Performance of the Intumescent Coatings in Structural Fire via ANN-Based Predictive Models

Abstract

In this paper, an Artificial Neural Network (ANN) is built to predict the performance of intumescent coatings subjected to the ISO 384 fire curve. The performance metric is called the Retention Loss Onset Time (RLOT) in the structural steel. The network receives the steel and coating thicknesses as input and provides RLOT as the performance of any intumescent coating in a fire accident with substantial accuracy. The required data for obtaining the model is provided by revisiting the recent attempts in this field, which include hybrid numerical and experimental methods. It is found that the trapped gas fraction parameter and empirical expansion ratio substantially affect the accuracy of predictive modelling. Therefore, a new, comprehensive dynamic model that numerically simulates the bubble expansion process has been developed. This novel method directly determines the expansion ratio of the thermal conductivity model. The Eurocode is then used with multi-layer models to predict the steel temperature profile for a 1 h duration ISO fire. The accuracy is improved by modelling the temperatures and thermal resistances at the centre of each divided layer. The effects of different coatings and steel thicknesses are also investigated to provide the required data. The results are verified and validated by comparing them with the recent numerical and empirical results available in the literature.

1. Introduction

Steel is widely used in construction across the globe due to its strength, light weight, endurance, versatility, recyclability and many other attractive properties, from design to maintenance and disposal [1]. When the structural steel is exposed to a fire, it loses strength and stiffness at elevated temperatures.

Structural steel starts to lose its yield strength at 350 °C, and the retention drops to approximately 10% at 800 °C. The critical temperature can be defined as the temperature at which the structural steel is expected to fail. It has a value between 350 °C and 800 °C depending on the degree of utilisation [2]. Buildings are likely to collapse when the steel temperature exceeds the critical temperature. This may lead to injuries and casualties when the individuals cannot evacuate before the building collapses. The responsible company will be affected by tremendous economic and social impacts. Further legal actions may be taken for breaching the criminal and civil laws in the UK. More importantly, the company’s reputation will be damaged, and future business opportunities will be reduced. It is therefore important to ensure the building’s structural integrity in a fire.

BS EN 1363-1 [3] has shown that the load-bearing capability can be maintained over a greater range of temperatures in fire resistance tests by applying insulations on the steel surface. The insulation is more commonly known as passive fire protection. It has a low thermal conductivity, which reduces the rate of steel temperature increase. This gives time for individuals to evacuate to a safe location and for the emergency shutdown of critical assets, i.e., chemical and nuclear plants.

Passive fire protections are available in intumescent coating, concrete encasement, calcium silicate or gypsum plaster boards, cementitious spray-on systems and flexible fire blankets [4]. Amongst all types of passive fire protection, intumescent coating has become increasingly popular in the past 20 years because of aesthetics, speed of construction, cost savings, corrosion protection, and better quality control [5].

Intumescent coating is a thin layer coating with a thickness of less than a few millimetres [4]. It is typically applied on the steel surface with a primer and a sealer coat. The primer is initially used on the steel surface to ensure good adhesion with the following intumescent coating and provide corrosion resistance. The second paint applied is the intumescent coating, which gives fire resistance to the steel. The sealer coat is painted last to provide a decorative finish and protection against the environment [5].

The intumescent coating contains an acid source, blowing agent and charring agent [4]. The intumescent process begins when the acid source undergoes thermal decomposition at approximately 200 °C. The coating surface melts and forms a viscous liquid [6]. The blowing agent then decomposes to release gas bubbles. This causes the coating to swell up to 100 times its original thickness [7]. The expansion continues until all blowing agents and decomposition products have reacted or the char layer cannot withstand the internal pressure increase. Char is formed as the temperature increases further. This is a carbonaceous material that confines the viscous liquid and gases generated. The char starts degrading when the coating has fully reacted and continues to heat up. The char is partially oxidised to form CO₂ or CO [8] and the colour gradually changes from grey to white due to the reduction in carbon content. The white colour can be caused by the other filler materials within the coating, i.e., TiO₂ [9]. The full intumescent process is also outlined graphically in Figure 1.

Swelling is the key stage for fire protection in the intumescent process. Gas bubbles are generated and expanded as the temperature rises. Gases have a lower thermal conductivity than solids and liquids. As the coating becomes more porous, the overall thermal conductivity of the coating decreases. This provides fire resistance to the steel underneath by reducing the steel’s temperature increase rate. Therefore, it is important to understand how the coating expands in the fire and its impact on the steel temperature. A common way of measuring the coating expansion is by using an expansion ratio, which compares the current coating thickness with its original value before the expansion. Although the expansion ratio is the key factor in modelling coating performances, very few studies have included it in their models.

In previous studies, an empirical parameter $E$ was introduced to represent the expansion ratio [10,11]. The expansion process is assumed to be fast, and the coating reaches its maximum thickness instantaneously. The model, therefore, uses the maximum expansion ratio and keeps it constant during the fire. This assumption is valid for coatings with a thin thickness, as the expansion process is more rapid. However, the accuracy of this method decreases as the coating becomes thicker. Therefore, the model does not capture the variation in the expansion ratio at different temperatures.

The model is improved by establishing the direct link between the expansion ratio and the thermal conductivity [12]. This allows the possibility of modelling the thermal conductivity accurately when the expansion ratio is provided. As a similar approach to previous models is adopted when simulating the coating expansion, the same concerns apply to the model, as the expansion ratio remains constant.

