# Towards the Use of Novel Materials in Shipbuilding: Assessing Thermal Performances of Fire-Doors by Self-Consistent Numerical Modelling

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material Properties

#### 2.1.1. Mechanical Tests

^{−1}was used (ASTM D1621 [17]) and the compression modulus E

_{c}was determined from the stress–strain curves in the 3–6% deformation range. Tests were performed on five specimens with an average diameter of 19.5 mm and an average height of 17.0 mm.

#### 2.1.2. Thermal Conductivity

_{ref}is the value at the reference temperature T

_{ref}(20 °C in this case) and T

_{i}is the temperature at which the thermal conductivity has to be found. f

_{T}is a scale factor tabulated in the standard ISO, in this study a factor for glassy materials (f

_{T}= 0.003) is adopted.

#### 2.1.3. Incombustibility

#### 2.2. Numerical Modelling of Fire Resistance Test

#### 2.2.1. Fire-Proof Door Characteristics and Test Conditions

^{−3}) with a thickness of 40 mm. The insulating material was selected by the door manufacturing company due to its lightness, efficiency and proper certifications. The steel sheets were glued to the insulator by a polymer-based glue. The door frame, consisting of steel profiles with a thickness of 4 mm, was screwed on the bulkhead with a spacing of 150 mm. The door was fixed to the frame by means of three hinges on one side and a lock was arranged on the opposite side. Figure 1a exemplifies the model geometry.

_{0}(in this case 18 °C) to 945 °C with a prescribed time–temperature relationship expressed by the following equation:

- (i)
- The final temperature of each thermocouple must not exceed 180 °C above T
_{0}; - (ii)
- The medium temperature of the five thermocouples must not exceed 140 °C above T
_{0}.

#### 2.2.2. FEM Modelling

_{2}is the average temperature between T

_{0}and that calculated on the fire-unexposed side without considering radiation and convection, ε is the steel emissivity, σ is the Stefan–Boltzmann constant in (W m

^{−2}K

^{−4}), L is the height of the door in (m) (characteristic dimension) and k

_{0}is the thermal conductivity of air in (W (m K)

^{−1}). The Nusselt number Nu is calculated for the vertical wall with turbulent and laminar flow [23] using the following equation:

^{−1}), α as the thermal diffusivity of air in (m s

^{−1}), β as the volumetric expansion coefficient of air in (K

^{−1}) and g as the gravity acceleration in (m s

^{−2}). The air properties are considered at the temperature T

_{2}[24].

^{2}K)

^{−1}[24] and it is assumed to receive uniform radiation heat flow with a view factor set to 1.0 [10], adopting the black body condition [25].

## 3. Results

#### 3.1. Foam Characterization

^{−3}was used, slightly lower than that of the rock wool (150 kg m

^{−3}) for which an elastic compression modulus of 3.4 ± 0.1 MPa was measured (for comparison, the rock wool modulus was 1 ± 0.1 MPa). The composition was chosen to satisfy the incombustibility test imposed by the FTP Code [1]. The foam was classified as incombustible material since the average of five tests showed a weight loss of 45.6 wt % (in compliance with the FTP Code, this must be ≤ 50 wt %) and flame persistence of 1.8 s (for the FTP Code, this must be ≤ 10 s).

^{−1}for the rock wool and the foam, respectively. The experimental thermal conductivity up to 200 °C fit fairly well to the conductivity vs. temperature ISO 10456: 2007 curve [20] (Figure 4); accordingly, Equation (1) was used for higher temperatures, both for the foam and the rock wool.

#### 3.2. Numerical Model Assessment

#### 3.2.1. Boundary Conditions for Thermal Analysis

_{2}= 568 K (see Equation (3)) was calculated for the fire-unexposed side using a simplified model, without considering radiation and convection. The resulting Nusselt (Equation (5)) and Rayleigh (Equation (7)) numbers were 447.4 and 6.2 × 10

^{10}, respectively. A Rayleigh number higher than 10

^{9}indicates a turbulent flow for which the correlation used for the Nusselt number is valid [26].

^{2}K)

^{−1}and 13.9 W (m

^{2}K)

^{−1}, respectively, were calculated. Thus, an equivalent convection coefficient of 22.9 W (m

^{2}K)

^{−1}, as obtained from Equation (8), was imposed on the fire-unexposed side of the door.

#### 3.2.2. Mesh Sensitivity Study

_{i}are the model values with i that decreases with the grid refinement and p is the solution convergence order expressed in the following equation:

_{r}is near to 1, the desired condition is satisfied.

_{r}equal to 1.005 was obtained, with GCI

_{1}and GCI

_{2}equal to 0.009 and 0.003, respectively.

#### 3.2.3. Validation of the Model

^{2}K)

^{−1}are reported; these values largely overestimate the experimental results.

_{n}) and experimental (F

_{e}) measurements and U is the norm between experimental (e

_{e}) and numerical error (e

_{n}), as expressed in the following equations:

_{0}is the ideal value calculated through the Richardson extrapolation [29] reported in the following equation:

#### 3.2.4. Effect of the Thermal Bridge

#### 3.2.5. Fire-Door with the Foam

## 4. Discussion

#### 4.1. Assessment of the FE Model

_{c}) on the fire-unexposed side of the door heavily affects the modelling—the final calculated temperature on the fire-unexposed side decreased by about 50 °C when h

_{c}was increased from 0 to 12 W (m

^{2}K)

^{−1}[13]. The Eurocode 1 [24] recommends the use of values 4 or 9 W (m

^{2}K)

^{−1}, when the effects of heat transfer by radiation are considered. The literature shows that h

_{c}ranging from 4 to 10 W (m

^{2}K)

^{−1}are used in order to fit the experimental results [2,9,10]. In this work the numerical results could not properly fit with the experimental data using either of the recommended values, as exemplified in Table 2. Noticeably, the Eurocode 1 recommended values could reasonably fit well to the data obtained on a door with different construction geometry (compare to caption of Figure 10), as observed from the analysis in the preliminary study [14]. This clearly shows the need for the appropriate and independent definition of the boundary conditions. With this aim, the methodology reported in Section 2.2.2 has been proposed to calculate the T

_{2}temperature and hence h

_{c}, using the geometry of the door and the physical properties of the employed materials as input data—this approach leads to a self-consistent procedure. Moreover, a criterion for the numerical validation of the results is proposed to confirm the reliability of the calculation. Noticeably, the presented self-consistent procedure can also properly fit to the results of the door analyzed in the preliminary study (data not reported for brevity) [14].

