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

reynoldsNumber	Dimensionless number relating inertial and viscous forces [-].
prandtlNumber	Dimensionless number relating viscosity and thermal diffusivity (temperaturleitfaehigkeit) [-].
porosity	The fraction of the porous medium which is void space.
formulation	Switch 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

dynamicViscosity	Dynamic (absolute) viscosity over density. http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s]
heatCapacity	Heat capacity at constant pressure. Specifies the energy change for a given temperature change [J / (kg K)]
thermalConductivity	Conductivity 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

darcyMagVelocity	The 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]
charcteristicLength	Typically, in the context of porous media flow, the mean grain size is taken as the characteristic length for calculation of Re. [m]
kinematicViscosity	Is 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

dynamicViscosity	Dynamic (absolute) viscosity over density. http://en.wikipedia.org/wiki/Viscosity#Dynamic_viscosity [m^2/s]
massDensity	Mass density of the considered phase. [kg / m^3]
diffusionCoefficient	Measure 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

schmidtNumber	Dimensionless number relating viscosity and mass diffusivity [-].
reynoldsNumber	Dimensionless number relating inertial and viscous forces [-].
formulation	Switch 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

thermalConductivity	A material property defining how well heat is transported via conduction [W/(mK)].
phaseDensity	The density of the phase for which the thermal diffusivity is to be calculated [kg/m^3].
heatCapacity	A 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:

dimensionlessnumbers.hh

Description

Static Public Member Functions

Member Function Documentation

◆ nusseltNumberForced()

◆ prandtlNumber()

◆ reynoldsNumber()

◆ schmidtNumber()

◆ sherwoodNumber()

◆ thermalDiffusivity()