Buildings.HeatTransfer.Conduction.BaseClasses.Examples

Name	Description
Temperature_u	Approximation of specific internal energy versus temperature curve with cubic hermite cubic spline

Approximation of specific internal energy versus temperature curve with cubic hermite cubic spline

Information

This example tests and demonstrates the implementation of the specific internal energy versus temperature T(u) relationship for phase-change problems. Cubic hermite interpolation and linear extrapolation is used to approximate the piece-wise linear T(u) relationship. A piece-wise linear T(u) relationship is assumed in all three characteristic regions (solid, mushy and liquid). The example uses the functions Buildings.HeatTransfer.Conduction.BaseClasses.der_temperature_u and Buildings.HeatTransfer.Conduction.BaseClasses.temeprature_u. The first function is used to compute the derivatives at the support points, and the second function computes the temperature for a given specific internal energy.

The example also demonstrates the use of cubic hermite spline interpolation with two different settings: One produces an approximation of the T(u) relationship that is monotone, whereas the other does not enforce monotonicity. The latter one is used by default in the Buildings library, since it produces a higher accuracy in the mushy region, especially for materials in which phase-change transformation occurs in a wide temperature interval (see the figure below). The curves errNonMonotone and errMonotone represent the relative error between approximated and exact temperatures obtained for different specific internal energy values (right hand side figure).

Parameters

Modelica definition

Type	Name	Default	Description
Generic	materialMonotone		Phase change material with monotone u-T relation
Generic	materialNonMonotone		Phase change material with non-monotone u-T relation
SpecificInternalEnergy	ud[Buildings.HeatTransfer.Conduction.nSupPCM]		Support points [J/kg]
SpecificInternalEnergy	udMonotone[Buildings.HeatTransfer.Conduction.nSupPCM]		Support points [J/kg]
Temperature	Td[Buildings.HeatTransfer.Conduction.nSupPCM]		Support points [K]
Temperature	TdMonotone[Buildings.HeatTransfer.Conduction.nSupPCM]		Support points [K]
Real	dT_du[Buildings.HeatTransfer.Conduction.nSupPCM]		Derivatives at the support points - non-monotone, default in Modelica PCM [kg.K2/J]
Real	dT_duMonotone[Buildings.HeatTransfer.Conduction.nSupPCM]		Derivatives at the support points for monotone increasing cubic splines [kg.K2/J]
TemperatureDifference	dTCha	materialMonotone.TSol + mate...	Characteristic temperature difference of the problem [K]

model Temperature_u "Approximation of specific internal energy versus temperature curve with cubic hermite cubic spline" extends Modelica.Icons.Example; // Phase-change properties parameter Buildings.HeatTransfer.Data.SolidsPCM.Generic materialMonotone( TSol=273.15+30.9, TLiq=273.15+40.0, LHea=1000000, c=920, ensureMonotonicity=true, d=1000, k=1, x=0.2) "Phase change material with monotone u-T relation"; parameter Buildings.HeatTransfer.Data.SolidsPCM.Generic materialNonMonotone(TSol=273.15+30.9, TLiq=273.15+40.0, LHea=1000000, c=920, ensureMonotonicity=false, d=1000, k=1, x=0.2) "Phase change material with non-monotone u-T relation"; parameter Modelica.SIunits.SpecificInternalEnergy ud[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points"; parameter Modelica.SIunits.SpecificInternalEnergy udMonotone[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points"; parameter Modelica.SIunits.Temperature Td[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points"; parameter Modelica.SIunits.Temperature TdMonotone[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points"; parameter Real dT_du[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false, each unit="kg.K2/J") "Derivatives at the support points - non-monotone, default in Modelica PCM"; parameter Real dT_duMonotone[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false, each unit="kg.K2/J") "Derivatives at the support points for monotone increasing cubic splines"; Modelica.SIunits.SpecificInternalEnergy u "Specific internal energy"; Modelica.SIunits.Temperature T "Temperature for given u without monotone interpolation"; Modelica.SIunits.Temperature TMonotone "Temperature for given u with monotone interpolation"; Modelica.SIunits.Temperature TExa "Exact value of temperature"; Real errMonotone "Relative temperature error between calculated value with monotone interpolation and exact temperature"; Real errNonMonotone "Relative temperature error between calculated value with non-monotone interpolation and exact temperature"; parameter Modelica.SIunits.TemperatureDifference dTCha = materialMonotone.TSol+materialMonotone.TLiq "Characteristic temperature difference of the problem"; protected function relativeError "Relative error" input Modelica.SIunits.Temperature T "Approximated temperature"; input Modelica.SIunits.Temperature TExa "Exact temperature"; input Modelica.SIunits.TemperatureDifference dTCha "Characteristic temperature difference"; output Real relErr "Relative error"; algorithm relErr :=(T - TExa)/dTCha; end relativeError; constant Real conFac(unit="1/s") = 1 "Conversion factor to satisfy unit check"; initial equation // Calculate derivatives at support points (non-monotone) (ud, Td, dT_du) = Buildings.HeatTransfer.Conduction.BaseClasses.der_temperature_u( c= materialNonMonotone.c, TSol=materialNonMonotone.TSol, TLiq=materialNonMonotone.TLiq, LHea=materialNonMonotone.LHea, ensureMonotonicity=materialNonMonotone.ensureMonotonicity); // Calculate derivatives at support points (monotone); (udMonotone, TdMonotone, dT_duMonotone) = Buildings.HeatTransfer.Conduction.BaseClasses.der_temperature_u( c= materialMonotone.c, TSol=materialMonotone.TSol, TLiq=materialMonotone.TLiq, LHea=materialMonotone.LHea, ensureMonotonicity=materialMonotone.ensureMonotonicity); equation u = 2.5e5+time*(1.5*materialMonotone.c*(materialMonotone.TLiq-273.15)+materialMonotone.LHea)*conFac; // Calculate T based on non-monotone interpolation T = Buildings.HeatTransfer.Conduction.BaseClasses.temperature_u( ud=ud, Td=Td, dT_du=dT_du, u=u); // Calculate T based on monotone interpolation TMonotone = Buildings.HeatTransfer.Conduction.BaseClasses.temperature_u( ud=udMonotone, Td=TdMonotone, dT_du=dT_duMonotone, u=u); //Relative errors of obtained temperatures by using monotone and non-monotone //interpolation against exact values of tempratures taken from the T(u) curve if time>=1.e-05 then if u <= materialMonotone.c*materialMonotone.TSol then // Solid region TExa = u/materialMonotone.c; elseif u >= materialMonotone.c*materialMonotone.TLiq+materialMonotone.LHea then // Liquid region TExa = (u-materialMonotone.LHea)/materialMonotone.c; else // Region of phase transition TExa=((u + materialMonotone.LHea*materialMonotone.TSol/(materialMonotone.TLiq - materialMonotone.TSol))/(materialMonotone.c + materialMonotone.LHea/( materialMonotone.TLiq - materialMonotone.TSol))); end if; else TExa =T; end if; errNonMonotone = relativeError(T=T, TExa=TExa, dTCha= dTCha); errMonotone = relativeError(T=TMonotone, TExa=TExa, dTCha= dTCha); end Temperature_u;