^{−3}) [31,32].

^{2}K)

^{−1}, recommended by Eurocode 1 [25] for the convection coefficient on the fire-exposed side of the door, is quite consistent with the high temperature inside the furnace (945 °C) and clearly does not depend on the door employed.

#### 4.2. Effect of the Construction Geometry and Use of Innovative Insulation Material

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Model geometry; (

**b**) test measuring points: location of thermocouples (1, 2, 3, 4, 5) and displacements measurements point (A, B, C, D, I).

**Figure 4.**Thermal conductivity vs. temperature; comparison between ISO curves and experimental data.

**Figure 5.**Cell dimension vs. displacements calculated at point B. Comparison with the experimental displacements (RINA test report n. 2016CS011021/14 2017).

**Figure 6.**Temperature distribution on the fire-unexposed side of the door at the final stage of the fire test (60 min); scale in (K).

**Figure 7.**Displacements field at the final stage of the fire test (60 min); scale in (m). Displacements perpendicular to the door face, along y direction.

**Figure 8.**Measurement point 1: decomposition of the heat flow as a function of time along the Cartesian directions (x, y, z).

**Figure 9.**Transversal temperature profile at the final stage of the fire test (60 min) along the line passing through measuring point 5 (see black line in Figure 1b): comparison between foam and rock-wool model.

**Figure 10.**(

**a**) Decomposition of the heat flow as a function of time along the Cartesian directions (x, y, z) at measuring point 1: (

**b**) Transversal temperature profile at the final stage of the fire test (60 min) along the line passing through measuring point 5 (see black line in Figure 1b). Door characteristics: 117 mm width, 2117 mm height, steel sheet of 5.0 mm and 1.5 mm for the exposed side with rock-wool, 60 mm thick, and 180 kg m

^{−3}density as insulator. Adapted from [14].

Material | Young Modulus E (GPa) | Poisson Ratio ν (-) | Density ρ (kg m ^{−3}) | Thermal Conductivity k (W (m K) ^{−1}) |
---|---|---|---|---|

Carbon steel | 190 | 0.3 | 7800 | 27.29 |

Rock wool | 1.0 × 10^{−3} | 0.0 | 150 | 0.035 |

Foam | 3.4 × 10^{−3} | 0.0 | 130 | 0.045 |

**Table 2.**Comparison between modelled—using, respectively, Eurocode and self-consistent convection coefficient—and measured temperatures; error estimation and validation of the self-consistent model.

Point | T Eurocode Model (°C) | T Self-Consistent Model (°C) | T Measured * (°C) | |E| | U |
---|---|---|---|---|---|

1 | 206.0 | 100.0 | 99.0 | 1.0 | 1.4 |

2 | 206.0 | 100.0 | 100.0 | 0.0 | 1.0 |

5 | 204.0 | 99.0 | 99.0 | 0.0 | 1.0 |

3 | 206.0 | 100.0 | 101.0 | 1.0 | 1.4 |

4 | 206.0 | 100.0 | 101.0 | 1.0 | 1.4 |

Point | d Model (mm) | d Measured * (mm) | |E| | U |
---|---|---|---|---|

A | 0.5 | 0.0 | 0.5 | 1.0 |

B | 19.2 | 19.0 | 0.2 | 1.0 |

C | 19.2 | ND | / | / |

D | 0.5 | ND | / | / |

I | 17.9 | 17.0 | 0.9 | 1.3 |

Position | T Foam Model (°C) | T Rock Wool Model (°C) | ∆T (°C) |
---|---|---|---|

1 | 120 | 100 | 20 |

2 | 120 | 100 | 20 |

5 | 119 | 99 | 20 |

3 | 120 | 100 | 20 |

4 | 120 | 100 | 20 |

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**MDPI and ACS Style**

Kyaw Oo D’Amore, G.; Mauro, F.; Marinò, A.; Caniato, M.; Kašpar, J.
Towards the Use of Novel Materials in Shipbuilding: Assessing Thermal Performances of Fire-Doors by Self-Consistent Numerical Modelling. *Appl. Sci.* **2020**, *10*, 5736.
https://doi.org/10.3390/app10175736

**AMA Style**

Kyaw Oo D’Amore G, Mauro F, Marinò A, Caniato M, Kašpar J.
Towards the Use of Novel Materials in Shipbuilding: Assessing Thermal Performances of Fire-Doors by Self-Consistent Numerical Modelling. *Applied Sciences*. 2020; 10(17):5736.
https://doi.org/10.3390/app10175736

**Chicago/Turabian Style**

Kyaw Oo D’Amore, Giada, Francesco Mauro, Alberto Marinò, Marco Caniato, and Jan Kašpar.
2020. "Towards the Use of Novel Materials in Shipbuilding: Assessing Thermal Performances of Fire-Doors by Self-Consistent Numerical Modelling" *Applied Sciences* 10, no. 17: 5736.
https://doi.org/10.3390/app10175736