Author Contributions
Conceptualization, J.Y.C. and S.C.; methodology, S.K., J.Y.C. and S.C.; software, S.K.; validation, J.Y.C. and S.C.; formal analysis, S.K. and S.C.; investigation, S.K. and S.C.; resources, J.Y.C.; data curation, S.K.; writing—original draft preparation, S.K.; writing—review and editing, J.Y.C. and S.C.; visualization, S.K.; supervision, S.C.; project administration, S.C.; funding acquisition, J.Y.C.
Figure 1.
Thermo-physical behaviour of an inorganic-based intumescent coating tested with cone calorimetry: (a) in the initial state and (b) in the fully-expanded state, where ℓ, h, and SEM refer to the thickness of the coating, the distance between the heater and the coating’s exposed surface, and scanning electron microscopy while the subscript 0 indicates the initial state.
Figure 1.
Thermo-physical behaviour of an inorganic-based intumescent coating tested with cone calorimetry: (a) in the initial state and (b) in the fully-expanded state, where ℓ, h, and SEM refer to the thickness of the coating, the distance between the heater and the coating’s exposed surface, and scanning electron microscopy while the subscript 0 indicates the initial state.
Figure 2.
Modelling scheme for heat transfer in porous media: (a) definition of the structural components of a cell, (b) conceptual solid skeleton superimposed on an SEM image, and (c) heat flows in a representative elemental cell (REC).
Figure 2.
Modelling scheme for heat transfer in porous media: (a) definition of the structural components of a cell, (b) conceptual solid skeleton superimposed on an SEM image, and (c) heat flows in a representative elemental cell (REC).
Figure 3.
Tortuosity of the solid skeleton: (a) outermost inline, (b) in-between, and (c) outermost staggered formations.
Figure 3.
Tortuosity of the solid skeleton: (a) outermost inline, (b) in-between, and (c) outermost staggered formations.
Figure 4.
Data obtained from the electric furnace tests: (a) schematics of the porosity distribution along the coating thickness, (b) the SEM images, and (c) the corresponding void boundaries.
Figure 4.
Data obtained from the electric furnace tests: (a) schematics of the porosity distribution along the coating thickness, (b) the SEM images, and (c) the corresponding void boundaries.
Figure 5.
Probability distributions of cell size and volume: (a) relative frequency densities of δcell and Vcell, and (b) relative class frequency of Vcell.
Figure 5.
Probability distributions of cell size and volume: (a) relative frequency densities of δcell and Vcell, and (b) relative class frequency of Vcell.
Figure 6.
Initial setup of the specimen placement with a dry-film thickness of (a) 2 mm, (b) 3 mm, and (c) 4 mm.
Figure 6.
Initial setup of the specimen placement with a dry-film thickness of (a) 2 mm, (b) 3 mm, and (c) 4 mm.
Figure 7.
Heat transfer mechanism through the porous specimen in cone calorimeter testing.
Figure 7.
Heat transfer mechanism through the porous specimen in cone calorimeter testing.
Figure 8.
Modelling scheme for the radiation in pores.
Figure 8.
Modelling scheme for the radiation in pores.
Figure 9.
Unit enclosure, which is composed of N discrete elemental surfaces and filled with translucent media.
Figure 9.
Unit enclosure, which is composed of N discrete elemental surfaces and filled with translucent media.
Figure 10.
Scheme of the FEA modelling: (a) prototype (NPV = 1) and (b) multicellular-type (e.g., NPV = 5 in inline formation).
Figure 10.
Scheme of the FEA modelling: (a) prototype (NPV = 1) and (b) multicellular-type (e.g., NPV = 5 in inline formation).
Figure 11.
Effect of semi-transparent media on cavity radiation: (a) relation between and keff at = 50 kW/m2 and Φ = 0.90; (b) deviations from keff at = 1.0 with changes of , , and Φ; (c) deviations from keff at = 1.0 with changes of δcell and , at = 50 kW/m2 and Φ = 0.90; and (d) keff at = 50 kW/m2 and Φ = 0.90 with changes of δcell and .
Figure 11.