Zhang et al. [13,14] proposed a model that relates the thermal conditions to the expansion ratio. This addresses the problem in previous models and allows more accurate modelling of the coating behaviour during the expansion. The input parameters provided in their model are also independent of the fire conditions. The same set of input data can therefore be used to predict other fire conditions. The generated results are in close agreement with their cone calorimeter tests [13] and furnace fire tests [14]. The main concern in their model is the introduction of the trapped gas fraction $\beta$. They have assumed that only a fraction of the gas generated is trapped in the coating and contributes to the expansion. The rest of the gas will escape into the atmosphere by travelling through the coating, including the char layer. This is correct when more pores appear as the coating expands further at higher temperatures. The empirical expression for β was derived through a nonlinear regression of the experimental data. It is also only specific to the coating studied by Zhang et al. [13,14]. There is no high-fidelity model regarding coating expansion chemistry and physical behaviour to verify the expression for β.

Cirpici et al. [15] adopted the bubble expansion model from Amon et al. [16] to simulate the coating expansion process. In contrast to Zhang et al.’s model [13,14], they assumed all gas released contributes to the coating expansion process. They generated the insulated structural steel temperature profile using expansion ratio curves with different heat fluxes [15,17]. They have also proposed a multi-layer model which divides the coating into five equal layers with identical temperature and thermal resistance at each layer. This allows more accurate calculations of the coating temperature and non-uniform conductive heat transfer. Key parameters of the model, such as viscosity and bubble pressure, are dependent on the coating temperature. The solution quality can therefore be improved by accurately modelling the coating temperature. This can be shown by the close alignment between the generated results and the furnace test data produced by Zhang et al. [14]. However, the expansion ratio is obtained by linear interpolation between expansion ratio curves with different heat fluxes at the coating temperature. The expansion ratio in the thermal conductivity calculations is not directly linked to the bubble expansion model. The approach of selecting the expansion ratio based solely on the heat flux and coating temperature is also questionable. Furthermore, the multi-layer model claims that the temperature is identical within each layer, but the temperatures and thermal conductivities are calculated at the layer boundary. It is unclear whether the layer temperature is taken from the boundary inlet, outlet or an average of both values.

Recent research studies have shifted the focus to solving more complex differential equations using computational fluid dynamics (CFD) software [18,19]. The accuracy of these models is improved by solving the governing equations numerically using a finite-difference or finite-element method. However, the computation cost is relatively high and requires significant solution time. Consequently, the problem duration in some studies does not exceed 20 min [18]. A. Bilotta et al. [20] have produced experimental data for existing steel structures coated with restored intumescent paint under different fire conditions and compared them with finite-element simulation results. The results show good alignment in the first 90 min, but the error increases afterwards. de Silva et al. [21] have used empirical regression models for thermal conductivity in finite-element simulations. The results are in close agreement between experimental and simulation results. However, small-scale lab tests are required to generate the regression models for each case. The experimental results are also close to the simulation results from different regression models for different cases, which makes it difficult to determine which regression model should be followed. In summary, the expansion ratio is a key parameter in modelling coating performance in a fire. Researchers have made significant progress in developing models to simulate the coating behaviours, yet each model has limitations and shortcomings. There is a need to explore the topic with greater accuracy and without high computation costs. This paper will attempt to address the issues identified from previous models and propose a model that links the expansion ratio directly to thermal conductivity and structural steel temperature calculations.

In this article, we have updated the existing models followed by substantial simulations. Consequently, we have introduced an important parameter, known as Retention Loss Onset Time (RLOT), which accounts for the time that is required for the steel to start losing its yield stress at 350 °C.

Moreover, we have explained how, from each of these models, we can build an Artificial Neural Network (ANN) model to estimate RLOT. It should be remembered that the results of this paper are validated through the tests done by Zhang [14] only, since more experimental data is not available.

2. Methodology Overview

The numerical simulation was carried out on a 100 mm length × 100 mm width steel plate with different thicknesses. Different thicknesses of intumescent coating were simulated on the steel surface for insulation. The thermal effects on the steel plate were simulated under the ISO 834 [22] fire conditions for 1 h.

The coating material simulated was a commercially available water-based intumescent coating by Firetex, Leighs Paints, Leeds, UK. This was identical to the coating used by Zhang et al. [13]. The coating and steel plate thickness were also the same as in the cases studied by Zhang et al. [13]. Based on numerical simulation with considerations of the expansion ratio, the simulation results were compared with the empirical results produced by Zhang et al. [13]. The key parameters for each case are listed in

Table 1. Case study details.

Case	Fire Type	Steel Plate Thickness/mm	Coating Thickness/mm
A04ISO	ISO Fire	6	0.4
A08ISO	ISO Fire	6	0.8
A12ISO	ISO Fire	6	1.2
B04ISO	ISO Fire	10	0.4
B08ISO	ISO Fire	10	0.8
B12ISO	ISO Fire	10	1.2
C04ISO	ISO Fire	20	0.4
C08ISO	ISO Fire	20	0.8
C12ISO	ISO Fire	20	1.2

.

3. Expansion Ratio

3.1. Expansion Model Overview

The first step in the problem-solving process is to calculate the expansion ratio of the coating. When a fire occurs, the temperature of the coating rises, triggering chemical reactions once it reaches approximately 200 °C. These reactions produce gas bubbles that grow as the temperature increases, causing the coating to expand. The expansion reduces the thermal conductivity significantly, causing a slower growth of steel temperature during a fire. Therefore, coating expansion is the most important factor in modelling intumescent coating performance in a fire.

The coating is populated with a total of 60 × 60 × 10 (36,000 in total) uniform-sized bubbles, as recommended by Cirpici et al. [15]. Each bubble has a radius of $R$ and an outer shell with a radius of $S$. The outer shell is a supersaturated envelope where gas diffuses into the bubble.

