A Review of Ventilation Systems and Fire Incidents on Ships: A Bibliometric and Mathematical Modelling Approach

Each ship has its own set of terms and conditions governing the utilization of air ventilation systems. The ventilation of the air is tailored to the specific requirements of each compartment. Consequently, some vessels solely rely on natural ventilation, while others necessitate mechanical ventilation systems or fans. The fans are utilized for air intake or exhaust, depending on the compartment's AC requirements. As per regulations, compartments storing flammable or hazardous liquids or gases must be mechanically ventilated to dissipate heat. The air exchange process is conducted per the room's cooling requirements to mitigate the fire risk [1]. An air handling unit (AHU) is a commonly employed ventilation system that offers ease of maintenance and operates at a low noise level.

The equipment compartment that has been installed requires air to operate effectively. This air is provided through mechanical air vents or supply air vents. The purpose of this air supply is essential for powering the cooling equipment. Natural ventilation is sufficient for areas that do not necessitate specialized equipment. Air vents are installed in passenger or engine rooms to enhance passengers and crew members' comfort. It is recommended that the amount of fresh air flow be at least 40% of the total number of individuals in the room [2]. Conversely, the air is either directly expelled or not circulated in areas such as toilet rooms, kitchen rooms, dining rooms, and washing rooms.

Research on ship fires is a prevalent topic within the shipping industry. The most common cause of these fires is reported to be unintentional accidents involving flammable fuels [3]. Specifically, fuel leaks from reserve fuel tanks are a frequent ignition source. Due to the high flammability and weight of the fuel, it is highly susceptible to burning [4]. While a ship engine fire may not immediately impact the vessel's movement, it can eventually restrict mobility. The primary concern, however, is the potential for harm to the crew, including injuries and fatalities [5]. Most fire incidents, approximately 56%, occur in the main engine room or auxiliary engines. The progression of a ship fire can be categorized into five stages: initial flame at the leak site, fire spreading, intense combustion, decay, and finally, extinguishment [6].

The shipping or maritime sector is confronted with considerable challenges, among which fires represent a significant source of loss, as evidenced by statistical analyses [7]. These fires typically occur in ship engine rooms, where various types of propulsion engines and flammable fuel are present [8]. Consequently, when a fire breaks out in the engine room, the resulting losses and impacts on the ship are substantial [9]. As a last resort, engine room insulation is frequently utilized to combat ship fires. This approach aims to create a sealed environment where the fire is contained within a closed room. As a result, a closed fire experiences higher pressure compared to an open fire [10]. Moreover, oxygen concentration gradually decreases as the combustion process unfolds. Previous studies have highlighted that fire behavior in enclosed ship spaces differs significantly from smoke or fire in open spaces, particularly in rooms equipped with ventilation systems [11-14].

Simplifying concerns regarding fires in ship engines can be achieved by isolating the fire location in the center of the engine room. Previous studies have extensively investigated the smoke or fire formation process in locked or sealed rooms [15-18]. Some studies have also reported on and described unsealed machine rooms [19-22]. In the event of a closed fire, plumes of fire rise upward. This vertical smoke-rising process continues as the hot smoke, denser than the surrounding air, enters the surrounding air. Once the smoke reaches the ceiling, it spreads horizontally to other room parts [20]. Consequently, the model of fire smoke filling the ship's engine room can be depicted as an enhanced filling box model [23].

Researchers have recently examined multiple reviews concerning ventilation fire safety on ships. In 2017, the ship's ventilation system was scrutinized alongside structural hydrodynamics and the implications of scaling on ventilation design [24]. A review in 2021 delved into the key aspects of regulations and standards governing shipboard air ventilation systems, particularly in light of the pandemic [25]. In 2022, a systematic review was conducted on methods and solutions for evacuating ship passengers during emergencies or accidents [26]. From 2023 onward, virtual reality (VR) was evaluated in firefighting processes and evacuation procedures during ship accidents [27]. This review aims to assess the regulations and air ventilation systems on ships and analyze air ventilation using a bibliometric approach with the assistance of VOSviewer to allow readers to explore the innovative aspects of air ventilation and fire research on ships.

