Legacy · Thermal validation · Português
Thermal validation
Step-by-step check of the thermal benchmarks in the reference article: the same heat-generation curve, the same geometries, the same activation energy. It is for anyone running a finite-element program who needs to confirm that thermal activation and the analytical solution are in place.
Reference article (Portuguese): Adiabatic–isothermal interconversion of internal heat generation · DOI 10.70271/rirc.v1n2a008
The figures below are those of the manuscript. Acceptance for each case is what the article already closed (RMSE and invariance of ΔT). The adiabatic × isothermal FlexPDE comparison at three placement temperatures on the infinite slab is marked in the text as a complement, and is not a closed benchmark here.
Shared protocol
For the cases with an analytical solution, the check has five steps.
- Fix the isothermal curve Q(teq) of the material. It is not recalibrated when the placement temperature or the geometry changes.
- Solve the thermal problem by finite elements with thermal activation, using the calibrated Ea/R.
- Read T(t) at the point of maximum temperature.
- In the analytical solution, regress the particular curve TQ(t) with Ea/R = 0, from the rate ∂TQ/∂t at that point.
- Compare the history T(t), the map at the instant of Tmax, and the section profile. The amplitude ΔT of the particular curve must remain that of the isothermal calibration.
1. Adiabatic–isothermal interconversion
This is the material case, before the geometries. The two formulations describe the same concrete, with distinct reference temperatures in the Arrhenius function.
- Start from the reference adiabatic rise (Hill, one term): a = 12 h, c = 2.0, ΔTadi∞ = 30 °C, T0 = 25 °C, Ea/R = 4000 K, T0refadi = 25 °C.
- Obtain the equivalent two-term isothermal curve at the same 25 °C reference: a1 = 10.32 h, c1 = 1.93, ΔT1 = 16.4 °C; a2 = 32.28 h, c2 = 1.57, ΔT2 = 13.6 °C.
- Check that the sum of the isothermal amplitudes recovers the adiabatic 30 °C and that a single Ea serves both the paste and the concrete as a whole.
Thermal properties used in the canonical slab that follows: k = 9.36 kJ/(m·h·°C), ρ = 2300 kg/m³, ce = 0.900 kJ/(kg·°C), h² = 4.52×10−3 m²/h, cement content C = 300 kg/m³.
2. Canonical infinite slab
Thickness L = 1.0 m. Opposite faces at T∞ = 25 °C. Initial temperature T0 = 25 °C. Monitored point at x = L/2. Four routes for the same material as the previous section.
- Finite elements with the adiabatic formulation, Ea/R = 4000 K.
- Finite elements with the two-term isothermal formulation, using the interconversion parameters.
- Finite elements with the particular generation TQ(t) of the Tmax point and Ea/R = 0.
- Analytical solution of the infinite slab with that same particular generation and Ea/R = 0.
What is examined: routes (iii) and (iv) follow the full thermochemical model. For the idealised case T0 = T∞ = 25 °C, the analytical–numerical agreement is visually close. The field at the instant of Tmax is one-dimensional.
3. Geometry × placement-temperature matrix
A reference material distinct from the canonical case: one-term Hill, ΔTadi∞ = 50 °C, a1 = 40.10873 h, c1 = 1.54782, Ea/R = 4000 K, Tref = 20 °C. Only T0 ∈ {20, 30, 40} °C is varied. Cold faces and soil at T∞ = 20 °C. Horizon tf = 480 h. Nine runs.
Concrete: k = 9 kJ/(m·h·°C), ρ = 2400 kg/m³, ce = 0.75 kJ/(kg·°C), h² = 0.005 m²/h. Soil, where it enters: k = 3 kJ/(m·h·°C), ρ = 1500 kg/m³, ce = 0.5 kJ/(kg·°C), h² = 0.004 m²/h.
3.1 Cylindrical shaft
Radius r = 0.5 m. Tmax point on the axis. For each T0, apply the shared protocol.
| T0 (°C) | regressed ΔT (°C) | τ (h) | RMSE (°C) | Tmax FEM / anal. (°C) |
|---|---|---|---|---|
| 20 | 50.00 | 23.63 | 0.012 | 43.3 / 43.3 |
| 30 | 50.00 | 15.38 | 0.090 | 52.4 / 52.6 |
| 40 | 50.00 | 10.35 | 0.166 | 62.8 / 63.2 |
Acceptance: regressed ΔT remains 50 °C (deviation below 0.01%). RMSE stays below 0.17 °C. The largest deviation in the whole matrix is this shaft at 40 °C.
3.2 Slab on a foundation
Slab thickness LY = 2.0 m, width LX = 1.0 m, semi-infinite soil. Tmax point at (0, LY/2).
| T0 (°C) | regressed ΔT (°C) | τ (h) | RMSE (°C) | Tmax FEM / anal. (°C) |
|---|---|---|---|---|
| 20 | 50.00 | 13.73 | 0.044 | 59.0 / 59.0 |
| 30 | 50.00 | 12.69 | 0.071 | 68.0 / 68.1 |
| 40 | 50.00 | 10.40 | 0.091 | 78.5 / 78.6 |
3.3 Infinite slab in the same matrix
Thickness L = 1.0 m, opposite faces at T∞. Tmax point at L/2. Same layout as the canonical slab, with the 50 °C calibration and a variable T0.
| T0 (°C) | regressed ΔT (°C) | τ (h) | RMSE (°C) | Tmax FEM / anal. (°C) |
|---|---|---|---|---|
| 20 | 50.00 | 34.27 | 0.008 | 34.7 / 34.7 |
| 30 | 50.00 | 32.72 | 0.027 | 38.0 / 38.1 |
| 40 | 50.00 | 30.68 | 0.054 | 43.1 / 43.1 |
Throughout the matrix, τ decreases as T0 increases. The input isothermal curve does not change.
4. Spillway pier — UHE São Manoel
Inverse problem. A measured history is available; the particular generation is recovered, and from it the general generation.
- Take spillway pier P4 as the reference (spillway monitoring; site visit 21–25 Sep 2015).
- In the analytical back-analysis, keep Ea/R = 0: thermal activation is already in the in-situ particular history.
- Regress the particular curve of the process from the temperature history at the centre point.
- Compare the infinite-slab analytical solution with FlexPDE and with the ANSYS model of the pier.
Published acceptance: RMSE of 0.27 °C on pier P4.