Equation (1) is the governing equation for modelling the bubble growth. The equation was developed to find bubble growth in foaming [16] and can be adapted to find the expansion ratio for intumescent coatings [15]. The formula shows that the bubble growth is primarily driven by the pressure increase due to the gas released in the reactions. The resistance forces to the bubble growth are the viscosity ($\mu$) and the surface tension ($\gamma$).

$${\displaystyle \frac{dR}{dt}}=R\left[{\displaystyle \frac{P_g-P_a-\left(\frac{2\gamma }{R}\right)}{4\mu }}\right]\left({\displaystyle \frac{S^3-R^3}{S^3}}\right)$$

(1)

Equation (1) is also a differential equation where $P_g$, $\gamma$ and $\mu$ are all temperature- and time-dependent. This makes the equation challenging to solve. A numerical approach is therefore adopted to estimate $∆R_n$ by taking a small time step, where $n$ is the number of steps taken. The algebraic representation is shown in Equation (2). The time step size is selected to be 1 s to provide sufficient solution accuracy

$$∆R_n=R_n\left[{\displaystyle \frac{P_{g,n}-P_a-\left(\frac{2{\gamma }_n}{R_n}\right)}{4{\mu }_n}}\right]\left({\displaystyle \frac{{S_n}^3-{R_n}^3}{{S_n}^3}}\right)∆t$$

(2)

The outer shell radius for the next time step can be determined using the new bubble radius by Equation (3). The initial outer shell diameter ($S_0$) can be calculated using Equation (4). The initial bubble diameter ($R_0$) is selected to be half of the initial outer shell diameter. As the fire progresses, the bubble diameter increases and reaches close to the outer shell diameter. Therefore, the initial bubble diameter does not contribute significantly to the results. The expansion ratio can be determined using Equation (5).

$$S_{n+1}={\left({S_0}^3+{R_{n+1}}^3-{R_0}^3\right)}^{{\displaystyle \frac{1}{3}}}$$

(3)

$$S_0={\left[{\displaystyle \frac{3}{4\pi N_{cell}}}\right]}^{{\displaystyle \frac{1}{3}}}$$

(4)

$$E_{S,n}={\displaystyle \frac{S_{n+1}}{S_0}}$$

(5)

3.2. Expansion Model Assumptions

Several assumptions are made when deriving Equation (1) [16]. It is assumed that an envelope of supersaturated solution will be created outside each bubble to allow gas diffusion into the bubble. The envelope is assumed to have uniform concentration and temperature. The bubble nucleation is assumed to be instantaneous and heterogeneous. The effect of coalescence is assumed to be negligible and the number of bubbles per unit mass is assumed to be constant throughout the growth. The gas inside the bubble is assumed to be an ideal gas. The thermodynamic equilibrium is assumed to follow Henry’s Law at all times at the gas–liquid interface. The outer shell solution is assumed to be an incompressible Newtonian fluid. It is also assumed that no gas is lost to the surroundings.

3.3. Bubble Pressure

The bubble pressure ($P_g$) increases as more gas is produced at higher temperatures. The gas pressure is related to the volume through the ideal gas law. Mass is also linked to volume via density. This suggests that the bubble pressure can be derived by monitoring the bubble mass at different temperatures.

Zhang et al. [13] have produced empirical results for the mass loss of intumescent coating at different temperatures, as shown in Figure 2. This mass loss refers to the gas mass generated in the experiment and escaped to the surroundings. By comparing the remaining mass of the coating with the original value before heating, the result for % mass loss at different temperatures is produced. The gas mass generated from the coating is transferred to the bubble via the outer shell. This causes a mass increase in the bubble, leading to an increase in bubble volume and pressure.

Equation (6) is derived from the ideal gas law [15]. It uses the gas volume increase in each bubble derived from the empirical results [13] and the initial bubble volume to determine the pressure increase. This equation is used to calculate the bubble pressure at each time step.

$$P_{g,n}={\displaystyle \frac{∆V_{G,Bubble,n}+V_0}{V_0}}{P}_a$$

(6)

3.4. Viscosity

The intumescent coating viscosity data from Zhang et al. [13,14] is unavailable. The viscosity data is estimated using a similar intumescent coating material instead. Morice et al. [23] produced empirical results for a coating consisting of polypropylene (PP) and ammonium polyphosphate (APP)/pentaerythritol (PER), as shown in Figure 3. The mass loss for the PP/APP-PER system is similar to the mass loss obtained by Zhang et al. [13]. Based on the similar mass loss characteristics, the viscosities for both coating materials are assumed to be the same. The measured viscosity–temperature relationship on the PP/APP/polyurethane (PUR) system by Bugajny et al. [24], shown in Figure 4, is used in the expansion ratio calculations. This viscosity estimation approach is also adopted by Cirpici et al. [15].

3.5. Surface Tension

The intumescent coating surface tension is also unavailable from Zhang et al. [13,14]. Previous research shows that surface tension is negligible to the expansion ratio via sensitivity analysis [15,16]. The surface tension for polyethylene is used for the expansion ratio calculations [25].

4. Effective Thermal Conductivity

The porous material thermal conductivity model, shown in Equation (7), can be used to calculate the apparent intumescent coating thermal conductivity [12]. To account for the expansion effects, the apparent thermal conductivity can be divided by the expansion ratio to obtain the effective thermal conductivity, as shown in Equation (8) [17].

