HeatEquation1D

Solves 1D heat equation for each boundary point. Can be used for temperature boundary condition.

Solves the 1D heat equation

\begin{align} \varrho \, c_v \frac{\partial T}{\partial t} = \frac{\partial}{\partial x} \left(\lambda \frac{\partial T}{\partial x} \right) \end{align}

for each boundary point. The 1D points are always equidistantly distributed and the number of points can be controlled by the common variable NB_POINTS_BC_HEAT_EQUATION_1D. The boundary conditions for the 1D equation are Robin type conditions. At the interface MESHFREE - 1D we use

\begin{align} \lambda_{\mathrm{1D}} \frac{\partial T}{\partial x} = \alpha_{\mathrm{in}} (T_{\mathrm{Fluid}} - T_{\mathrm{1D}})\end{align}

and at the interface 1D - outer surrounding we use

\begin{align} \lambda_{\mathrm{1D}} \frac{\partial T}{\partial x} = \alpha_{\mathrm{out}} (T_{\mathrm{1D}} - T_{\mathrm{ambient}})\end{align}

The physical properties for the 1D equation are specified in the following sense: Syntax:

HEAT_EQ_1D($BC_wall$,%ind_xxx%) = (TotalLength, EndOfFirstInterval, PhysicalPropFirstInterval, EndOfSecondInterval, PhysicalPropSecondInterval,...)

where %ind_xxx% can be %ind_T%, %ind_LAM%, %ind_r%, or %ind_CV%.

TotalLength defines the overall length of the 1D domain
EndOfFirstInterval marks the endpoint of the first segment of the 1D domain
PhysicalPropFirstInterval is the constant value of %ind_xxx% within the first segment
EndOfSecondInterval marks the endpoint of the second segment of the 1D domain
PhysicalPropSecondInterval is the constant value of %ind_xxx% within the second segment

Exception for %ind_T%: The temperature needs an additional parameter for the outer surrounding which must be always the last entry in HEAT_EQ_1D($BC_wall$,%ind_T%). Example: Modelling of an insulation liner around a cylinder, which consists of two different materials.

HEAT_EQ_1D($BC_wall$,%ind_T%) = (&liner_thickness&, &liner_end_interval1&, &TEMP_alu&, &liner_end_interval2&, &TEMP_carbon&, &TEMP_ambient&) HEAT_EQ_1D($BC_wall$,%ind_LAM%) = (&liner_thickness&, &liner_end_interval1&, &LAM_alu&, &liner_end_interval2&, &LAM_carbon&) HEAT_EQ_1D($BC_wall$,%ind_r%) = (&liner_thickness&, &liner_end_interval1&, &RHO_alu&, &liner_end_interval2&, &RHO_carbon&) HEAT_EQ_1D($BC_wall$,%ind_CV%) = (&liner_thickness&, &liner_end_interval1&, &CV_alu&, &liner_end_interval2&, &CV_carbon&)

The heat transfer coefficients $ \alpha_{\mathrm{in}}, \alpha_{\mathrm{out}}$ must be specified by

HEAT_EQ_1D_TRANSFER_COEFF_INTERNAL($BC_wall$) = [&alpha_in&] HEAT_EQ_1D_TRANSFER_COEFF_EXTERNAL($BC_wall$) = [&alpha_out&]

The results of all 1D heat equations are stored in %ind_T1D(i)%, i = 1:NB_POINTS_BC_HEAT_EQUATION_1D+1. To use these results in the MESHFREE boundary conditions, use the keyword %HEAT_EQ_1D_BC% (see %BND_ROBIN% and %BND_DIRICH%). Then the corresponding temperature value in the MESHFREE boundary condition is replaced by %ind_T1D(1)%. Example:

BC_T($BC_wall$) = (%BND_DIRICH%, [&T_BND&], 0.0, %HEAT_EQ_1D_BC%)

Note: MESHFREE does not support negative values in the indices, %ind_T1D(i)%. Such cases will result in a termination with an error message.