Effect of semi-transparent media on cavity radiation: (a) relation between and keff at = 50 kW/m2 and Φ = 0.90; (b) deviations from keff at = 1.0 with changes of , , and Φ; (c) deviations from keff at = 1.0 with changes of δcell and , at = 50 kW/m2 and Φ = 0.90; and (d) keff at = 50 kW/m2 and Φ = 0.90 with changes of δcell and .
Figure 12.
Scheme of FEA multicellular modelling: (a) inline and (b) staggered formations.
Figure 12.
Scheme of FEA multicellular modelling: (a) inline and (b) staggered formations.
Figure 13.
Φ-keff relationships obtained from FEA multicellular simulations with existing generic models.
Figure 13.
Φ-keff relationships obtained from FEA multicellular simulations with existing generic models.
Figure 14.
Schematic of concept of the FEA multilayer model for intumescence over time in association with the electric furnace test.
Figure 14.
Schematic of concept of the FEA multilayer model for intumescence over time in association with the electric furnace test.
Figure 15.
Experimental measurements of substrate-temperature (Ts) and thickness expansion (ℓz) over time: (a) at DFT = 3 mm and (b) at = 50 kW/m2.
Figure 15.
Experimental measurements of substrate-temperature (Ts) and thickness expansion (ℓz) over time: (a) at DFT = 3 mm and (b) at = 50 kW/m2.
Figure 16.
Experimental measurements and numerical predictions of the upper and lower limits of substrate-temperature (Ts) and thickness expansion (ℓz) over time: (a) at DFT = 3 mm and (b) at = 50 kW/m2.
Figure 16.
Experimental measurements and numerical predictions of the upper and lower limits of substrate-temperature (Ts) and thickness expansion (ℓz) over time: (a) at DFT = 3 mm and (b) at = 50 kW/m2.
Figure 17.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to δcell at L = 3 mm, Φ = 0.895, and = 50 kW/m2.
Figure 17.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to δcell at L = 3 mm, Φ = 0.895, and = 50 kW/m2.
Figure 18.
Variations in and ΔT according to δcell, at L = 3 mm, Φ = 0.895, and = 50 kW/m2.
Figure 18.
Variations in and ΔT according to δcell, at L = 3 mm, Φ = 0.895, and = 50 kW/m2.
Figure 19.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to Φ at L = 3 mm, δcell = 258 μm, and = 50 kW/m2.
Figure 19.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to Φ at L = 3 mm, δcell = 258 μm, and = 50 kW/m2.
Figure 20.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to at L = 3 mm, δcell = 258 μm, and Φ = 0.895.
Figure 20.
Variations in the keff/ksolid ratio and the individual contributions of the heat transfer modes according to at L = 3 mm, δcell = 258 μm, and Φ = 0.895.
Table 1.
Percentage probability distributions of cell volume.
Table 1.
Percentage probability distributions of cell volume.
Temperature (°C) | Probability Distributions, Vpro (%) | Sum |
---|
Classification of Cell Diameter (μm) |
---|
I (5–100) | II (100–200) | III (200–300) | IV (300–400) |
---|
300 | 22.8 | 18.6 | 15.6 | 22.5 | 79.5 |
600 | 27.5 | 21.1 | 17.4 | 23.8 | 89.8 |
Table 2.
Experimental data of fully expanded thickness (units of mm) [
14].
Table 2.
Experimental data of fully expanded thickness (units of mm) [
14].
| (kW/m2) | 35 | 50 | 65 |
---|
DFT (mm) | |
---|
2 | Upper | 22.3 (17.7) | 22.4 (17.6) | 22.9 (17.1) |
Lower | 19.2 (20.8) | 20.2 (19.8) | 20.5 (19.5) |
3 | Upper | 30.3 (24.7) | 31.7 (23.3) | 31.2 (23.8) |
Lower | 29.0 (26.0) | 29.7 (25.3) | 31.0 (24.0) |
4 | Upper | - | 41.2 (18.8) | - |
Lower | - | 39.6 (20.4) | - |
Table 3.
View factors [
38] and the corresponding true radiant heat absorptions (
) at emissivity (
ε) = 0.77.
Table 3.