A novel approach is adopted by directly inputting Section 3’s expansion ratio into the thermal conductivity calculations. This differs from the approach of Cirpici et al. [17], where they generate expansion ratio curves at different heat fluxes and carry out linear interpolation to obtain the expansion ratio at each time step. The new approach inputs the expansion ratio directly and does not rely on linear interpolations, and this improves the solution’s accuracy.

$${{\lambda }^{*}}_n={\lambda }_s{\displaystyle \frac{\frac{{\lambda }_{g,n}}{{\lambda }_s}{{\epsilon }_n}^{\frac{2}{3}}+1-{{\epsilon }_n}^{\frac{2}{3}}}{\frac{{\lambda }_{g,n}}{{\lambda }_s}\left({{\epsilon }_n}^{\frac{2}{3}}-{\epsilon }_n\right)+1-{{\epsilon }_n}^{\frac{2}{3}}+{\epsilon }_n}}$$

(7)

$${\lambda }_{effective,n}={\displaystyle \frac{{{\lambda }^{*}}_n}{E_{S,n}}}$$

(8)

The solid thermal conductivity (${\lambda }_s$) is contributed by the coating’s solid char layer. As the coating temperature increases, the char layer oxidises to form a stable solid protective insulating layer on the coating surface [13], and the solid thermal conductivity remains approximately constant. For simplicity, it is assumed the char layer oxidation is instantaneous, and the solid thermal conductivity is 0.5 W m⁻¹ K⁻¹ [17].

The gas thermal conductivity (${\lambda }_{g,n}$) considers the heat transfer through gas conduction and radiation. This relationship is shown in Equation (9). The gas conduction part can be calculated using Equation (10) [26], and the radiation part can be found using Equation (12) [12].

The current pore diameter ($d_{pore,n}$) is required to solve Equation (12). This can be obtained by assuming the current pore diameter to the maximum pore diameter ratio is identical to the current expansion to the maximum expansion ratio. The maximum pore diameter is considered to be 3.5 mm [12]. The maximum expansion ratio is initially assumed to be any random number. The revised maximum expansion ratio is calculated numerically using Excel Solver by comparing the initial value with the expansion ratio at the last time step. Using the maximum pore diameter, maximum expansion ratio and the current expansion ratio obtained from Section 3, the current pore diameter can be calculated using Equation (11).

$${\lambda }_{g,n}={\lambda }_{g, cond,n}+{\lambda }_{g, rad,n}$$

(9)

$${\lambda }_{g, cond,n}=4.815\times {10}^{-4}\times {T_{coating, n}}^{0.717}$$

(10)

$$d_{pore,n}=d_{pore, max}{\displaystyle \frac{E_{S,n}}{E_{S,max}}}$$

(11)

$${\lambda }_{g, rad,n}={\displaystyle \frac{2}{3}}\times 4d_{pore,n}e\sigma {T_{coating,n}}^3$$

(12)

The porosity (${\epsilon }_n$) is often referred to as the void fraction. This represents the ratio between the void volume and the total volume. For intumescent coating, the void volume is the gas volume occupied in the gas–liquid mixture. As all the bubbles in the coating are assumed to have identical size and chemical composition, the coating porosity can be considered the same as the bubble shell porosity. The initial volume of the bubble shell has a large liquid content. The gas content in the shell increases as the coating temperature increases. The gas volume is therefore the current shell volume minus its initial value before heating. The porosity can be obtained by dividing the gas volume by the current shell volume, as shown in Equation (13). The expression can be further simplified to only relate to the outer shell expansion ratio obtained in Section 3. This expression is slightly different to the approach from Cirpici et al. [17] which uses the bubble radius instead of the volume. Equation (13) gives greater accuracy in modelling the porosity as it calculates the void volume fraction instead of the radius ratio.

$${\epsilon }_n={\displaystyle \frac{{{\frac{4}{3}\pi S}_{n+1}}^3-\frac{4}{3}\pi {S_0}^3}{{{\frac{4}{3}\pi S}_{n+1}}^3}}=1-{\displaystyle \frac{{S_0}^3}{{S_{n+1}}^3}}=1-{\displaystyle \frac{1}{{E_{S,n}}^3}}={\displaystyle \frac{{E_{S,n}}^3-1}{{E_{S,n}}^3}}$$

(13)

The expansion ratio is used to find the effective thermal conductivity, gas radiation conductivity and porosity in Section 4. The expansion process, therefore, significantly impacts the thermal conductivity numerically. This reinforces the importance of the expansion process in the modelling process.

5. Structural Steel Temperature

The Eurocode [27] shows that the change in structural steel temperature over time can be calculated using Equation (14). The model assumes the insulation is thick, so the structural steel is only affected by conduction heat transfer through the insulation coating. This assumption is only valid after the coating has expanded significantly in a fire. Radiation and convection heat transfers should also be considered to accurately model the steel temperature, as they are dominant at the start of the fire. Therefore, Equation (15) includes the overall heat transfer coefficient ($h_n$) to account for this effect. Equation (15) is also used as the governing equation for modelling changes in the structural steel temperature over time.

$$\mathsf{\Delta }{T}_{S,n}={\displaystyle \frac{\left(T_{g,n}-T_{S,n}\right)\frac{A_p}{V_s}}{\frac{x}{{\lambda }_{effective,n}}{C}_S{\rho }_S\left(1+\frac{\varphi }{3}\right)}}\mathsf{\Delta }t-\left(e^{{\displaystyle \frac{\varphi }{10}}}-1\right)\mathsf{\Delta }{T}_{g,n}$$

(14)

$$\mathsf{\Delta }{T}_{S,n}={\displaystyle \frac{\left(T_{g,n}-T_{S,n}\right)\frac{A_p}{V_s}}{\left(\frac{1}{h_n}+\frac{x}{{\lambda }_{effective,n}}\right)C_S{\rho }_S\left(1+\frac{\varphi }{3}\right)}}\mathsf{\Delta }t-\left(e^{{\displaystyle \frac{\varphi }{10}}}-1\right)\mathsf{\Delta }{T}_{g,n}$$

(15)

Equations (14) and (15) are a key part of the model. Moreover, the boundary conditions of the unsteady heat conduction equations are also embedded in them by Eurocode [27] for passive fire insulation.