The main motivation of this research review is to examine and explore the air ventilation system implemented on ships in order to mitigate the risks associated with leakage and fire incidents. The impact of air ventilation on the potential dangers of fire and leakage aboard a ship is significant. Hence, conducting this review holds utmost importance. The most significant aspect of this review is to scrutinize mathematical modeling in calculating each parameter, performance-related variables, energy analysis, and exergy analysis in the study of air ventilation and fire on ships. Bibliometric findings help researchers identify research gaps, novelty, and current trends. Conversely, a mathematical model review offers a comprehensive examination of variables and parameters pertinent to advancing research in a manner akin to previous studies. Consequently, integrating bibliometrics and mathematical modeling proves invaluable for researchers involved in the future design, construction, and enhancement of air ventilation systems and leak prevention mechanisms on maritime vessels.

3.1 General requirements for air ventilation

Ships' ventilation systems typically comprise various components [28], such as a ventilation system in the room and engine room of the vessel. Additionally, ventilation is required in areas like the kitchen, sleeping room, service room, weather dampers, fire dampers, ventilation ducts, supporting equipment, cargo spaces, and the surrounding environment. In engine rooms falling under category A and special passenger rooms, the kitchen room must be separated from the ventilation system according to the regulations set by the classification society and following IMO rules [33]. The galley air ventilation system is not mandatory for small ships with a maximum passenger capacity of 36 individuals and cargo ships below a 4000 gross tonnage threshold. Ensuring that the ventilation duct penetration section remains water- and fire-tight is crucial for the safety and prevention of hazards. Thin penetration channels pose risks regarding temperature and fire; therefore, their thickness and length must adhere to IMO standards [33] to guarantee safety.

The positioning of ventilation systems below ground level and their ability to prevent flooding risks comply with the terms and conditions outlined in references [34-36]. The regulations for integrated ventilation requirements on ships measuring 80 m and above are based on the guidelines set by the Association of International Classification Societies (UR) S27 [37]. As per these regulations, the ventilation system on the ship is located at the front of the deck. An explosion-proof shield is installed in the appropriate position for the ventilation system to mitigate the risk of explosions within the ship's vicinity. This shield is equipped with a square mesh protective screen measuring 13 mm to prevent any objects from entering the fan [38].

3.2 Main requirements and compartments

When the air vent is installed, the ship's engine can reach a maximum temperature of 45^oC. The air ventilation system in the engine room consists of an air supply fan and an air exhaust fan, both of which have a high airflow rate [39]. This ventilation system ensures that the engine room's air conditions remain stable per established standards [40]. In cases where ships are not certified or do not adhere to temperature standards in the engine room, higher temperatures can be tolerated [41]. According to the International Organization for Standardization (ISO) 8861 standard, the air ventilation in the engine room must meet the required air flow rate. To achieve proper air ventilation in the engine room, it is necessary to conduct initial research activities such as simulations or CFD approaches [42].

The ventilation system's separation is necessary for the fuel room and engine tank. The engine room requires a minimum air speed of 30 changes per hour, achieved by implementing an extraction-type ventilation system [43]. Additionally, the ship's engine control room is equipped with an air ventilation and conditioning system to maintain a temperature below 27℃. This ventilation system extracts hot air from the room and replaces it with fresh air from the outside [44]. The air change process is conducted every hour in the flammable and dangerous liquid storage compartment.

The oxygen-acetylene and carbon dioxide (CO₂) storage rooms maintain the ventilation system by implementing hourly air changes. On the other hand, the refrigeration machine room and battery room have fresh air changed either every hour or at intervals shorter than an hour. Installing a fan within less than 3 meters of the air exhaust duct is considered safe, adhering to the specified fan model [45]. The kitchen has a mechanical ventilation system that complies with ISO regulations, providing fresh air and exhaust fans [46]. In order to ventilate the wheelhouse, an air conditioner is installed following the guidelines outlined in ISO 8864 [44]. Consequently, the general standard for air ventilation is maintaining a low air pressure within the room where ventilation or AC systems are installed [44].

