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.
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 [-].
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
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.
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]
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
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 [-].
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:
1.9.3