The overall heat transfer coefficient consists of two parts, radiation and convection. The radiation heat transfer coefficient is calculated using Equation (16), and the convection heat transfer coefficient is taken as 25 W m⁻² K⁻¹ [28].

$$h_{rad,n}=e\sigma \left({T_{g,n}}^2+{T_{S,n}}^2\right)\left(T_{g,n}+T_{S,n}\right)$$

(16)

$$h_n=h_{rad,n}+h_{conv,n}$$

(17)

$\varphi$ is the ratio of the heat stored in the coating relative to the heat stored in the structural steel. Equation (18) can be used to calculate this parameter.

$$\varphi ={\displaystyle \frac{C_c{\rho }_c{A}_{p}x}{C_s{\rho }_s{V}_s}}$$

(18)

6. Single-Layer Model

The coating temperature is the key parameter in finding the coating pressure, viscosity, surface tension and thermal conductivity. It is therefore important to update the parameter accurately after each time step. The single-layer model is used to calculate the coating temperature [17]. The heat flux is constant throughout the layer if the heat transfer is considered to be in a steady state. The heat flux from the fire to the steel can therefore be assumed to be identical to the heat flux from the fire to the coating surface. This relationship can be presented using Equation (19). The coating temperature can be modelled as the mean temperature between the coating surface and the steel, as shown in Equation (20).

$$T_{surface,n+1}=T_{g,n}-{\displaystyle \frac{\frac{\left(T_{g,n}-T_{S,n+1}\right)}{\left(\frac{1}{h_n}+\frac{x}{{\lambda }_{effective,n}}\right)}}{h_n}}$$

(19)

$$T_{coating,n+1}={\displaystyle \frac{T_{surface,n+1}+T_{S,n+1}}{2}}$$

(20)

7. Multi-Layer Model

The single-layer model assumes steady-state conduction heat transfer across a single coating layer. This method provides a simple yet effective way of estimating the temperature profile across the coating. However, the actual temperature profile is non-uniform across the coating. Additionally, the coating temperature is an important parameter in determining the coating pressure, viscosity, surface tension and thermal conductivity. The conduction heat transfer across the coating also affects the change in structural steel temperature over time. It is therefore necessary to model the heat transfer more accurately.

The multi-layer model divides the coating into five equal-distance layers, as shown in Figure 5. The temperature and thermal resistance are calculated at the centre and layer boundary. The temperature and thermal resistance at the centre of each layer are used for calculating the coating temperature and the overall conductive thermal resistance, respectively. The multi-layer model accounts for the difference in thermal conductivity in different positions within the coating. This improves the model’s accuracy and solution quality under non-uniform heat transfer conditions.

The temperatures and thermal resistances in the multi-layer model can be determined after the thermal conductivity calculations and before the structural steel temperature calculations. The multi-layer model calculations start with finding the coating surface temperature using Equations (21)–(23). The method in this step is similar to finding the coating surface temperature in the single-layer model. The expansion ratio and thermal conductivity are calculated following the methodology outlined in Section 3 and Section 4.

$$q_n={\displaystyle \frac{\left(T_{g,n}-T_{S,n}\right)}{\left(\frac{1}{h_n}+\frac{x}{{\lambda }_{effective,n}}\right)}}$$

(21)

$$R_{g,n}={\displaystyle \frac{1}{h_n}}$$

(22)

$$T_{\mathrm{0,1},n}=T_{g,n}-R_{g,n}{q}_n$$

(23)

The effective thermal conductivity at the inlet boundary (${\lambda }_{x-1,x,n}$) can be calculated using the coating surface temperature ($T_{\mathrm{0,1},n}$) following the method outlined in Section 4. This value can be inputted into Equation (24) to obtain the conductive thermal resistance at the surface. The temperature at the boundary between Layers 1 and 2 is initially calculated using Equation (25). The temperature at the centre of Layer 1 is determined by taking the mean value between the Layer 1 boundaries shown in Equation (26). The thermal resistance at the Layer 1 centre can be obtained using the temperature at the centre of Layer 1, following Equation (27). The temperature at the boundary between Layers 1 and 2 can be revised using the thermal resistance at the Layer 1 centre shown in Equation (28).

The calculation method is repeated for the following layers to obtain all the temperatures and thermal resistances shown in Figure 5.

The temperature modelling approach iterates the boundary temperatures and averages the boundary temperatures to obtain the centre temperature. This allows the temperatures and thermal resistance to be calculated more accurately.

$$R_{x-1,x,n}={\displaystyle \frac{∆x_x}{{\lambda }_{x-1,x,n}}}$$

(24)

$$T_{x,x+1,n}=T_{x-1,x,n}-R_{x-1,x,n}{q}_n$$

(25)

$$T_{x,n}={\displaystyle \frac{{T_{x-1,x,n}+T}_{x,x+1,n}}{2}}$$

(26)

$$R_{x,n}={\displaystyle \frac{∆x_x}{{\lambda }_{x,n}}}$$

(27)

$$T_{x,x+1 new,n}=T_{x-1,x,n}-R_{x,n}{q}_n$$

(28)

The structural steel temperature in the multi-layer model is calculated slightly differently from the single-layer model. The thermal resistances are summed together to form an overall thermal resistance, as shown in Equation (29). This overall thermal resistance replaces the ${\displaystyle \frac{1}{h_n}}+{\displaystyle \frac{x}{{\lambda }_{effective,n}}}$ term in the adopted single-layer model from Eurocode Part 3 [27]. The change in structural steel temperature over time can be calculated using the revised formula shown in Equation (30).