3.3 Additional requirements for air ventilation

Mechanical ventilation is essential for enclosed spaces filled with livestock to ensure their comfort. It is recommended that the air in these spaces circulate 20 times per hour. Similarly, in the case of oil tanker cargo ships and floating liquefied natural gas storage (FLS) tankers, air changes are carried out 20 times within one hour. However, for oil recovery vessels, air changes are conducted approximately eight times per hour [47]. In order to maintain safety, ships carrying liquefied gas are equipped with air vents positioned 10 meters away from hazardous areas in the horizontal direction. On the other hand, passenger ships follow the regulations set by the Maritime Safety Committee (MSC) 1034, which require the extraction ventilation system to remove smoke within 10 minutes [48, 49]. Inland navigation passenger ships with two gas-tight engine rooms must have separate air ventilation systems, and special independent ventilation must be prepared [50].

The air ventilation in cargo spaces of ro-ro ships is regulated and calculated according to ISO 9785 rules [51]. As per the International Convention for Maritime Safety (SOLAS), the air ventilation system on ships ensures that the air changes 10 times an hour or 20 times, depending on the air conditions. For ships sailing in polar waters, air vents are installed on both sides of the vessel as deemed appropriate [51].

5.1 Eco-friendly marine power system using liquefied natural gas (LNG) waste with cold air ventilation

The investigation conducted by Ahn et al. [76] focused on refrigerating LNG waste to remove heat and maintain a safe condition in the cargo space. The cargo ship in question utilized a ship power system that combines steam and electricity from gas turbine systems. A cold air ventilation flow system was employed, which underwent energy and exergy analysis and a greenhouse gas emission approach. The efficiency of the cooled air ventilation system is remarkably high, boasting a coefficient of performance that is twice as efficient as conventional systems. Moreover, the exergy value experiences an increase as the ventilation airflow is augmented. Consequently, using waste LNG energy as a cooling source proves to be highly beneficial in reducing greenhouse gas emissions. The electrical efficiency calculation (ξ_elec) takes into account the lower heating value (LHV) of LNG, which is approximately 42,850 kJ/kg as follows:

$\xi_{{elec }}=\frac{\dot{W}_{{net }}}{\dot{m}_f \times L H V_f} \times 100 \%$           (1)

where, $\dot{W}_{n e t}$ (kW) is the net power released in this system, $\dot{m}_f$ (kg/s) is the mass flow of fuel consumption in the marine propulsion system. $\dot{W}_{n e t}$ can be calculated as follows:

$\dot{W}_{ {net }}=\dot{W}_{G T}+\dot{W}_{S T}- \left(\dot{W}_{ {R-pump }}+\dot{W}_{f- { pump }}+\dot{W}_{E G W}\right)$             (2)

$\dot{W}_{ {net }}=\dot{W}_{G T}+\dot{W}_{S T}- \left(\dot{W}_{ {R-pump }}+\dot{W}_{f- { pump }}+\dot{W}_{ {sea-pump }}+\dot{W}_{ {air-comp }}\right)$               (3)

The equation for power consumption ($\dot{W}_X$) on a rotary machine with rotary equipment type (X) is as follows:

$\dot{W}_X=\dot{m}\left(h_o-h_i\right)$             (4)

The energy equation for heat exchange in low- and high-temperature fluids is as follows:

$\dot{m}_{ {hot }, { in }} h_{ {hot }}+\dot{m}_{ {cold,in }} h_{ {cold,in }} =\dot{m}_{ {hot,out }} h_{ {hot,out }}+\dot{m}_{ {cold,out }} h_{ {cold,out }}$            (5)

The coefficient of performance (COP) is the ratio between the input power and the heat dissipation power in the cargo hold, which can be calculated as follows:

$COP =\frac{\dot{Q}_{ {output }}}{\dot{W}_{ {input }}}$                 (6)

The exergy calculation in this system [77] is as follows:

$\dot{E}_x=\dot{m}\left(e x_{p h}+e x_{c h}\right)$              (7)

$e x_{p h}=\left(h-h_0\right)-T_0\left(s-s_0\right)$             (8)

$e x_{c h}=\sum \frac{x_i \times e_{i, c h}^{\sim 0}}{M_i}$              (9)

$\xi_{{exergy }}=\frac{\dot{E} x_{ {out }}}{\dot{E} x_{ {input }}} \times 100 \%$             (10)

where, ex_ch is the specific chemical exergy, ex_ph is the specific physical exergy of the system, M_i is the molecular weight, $e_{i, c h}^{\sim 0}$ is the standard molar chemical exergy, x_i is the function of the mole fraction, and ξ_exergy is the efficiency exergy, which is the ratio between exergy input ($\dot{E} x_{ {input }}$) and output ($\dot{E} x_{ {out}}$) of the system.

