modelOneDHeatTransferTI_Check

analytical solution for TI models
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Information

In the verification of OneDHeatTransferTI_FD and OneDHeatTransferTI_Modelica, mid-nodes of the models are selected, and verified by the analytical solution. The temperature at the middle of the model implemented by analytical solution, the temperature at the mid-node of the model implemented by equations and the temperature of the heat capacitor at the middle of the model implemented by Modelica.Thermal library, are compared.

Analytical solutions are based on the solutions in “Handbook of Linear Partial Differential Equations for Engineers and Scientists” by Andrei D. Polyanin. If we consider one dimensional partial differential heat equation in the form:

In HeatConductionTI models, one end of the rod is exposed to a fixed temperature while the other end is insulated. At the insulated end, heat flux is zero therefore derivative of the temperature will be zero. The following conditions are prescribed for the model:

Solution:

where:

Parameters Comment
L length of the rod
N number of nodes
TN fixed temperature at the last node
T0 Initial temperature of the heat capacitors
cp material specific heat capacity
lambda material thermal conductivity
rho material density
A cross section area of the rod

Parameters

TypeNameDefaultDescription
SIunits.LengthL0.2Length of the rod
IntegerN30Number of nodes
SIunits.TemperatureTN330Fixed temperature at the last node
SIunits.TemperatureT0273.15Initial temperature
SIunits.SpecificHeatCapacitycp910Material Heat Capacity
SIunits.ThermalConductivitylambda237Material thermal conductivity
SIunits.Densityrho2712Material Density
SIunits.AreaA0.05cross section area

Components

TypeNameDefaultDescription
Modelica.Units.SI.TemperatureT_mid_exact
Modelica.Units.SI.TemperatureT_mid_numerical
Modelica.Units.SI.TemperatureT_mid_modelica
Models.OneDHeatTransferTI_FDexample
Models.OneDHeatTransferTI_Modelicaexample2

Contents

NameDescription
f