$$R_{Total,n}=R_{g,n}+R_{1,n}+R_{2,n}+R_{3,n}+R_{4,n}+R_{5,n}$$

(29)

$$\mathsf{\Delta }{T}_{S,n}={\displaystyle \frac{\left(T_{g,n}-T_{S,n}\right)\frac{A_p}{V_s}}{R_{Total,n}{C}_S{\rho }_S\left(1+\frac{\varphi }{3}\right)}}\mathsf{\Delta }t-\left(e^{{\displaystyle \frac{\varphi }{10}}}-1\right)\mathsf{\Delta }{T}_{g,n}$$

(30)

The coating temperature in the multi-layer model can be determined by the mean between the centre temperature of each layer and the steel temperature shown in Equation (31).

$$T_{coating,n+1}={\displaystyle \frac{T_{1,n}+T_{2,n}+T_{3,n}+T_{4,n}+T_{5,n}+T_{s,n}}{6}}$$

(31)

8. Comparative Analysis with Existing Models and Experimental Data

The calculated steel temperature profile and thermal conductivity are verified by comparing them with an adopted model of Cirpici et al. [17]. It involves calculating the change in coating temperature at each time step to determine the heat flux. The results are compared with expansion ratio curves to obtain the expansion ratio [17]. Linear interpolations are also performed when the heat flux values are between the associated curves. The expansion ratios are input into the thermal conductivity and steel temperature model outlined in Section 4, Section 5, Section 6 and Section 7 for generating solutions.

The calculated steel temperature profile will also be validated by comparing it with the empirical results of Zhang et al. [14]. The experimental results come from furnace tests on the same scenarios as this study. The furnace fire temperature is controlled by matching the desired fire temperature with the measured average temperature from six bead thermocouples, which are situated at the centre, left side and right side of two insulated steel plates [14]. The steel temperature is also measured throughout the 1 h ISO fire, ranging from approximately 25 °C to 950 °C. This ensures the variables in the case studies are consistent between the simulation and the empirical results and allows for an efficient validation process to be carried out.

9. Results and Discussion

9.1. Steel Thickness

The calculated steel temperature profile is compared with the profile without the intumescent coating to highlight the significance of passive fire protection during a fire. The breakpoint where the coating starts becoming effective is also circled in Figure 6 to show its importance. The key temperatures for steel plastic deformation are also plotted in Figure 6, Figure 7, Figure 8 and Figure 9. Structural steel starts to lose its yield strength at 350 °C, and the retention drops to approximately 10% at 800 °C [27]. It is therefore important to capture the fire duration to reach these milestone temperatures to estimate the evacuation time available for personnel in a fire. The performance of the coating can be determined by comparing the fire duration to reach these milestone temperatures with and without the coating. The longer it takes to approach these temperatures, the more effective the coating is.

Figure 6 and Figure 7 show the structural steel temperature profiles with different coating thicknesses. The steel temperature increases more slowly when a thicker coating is applied. The thicker coating allows less conduction heat transfer from the fire to the steel surface [27]. This reduces the overall rise in steel temperature during the fire and increases the time to reach 350 °C and 800 °C. The thicker coating curves are also further apart from the curves of steel plates without coating. All of the above indicates the coating’s passive fire protection performance increases as the layer becomes thicker. This trend remains the same for different steel plate thicknesses, as shown in Figure 6 and Figure 7.

9.2. Results Verification and Validation

Figure 8 and Figure 9 show the calculated structural steel temperature profile compared to the simulation results from the adopted model of Cirpici et al. [17] and the empirical results from Zhang et al. [14]. The results are provided from Case A04ISO and A12ISO for clarity. Figure 8 and Figure 9 show that the predicted results have a similar trend to the test results. The simulation results are in slightly closer agreement with the test results than the adopted model of Cirpici et al. [17]. The steel temperature from the test result is also higher than the simulation results in Figure 9. This may be caused by cracks developed in the coating during the furnace fire test at higher temperatures [17]. The gas generated may have escaped through the cracks, causing the thermal conductivity to increase. This results in a higher steel temperature in the test. The current mathematical model cannot capture this feature; thus, the predicted results tend to be lower than the test results. The multi-layer model results in Figure 9 show closer alignment with the test results than the single-layer model. This could be caused by the coating temperature and heat transfer being captured more accurately in the multi-layer model, as discussed in Section 7.

In Figure 6, Figure 7, Figure 8 and Figure 9, it is shown how various models and tests can indicate important fire safety parameters that are measured by time. These models can overestimate or underestimate the time when we compare them to the tests. It is obvious that underestimated models are more conservative since, referring to that time, personnel can be evacuated earlier.

We have introduced a new safety parameter and named it Retention Loss Onset Time (RLOT), the time that is required for steel to reach 350 °C in which yield stress drops. The examination of Figure 9 shows how some of these methods overestimate or underestimate RLOT compared to the testing. In the subsequent section that we have added, we have explained more about obtaining ANN-based models for RLOT.

9.3. Data Analysis with an ANN Model

Application of ANN in fire-related research can be found in the recent literature [29]. In this paper, we have used the results of 36 simulations, and the RLOT for the multi-layer model is plotted in Figure 10. The x-axis indicates the paint thickness in mm, and the y-axis indicates the steel thickness. The z-axis shows the RLOT in minutes.

It is obvious that increasing the paint thickness and steel plate thickness increases RLOT significantly.