View factors [
38] and the corresponding true radiant heat absorptions (
) at emissivity (
ε) = 0.77.
z (mm) | Fh-top * | (kW/m2) |
---|
= 35 kW/m2 | 50 kW/m2 | 65 kW/m2 |
---|
30 | 0.2382 | 24.75 | 35.36 | 45.97 |
25 | 0.2253 | 23.42 | 33.45 | 43.49 |
20 | 0.2126 | 22.10 | 31.57 | 41.04 |
15 | 0.2002 | 20.81 | 29.73 | 38.65 |
Table 4.
Thermo-physical properties of the materials and convective coefficients in cone calorimeter testing.
Table 4.
Thermo-physical properties of the materials and convective coefficients in cone calorimeter testing.
| Properties | Conductivity k (W/(mK)) | Density ρd (kg/m3) | Specific Heat c (J/(kgK)) | Emissivity ε | Convective Coefficient h (W/(m2 K)) |
---|
Materials | |
---|
Intumescent coating | 1.56 | 2077 | 1780 | 0.77 4 | 14.6 5 |
Steel plate 1 | 53.30 | 7870 | 440 | - |
Air 2 | 0.03 | 1.16 | 1007 | - |
Insulation board 3 | 0.21 | 900 | 1000 | 0.90 |
Table 5.
Geometric variables used in the supplementary FEA simulations.
Table 5.
Geometric variables used in the supplementary FEA simulations.
Classification | Φ 1 | L1 (μm) | NPV1 | δunit2 (μm) | δcell2 (μm) | δwall2 (μm) |
---|
I | 0.913 | 3000 | 60 | 50 | 48 | 1 |
II | 20 | 150 | 143 | 3 |
III | 12 | 250 | 239 | 6 |
IV | 8 | 375 | 358 | 8 |
Table 6.
Results of supplementary simulations at Φ = 0.913 and = 50 kW/m2.
Table 6.
Results of supplementary simulations at Φ = 0.913 and = 50 kW/m2.
Heat Transfer Course | Effective Thermal Conductivity, keff (W/(mK)) |
---|
Class I | Class II | Class III | Class IV |
---|
A-A’ | 0.1080 | 0.1105 | 0.1130 | 0.1160 |
B-B’ | 0.1061 | 0.1050 | 0.1040 | 0.1027 |
C-C’ | 0.0937 | 0.0932 | 0.0927 | 0.0922 |
D-D’ | 0.0953 | 0.0979 | 0.1005 | 0.1037 |
Table 7.
keff* for inorganic intumescent coating at Φ = 0.913 and = 50 kW/m2.
Table 7.
keff* for inorganic intumescent coating at Φ = 0.913 and = 50 kW/m2.
Temperature (°C) | keff* (W/(mK)) | Upper Bound | Lower Bound |
---|
Heat Transfer Course |
---|
A-A’ | B-B’ | C-C’ | D-D’ |
---|
300 | 0.1118 | 0.1045 | 0.0930 | 0.0993 | 0.1118 | 0.0930 |
600 | 0.1117 | 0.1045 | 0.0930 | 0.0991 |
Table 8.
Geometric variables for multicellular modelling.
Table 8.
Geometric variables for multicellular modelling.
L1 (μm) | Φ 1 | NPV1 | δunit2 (μm) | δcell2 (μm) | δwall2 (μm) |
---|
3000 | 0.865 | 1 | 3000.0 | 2790.2 | 104.9 |
2 | 1500.0 | 1395.1 | 52.5 |
3 | 1000.0 | 930.1 | 35.0 |
6 | 500.0 | 465.0 | 17.5 |
11 | 272.7 | 253.7 | 9.5 |
50 | 60.0 | 55.8 | 2.1 |
0.895 | 1 | 3000.0 | 2838.1 | 80.9 |
2 | 1500.0 | 1419.1 | 40.5 |
3 | 1000.0 | 946.0 | 27.0 |
6 | 500.0 | 473.0 | 13.5 |
11 | 272.7 | 258.0 | 7.4 |
50 | 60.0 | 56.8 | 1.6 |
0.930 | 1 | 3000.0 | 2893.1 | 53.5 |
2 | 1500.0 | 1446.5 | 26.7 |
3 | 1000.0 | 964.4 | 17.8 |
6 | 500.0 | 482.2 | 8.9 |
11 | 272.7 | 263.0 | 4.9 |
50 | 60.0 | 57.9 | 1.1 |