Modelica.Fluid.Dissipation.Utilities.Functions.General

Package with utility functions

Information

Extends from Modelica.Icons.FunctionsPackage (Icon for packages containing functions).

Package Content

Name	Description
CubicInterpolation_DP
CubicInterpolation_MFLOW
LambertW	Closed approximation of Lambert's w function for solving f(x) = x exp(x) for x
LambertWIter	Iterative form of Lambert's w function for solving f(x) = x exp(x) for x
PrandtlNumber	calculation of Prandtl number
ReynoldsNumber	calculation of Reynolds number
SmoothPower	Limiting the derivative of function y = if x>=0 then x^pow else -(-x)^pow
SmoothPower_der	The derivative of function SmoothPower
Stepsmoother	Continuous interpolation for x
Stepsmoother_der	Derivative of function Stepsmoother

Modelica.Fluid.Dissipation.Utilities.Functions.General.CubicInterpolation_DP

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Re_turbulent
Re1	[1]
Re2	[1]
Delta
lambda2

Outputs

Name	Description
Re	[1]

Modelica.Fluid.Dissipation.Utilities.Functions.General.CubicInterpolation_MFLOW

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
Re	[1]
Re1	[1]
Re2	[1]
Delta

Outputs

Name	Description
lambda2

Modelica.Fluid.Dissipation.Utilities.Functions.General.LambertW

Closed approximation of Lambert's w function for solving f(x) = x exp(x) for x

Information

This function calculates an approximation of the inverse for

    f(x) = y = x * exp( x )

within ∞ > y > -1/e. The relative deviation of this approximation for Lambert's w function x = W(y) is displayed in the following graph.

LambertW_deviation

For y > 10 and higher values the relative deviation is smaller 2%.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
y	f(x)

Outputs

Name	Description
x	W(y)

Modelica.Fluid.Dissipation.Utilities.Functions.General.LambertWIter

Iterative form of Lambert's w function for solving f(x) = x exp(x) for x

Information

This function calculates an approximation of the inverse for

    f(x) = y = x * exp( x )

within ∞ > y > -1/e. Please note, that for negative inputs two solutions exists. The function currently delivers the result x = -1 ... 0 for that particular range.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
y	f(x)

Outputs

Name	Description
x	W(y)
iter

Modelica.Fluid.Dissipation.Utilities.Functions.General.PrandtlNumber

calculation of Prandtl number

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
cp	specific heat capacity of fluid at constant pressure [J/(kg.K)]
eta	dynamic viscosity of fluid [Pa.s]
lambda	thermal conductivity of fluid [W/(m.K)]

Outputs

Name	Description
Pr	Prandtl number [1]

Modelica.Fluid.Dissipation.Utilities.Functions.General.ReynoldsNumber

calculation of Reynolds number

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
A_cross	Cross sectional area [m2]
perimeter	Wetted perimeter [m]
rho	Density of fluid [kg/m3]
eta	Dynamic viscosity of fluid [Pa.s]
m_flow	Mass flow rate [kg/s]

Outputs

Name	Description
Re	Reynolds number [1]
velocity	Mean velocity [m/s]

Modelica.Fluid.Dissipation.Utilities.Functions.General.SmoothPower

Limiting the derivative of function y = if x>=0 then x^pow else -(-x)^pow

Information

The function is used to limit the derivative of the following function at x=0:

   y = if x ≥ 0 then x^pow else -(-x)^pow;  // pow > 0

by approximating the function in the range -deltax< x < deltax with a third order polynomial that has the same derivative at abs(x)=deltax, as the function above.

Example

In the picture below the input x is increased from -1 to 1. The range of interpolation is defined by the same range. Displayed is the output of the function SmoothPower compared to

y=x*|x|

For |x| > 1 both functions return identical results.

SmoothPower

References

ThermoFluid Library: http://sourceforge.net/projects/thermofluid/

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
x	input variable
deltax	range for interpolation
pow	exponent for x

Outputs

Name	Description
y	output variable

Modelica.Fluid.Dissipation.Utilities.Functions.General.SmoothPower_der

The derivative of function SmoothPower

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
x	input variable
deltax	range of interpolation
pow	exponent for x
dx	derivative of x

Outputs

Name	Description
dy	derivative of SmoothPower

Modelica.Fluid.Dissipation.Utilities.Functions.General.Stepsmoother

Continuous interpolation for x

Information

The function is used for continuous fading of variable inputs within a defined range. It allows a differentiable and smooth transition between function outputs, e.g., laminar and turbulent pressure drop or correlations for certain ranges.

Function

The tanh-function is used, since it provides an existing derivative and the derivative is zero at the borders [nofunc, func] of the interpolation domain (smooth derivative for transitions).

In order to work correctly, the internal interpolation range in terms of the external arbitrary input x needs to be scaled such that:

f(func)   = 0.5 π
f(nofunc) = -0.5 π

Example

In the picture below the input x is increased from 0 to 1. The range of interpolation is defined by:

func = 0.75
nofunc = 0.25

Stepsmoother

References

Wischhusen, St.: Simulation von Kältemaschinen-Prozessen mit MODELICA / DYMOLA. Diploma thesis, Hamburg University of Technology, Institute of Thermofluiddynamics, 2000.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
func	input value for that result = 100%
nofunc	input value for that result = 0%
x	input variable for continuous interpolation

Outputs

Name	Description
result	output value

Modelica.Fluid.Dissipation.Utilities.Functions.General.Stepsmoother_der

Derivative of function Stepsmoother

Information

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
func	input for that result = 100%
nofunc	input for that result = 0%
x	input for interpolation
dfunc	derivative of func
dnofunc	derivative of nofunc
dx	derivative of x

Outputs

Name	Description
dresult

Automatically generated Thu Dec 19 17:20:12 2019.