To find a verification function for the data in Figure 10, we have found an Artificial Neural Network (ANN) architecture that is depicted in the following figure:

The architecture in Figure 11 consists of 27 neurons. Apart from two input neurons for the paint and steel thickness (in mm) and an output neuron for RLOT, there are 24 neurons in four hidden layers (6 neurons each). In this network, there are 126 weights (IDs 1–126) designated in Figure 11.

Apart from these weights, there are also biases. Some of these weights and biases are indicated in

Table 2. Input weights for paint thickness in mm and steel thickness in mm.

Input	Weight	Weight	Weight	Weight	Weight	Weight
Paint thickness	2.2269	1.0339	2.1020	1.7981	−3.6484	−3.1766
Steel thickness	−0.4752	−3.5265	0.7820	1.1425	−0.4571	0.2169

,

Table 3. Weight and bias numbers (IDs 13–38) for six neurons in hidden Layers 1 to 2.

Name	Weight	Weight	Weight	Weight	Weight	Weight	Bias
Neuron 1	0.7838	−1.3671	−0.3988	−1.1253	0.9328	0.2731	−2.1139
Neuron 2	1.2503	1.0484	0.5759	−0.0527	0.1855	1.4490	−1.5561
Neuron 3	−1.0423	−0.3203	1.3713	−0.2152	0.0085	0.4753	0.6749
Neuron 4	−1.1483	1.6754	1.1528	0.4155	−0.6588	0.8866	−0.0681
Neuron 5	−0.1845	0.2003	−1.0713	−1.3450	0.3356	−0.1257	1.5374
Neuron 6	0.7271	−1.0729	−0.1657	1.4165	−0.7026	−0.6534	1.5420

and

Table 4. Weight and bias numbers (IDs 121–126) from hidden Layer 4 to RLOT.

Layer 4	Weight	Weight	Weight	Weight	Weight	Weight
RLOT	−0.1509	−0.0333	−1.4112	−0.3811	−0.3464	−0.2777

.

The network that represents Table 3 is a deep learning network [30] with six layers. The refined output of the layer l − 1 is input to the layer l according to the following expressions:

$${\mathit{a}}^{\left(l\right)}=f^{\left(l\right)}\left({\mathit{W}}^{\left(l\right)}{\mathit{a}}^{\left(l-1\right)}+{\mathit{b}}^{\left(l\right)}\right)$$

(32)

In Equation (32), ${\mathit{W}}^{\left(l\right)}$ is the weight matrix for layer l, ${\mathit{a}}^{\left(l\right)}$ is the layer l output vector,${\mathit{a}}^{\left(l-1\right)}$ is the layer l − 1 output vector, and ${\mathit{b}}^{\left(l\right)}$ is the bias vector in layer l.

$${\mathit{a}}^{\left(l-1\right)}=f\left({\mathit{z}}^{\left(l-1\right)}\right)$$

(33)

In Equation (33) ${\mathit{z}}^{\left(l-1\right)}$ is the linear combination vector for the layer l −1, ${\mathit{a}}^{\left(l-1\right)}$ is the layer l − 1 output vector, and f is the activation function for the layer.

$${\mathit{z}}^{\left(l-1\right)}={\mathit{W}}^{\left(l-1\right)}{\mathit{x}}^{\left(l-1\right)}+{\mathit{b}}^{\left(l-1\right)}$$

(34)

In Equation (34), ${\mathit{W}}^{\left(l-1\right)}$ is the weight matrix for layer l − 1. ${\mathit{x}}^{\left(l-1\right)}$ is the input vector for layer l − 1, and ${\mathit{b}}^{\left(l-1\right)}$ is the bias vector in layer l − 1, while ${\mathit{z}}^{\left(l-1\right)}$ is the linear combination vector for the layer l − 1 [31].

In Figure 12, both the approximated RLOT from this network and the computed RLOT from the simulations are shown. It can be observed that the ANN surface fitted over the data, and intelligently folded over the non-smooth part of the data to cover it. Only in that fold, there are some estimation errors, as can be seen in Figure 12. This insignificant error is unavoidable due to the non-smoothness of the data shown in Figure 10. The regressor herein can produce close fitness according to the metrics that will be discussed in the next section.

The ANN can provide predictions for RLOT at different paint thicknesses and steel thicknesses that are within the range studied. This allows initial estimations for RLOT without carrying out lab testing, which can be repetitive and costly.

Obviously, this network is valid only when the intumescent paint is used for coating and protecting the structural steel. If other types of insulation are used, the underlying models will not be sophisticated, but the equivalent ANN model can be produced similarly for any desired output.

10. Model Limitations, Sensitivity Analysis and Optimisation

In this section, after discussing the novelty and the robustness of the model in this paper, the sensitivity of the parameters is investigated. The work herein is methodologically similar to Cirpici et al. [17], but the significant difference is the calculation of the expansion ratio from a direct simulation of the bubble growth model. However, in [17], the expansion ratio is determined from Figure 13.

Figure 13 is also shown in [17], but it is not explained how this figure was obtained. We are not using this figure in our paper. Our method is the calculation of the expansion ratio directly from the bubble growth model.

The first limitation of the model is the experimental data for viscosity and surface tension. It has been taken from the only available source in the literature [23,24,25] and cannot be implemented for all types of intumescent paints. Therefore, a sensitivity analysis is required, as shown in Figure 14. In this figure, the viscosity variations of ±50% and surface tension variation of ±80% and their effect on RLOT are investigated. It is shown that, while surface tension variations cannot change the RLOT, the viscosity variations can change the RLOT and the viscosity underestimation results in a higher RLOT. We recommend that, for future work, a reliability-based approach for RLOT predictions should be carried out, similar to the research that is described in [32].

