber()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::reynoldsNumber ( const Scalar  darcyMagVelocity,
const Scalar  charcteristicLength,
const Scalar  kinematicViscosity 
)
inlinestatic

Calculate the Reynolds Number [-] (Re).

The Reynolds number is a measure for the relation of inertial to viscous forces. The bigger the value, the more important inertial (as compared to viscous) effects become. According to Bear [Dynamics of fluids in porous media (1972)] Darcy's law is valid for Re<1.

Source for Reynolds number definition: http://en.wikipedia.org/wiki/Reynolds_number

Parameters
darcyMagVelocityThe absolute value of the darcy velocity. In the context of box models this leads to a problem: the velocities are defined on the faces while other things (storage, sources, output) are defined for the volume/vertex. Therefore, some sort of decision needs to be made which velocity to put into this function (e.g.: face-area weighted average). [m/s]
charcteristicLengthTypically, in the context of porous media flow, the mean grain size is taken as the characteristic length for calculation of Re. [m]
kinematicViscosityIs defined as the dynamic (or absolute) viscos ity divided by the density. http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity. [m^2/s]
Returns
The Reynolds Number as calculated from the input parameters

◆ schmidtNumber()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::schmidtNumber ( const Scalar  dynamicViscosity,
const Scalar  massDensity,
const Scalar  diffusionCoefficient 
)
inlinestatic

Calculate the Schmidt Number [-] (Sc).

     The Schmidt Number is a measure for the relation of viscosity and mass diffusivity.

     It is defined as
     \f[
     \textnormal{Sc}= \frac{\nu}{D} = \frac{\mu}{\rho D}\, ,
     \f]
     with kinematic viscosity\f$\nu\f$, diffusion coefficient \f$D\f$, dynamic viscosity
     \f$\mu\f$ and mass density\f$\rho\f$. Therefore, Sc is a material specific property
     (i.e.: not a function of flow directly but only of temperature, pressure and fluid).

     source for Schmidt number definition: http://en.wikipedia.org/wiki/Schmidt_number
Parameters
dynamicViscosityDynamic (absolute) viscosity over density. http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s]
massDensityMass density of the considered phase. [kg / m^3]
diffusionCoefficientMeasure for how well a component can move through a phase due to a concentration gradient. [m^2/s]
Returns
The Schmidt Number as calculated from the input parameters.

◆ sherwoodNumber()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::sherwoodNumber ( const Scalar  reynoldsNumber,
const Scalar  schmidtNumber,
SherwoodFormulation  formulation 
)
inlinestatic

Calculate the Sherwood Number [-] (Sh).

     The Sherwood Number is a measure for the relation of convective- to diffusive mass exchange.

     The Sherwood number is defined as Sh = K L/D,
     with K= mass transfer coefficient, L=characteristic length, D=mass diffusivity (stagnant).

     However, the mass transfer coefficient from one phase to another is typically not known.
     Therefore, Sherwood numbers are usually given as *empirical* Sh(Reynolds, Schmidt) for a given flow
     field (and fluid).

     Often, even the Sherwood number is not known. By means of the Chilton-Colburn analogy it can be deduced
     from the Nusselt number. According to the Chilton-Colburn analogy in a known Nusselt correltion one
     basically replaces Pr with Sc and Nu with Sh. For some very special cases this is actually accurate.
     (Source: Course Notes, Waerme- und Stoffuebertragung, Prof. Hans Hasse, Uni Stuttgart)

     This function implements an *empirical* correlation for the case of porous media flow
     (packed bed flow as the chemical engineers call it).

     source for Sherwood number definition: http://en.wikipedia.org/wiki/Sherwood_number
Parameters
schmidtNumberDimensionless number relating viscosity and mass diffusivity [-].
reynoldsNumberDimensionless number relating inertial and viscous forces [-].
formulationSwitch for deciding which parametrization of the Sherwood number is to be used. Set via the property SherwoodFormulation.
Returns
The Nusselt number as calculated from the input parameters [-].

◆ thermalDiffusivity()

template<class Scalar >
static Scalar Dumux::DimensionlessNumbers< Scalar >::thermalDiffusivity ( const Scalar &  thermalConductivity,
const Scalar &  phaseDensity,
const Scalar &  heatCapacity 
)
inlinestatic

Calculate the thermal diffusivity alpha [m^2/s].

     The thermal diffusivity is a measure for how fast "temperature (not heat!) spreads".
     It is defined as \f$\alpha = \frac{k}{\rho c_p}\f$
     with \f$\alpha\f$: \f$k\f$: thermal conductivity [W/mK], \f$\rho\f$: density [kg/m^3],
     \f$c_p\f$: cpecific heat capacity at constant pressure [J/kgK].

     Source for thermal diffusivity definition: http://en.wikipedia.org/wiki/Thermal_diffusivity
Parameters
thermalConductivityA material property defining how well heat is transported via conduction [W/(mK)].
phaseDensityThe density of the phase for which the thermal diffusivity is to be calculated [kg/m^3].
heatCapacityA measure for how a much a material changes temperature for a given change of energy (at p=const.) [J/(kgm^3)].
Returns
The thermal diffusivity as calculated from the input parameters [m^2/s].

The documentation for this class was generated from the following file: