Legacy · Thermal validation

Infinite slab, four routes

Slab thickness $L=1.0\,\mathrm{m}$ , $T_0=T_{\infty}=25\,^{\circ}\mathrm{C}$ , reading at $x=L/2$ . Four calculations with the homogeneous-interconversion material.

FEM, adiabatic formulation, $E_a/R=4000\,\mathrm{K}$ .
FEM, two-term isothermal formulation, the same $Q(t_{eq})$ as the homogeneous material.
Particular curve $T^{Q}(t)$ taken at the $T_{\max}$ point, with $E_a/R=0$ .
Analytical infinite-slab solution with that particular curve.

Paper: Analytical and numerical validation of the thermochemical problem of early-age concrete · DOI 10.70271/rirc.v1n2a008

Input data

Item	Value
$L$	$1.0\,\mathrm{m}$
Monitor point	$x=L/2$ (column `0,0.5` in the FEM histories)
$T_0=T_{\infty}$	$25\,^{\circ}\mathrm{C}$
$T_{\mathrm{ref}}$	$25\,^{\circ}\mathrm{C}$
$k$	$9.36\,\mathrm{kJ/(m\cdot h\cdot^{\circ}C)}$
$\rho$	$2300\,\mathrm{kg/m^3}$
$c_e$	$0.900\,\mathrm{kJ/(kg\cdot^{\circ}C)}$
$h^{2}=k/(\rho c_e)$	$4.52\times 10^{-3}\,\mathrm{m^2/h}$
$C$	$300\,\mathrm{kg/m^3}$
Particular curve	$\Delta T^{Q}_{\infty}=30\,^{\circ}\mathrm{C}$ , $a=12.35\,\mathrm{h}$ , $c=1.678$ , $E_a/R=0$
Fourier modes	$10$ to $15$

The particular curve keeps the amplitude at $30\,^{\circ}\mathrm{C}$ . Parameters $a$ and $c$ absorb the history at the $T_{\max}$ point and the lack of thermal activation in the analytical equation.

Reference result

Peak temperatures at mid-depth, read from the series on this page:

Route	$T_{\max}$ (°C)
Isothermal FEM	41.35
Adiabatic FEM	41.38
Analytical, particular curve	41.04

The report shows the four routes on the figure and does not publish an RMSE for this slab. Comparing the series on this page, the isothermal route against the analytical solution has RMSE $0.27\,^{\circ}\mathrm{C}$ (maximum absolute difference about $1.0\,^{\circ}\mathrm{C}$ ). The adiabatic route has RMSE $0.27\,^{\circ}\mathrm{C}$ (maximum absolute difference about $0.98\,^{\circ}\mathrm{C}$ ).

Series t × Tmax × teq × Q(t)

In the first two files, Tmax_C is the FEM value at point 0,0.5, teq_h uses $E_a/R=4000\,\mathrm{K}$ and Q_kJ_kg is the two-term isothermal generation.

In the particular file, Tmax_C is the analytical centre temperature, teq_h $=t$ and Q_kJ_kg comes from the particular Hill law ( $a=12.35\,\mathrm{h}$ , $c=1.678$ , $\Delta T=30\,^{\circ}\mathrm{C}$ ).

serie_laje_canonica_particular.csv

Spatial histories from the report, for checks at other depths: IsoTeq, FormAdi, Qparticular and analytical centre and $x=0.25\,\mathrm{m}$ .

Report figures