Moreover, the sensitivity of the simulation time step size on the results is also an important issue. Since the bubble growth phenomenon is a fast event, the authors considered the finest 1 s time step. We have also investigated the effect of a coarse time step on RLOT results. Surprisingly, Figure 15 shows that, in some time steps, the RLOT can be up to 150% overestimated, which is not good for safety analysis. It is obvious in the coarse time step that the bubble expansion phase cannot be analysed correctly. Therefore, time steps of 1 s or less can be recommended.

Regarding optimisation, the architecture of the AI network in this paper is based on hyperparameter optimisation using the grid search method since the network is based on 36 simulations. However, in sophisticated networks, advanced optimisation techniques are required to find a proper architecture, as shown in [33]. The performance metrics and other important parameters of the network are tabulated in

Table 5. Performance metrics of the optimised ANN.

Error % Objective	MAE	RMSE	R2	Layer Range	Neuron Range	Epochs
10%	1.153712	−1.514361	0.988161	1–9	3–10	1000

.

By setting a 10% threshold for the maximum error, we want to ensure that the trained network is not overfitting. Therefore, according to Table 5, we have also limited the ranges for the grid search. The performance metric shows a reasonably good network.

Moreover, we have also tried traditional interpolation methods. The approximated interpolation function is a bilinear surface given by the following:

$$\mathrm{RLOT}\cong 1.9283t_s{t}_p-1.8576t_p+0.0739t_s+7.2302t_p^2+7.7518$$

(35)

In Figure 16, it is shown that a quadratic bilinear surface in Equation (35) does not provide the same fitting performance when compared to an ANN prediction in Figure 12.

11. Conclusions

A thorough investigation into the performance of intumescent coatings was conducted in ISO fire conditions. The review of the existing literature highlighted that the expansion ratio is the most crucial factor in modelling coating performance, which also aligns with the modelling details. The coating releases gases at elevated temperatures, causing the coating to expand. This reduces the thermal conductivity and the heat transfer from the fire to the steel surface. Therefore, the steel temperature increases at a slower rate, which causes a slower loss in yield strength. This results in a longer fire duration to reach the critical temperature and gives time for individuals to evacuate before the steel building collapses. This paper is written to develop a new multi-layer model that is verified by the previous tests, but does not consider the trapped gas fraction parameter (β). Current models use empirical expansion ratios, which are not always valid for every coating. The authors believe that directly simulating the bubble growth model in multiple layers provides sufficient accuracy to compensate for the absence of β, as it only represents the trapped gas.

The existing bubble expansion model is used to simulate the swelling behaviour of the coating. The model uses pressure as the drive for coating expansions, and the resistance forces are viscosity and surface tension. These parameters are found or derived from empirical data and are input into the coating expansion governing equation. The model’s differential equation is solved using a time step approach. The 1 h fire duration is divided into small steps, and the solution from the last time step is used for solving the next time step. This gradually generates the expansion ratio over the fire duration.

A novel approach is adopted to input the resulting expansion ratio directly into the thermal conductivity model. This differs from the current approach, where they generate expansion ratio curves at different heat fluxes and carry out linear interpolation to obtain the expansion ratio at each time step. The new approach inputs the expansion ratio directly and does not rely on linear interpolations, which improve the solution’s accuracy.

The Eurocode is then used with both single-layer and multi-layer models to predict the temperature of structural steel during a 1 h ISO fire. The multi-layer model divides the coating into five equal-thickness layers with identical temperatures and thermal resistances at each layer. The temperatures and thermal resistances are calculated at each layer to capture the non-uniform heat transfer characteristics. A novel approach is adopted to simulate the temperature and thermal resistance at the centre of each layer. The model’s accuracy is improved compared to the existing literature, which is unclear on whether the layer temperatures are taken from the boundary inlet, outlet or an average of both values.

The steel temperature profiles over the 1 h fire duration are plotted on graphs for different coating thickness scenarios. The insulation performance improves as the coating becomes thicker. The same trend also applies to different steel plate thicknesses. The simulated results are verified and validated by comparing the analytical data from the current adopted model and current empirical data, showing close agreement, especially with the multi-layer model.

We have introduced a new safety parameter and named it Retention Loss Onset Time (RLOT), the time that is required for steel to reach 350 °C, at which the yield stress drops. An Artificial Neural Network (ANN) is found to calculate the estimated RLOT with different steel thicknesses and coating thicknesses. The ANN shows close alignment with the simulation results. This allows predictions for RLOT at different steel thicknesses and coating thicknesses without lab testing, which can be repetitive and costly.

12. Recommendations for Future Work

This paper has covered the key areas for modelling the coating expansion ratio, thermal conductivity and structural steel temperature profile. However, certain aspects cannot be investigated due to the limitations on the availability of substantial experimental results in different fire conditions which can help to calibrate the bubble growth model and improve the simulation results. The coating’s exact viscosity and surface tension are also unavailable from the literature reviewed. Values from similar systems are used instead in the model. The model’s accuracy can be improved if these properties are obtained from chemical analysis. The recent advancements in AI and machine learning may also open up a new era for analysing the present data and proposing more accurate intumescent coating performance models. The validation in this paper relied on limited experimental data available. The authors also recommend that, for future work, they include more experimental data in training the AI models. Moreover, in future, the uncertainty of the key parameters and their effect on RLOT should also be investigated.

Finally, the dynamic model developed in this paper can be used to build digital twins of the structures potentially exposed to fire hazards. This can provide an online prediction of evacuation time when a fire has been detected in the building, and the sensors can report vital information. According to Eurocode [27], the RLOT herein is the time by which occupants should not be in a position near steel columns to avoid possible building collapse on them.