An eco-efficiency approach to ships is explored in the context of sustainable development in the maritime sector [78]. One method involves calculating the energy efficiency design index (EEDI) for the system [79], as described in the following calculation:

$E E D I_{ {reference \,line }}=174.22 \times(D W T)^{-0.201}$             (11)

$Required \,EEDI =\left(1-\frac{X}{100}\right) \times E E D I_{ {reference \,line }}$                (12)

$Attained \,EEDI =C_F \times \frac{S F C_{M E} \times \sum_i P_{M E}+S F C_{A E} \times P_{A E}}{0.7 \times D W T \times V_{r e f}}$               (13)

$P_{A E}=\left[ 0.025 \times\left(\sum_{i=1} M C R_{M E(i)}+\frac{\left(\sum_i P_{P T I(i)}\right)}{0.75}\right)\right]+250$              (14)

where, CF represents a dimensionless CO₂ emission conversion factor, DWT denotes ship weight, and SFCAE and SFCAE are the specific fuel consumption levels in kg/kWh for the main and auxiliary propulsion engines [80].

5.2 The effect of mechanical air ventilation on ship engine room temperature on a large scale

Wu et al. [81] conducted a study on the combustion rate and smoke temperature in the air ventilation of ship engine rooms. The height variations of the inlet air ventilation duct range from 0.1 meter to 1.0 meter, while the time variations span six levels from 38 per hour to 126 per hour. The research findings indicate that the average ventilation level impacts the combustion temperature rate in the engine room. As the ventilation air rate increases, the average temperature of the engine room decreases. Mathematical models are employed to forecast the average temperature of the engine room based on the energy balance principle. Mechanical ventilation in ship engine rooms has two distinct heat transfer effects on fires in ship engine rooms: the cooling effect and the combustion intensification effect. The fire conservation heat equation in a ship's engine room can be calculated as follows:

$\dot{Q}=\dot{Q}_{g a s}+\dot{Q}_e+\dot{Q}_w$          (15)

The gas heating rate, denoted as $\dot{Q}_{gas}$, is accompanied by the heat dispersion rate of smoke in the exhaust, represented as $\dot{Q}_{e}$, the heat dispersion rate of the insulation, indicated as $\dot{Q}_{W}$, and the heat release process in a room fire, labeled as $\dot{Q}$. The average gas heat can be calculated as follows:

$\dot{Q}_{ {gas }}=d\left(c_p \rho_{ {gas }} V T_{ {gas }}\right) / d t$              (16)

$\dot{Q}_e=c_p \dot{m}_{e x}\left(T_{e x}-T_{i n}\right)$           (17)

$\dot{Q}_w=h_T A_T\left(T_{g a s}-T_{i n}\right)$              (18)

The volume of the ship's engine room, denoted as V, along with the heat capacity of the gas, denoted as c_p, and the density of the gas in the engine room, denoted as ρ_gas, play a crucial role in this calculation. Additionally, the average mass flow of smoke coming out of the engine room, represented as $\dot{m}_{e x}$, is taken into consideration. The heat transfer coefficient for the partition, h_T, and the total surface area of the partition in the room, A_T, are also important factors in this equation. By incorporating all these variables, the heat balance equation in the engine room can be calculated using the following equation:

$\dot{Q}=c_p \dot{m}_{e x}\left(T_{e x}-T_{i n}\right)+h_T A_T\left(T_{g a s}-T_{i n}\right)$            (19)

The heat conservation equation in the engine room can be calculated by balancing the mass flow of smoke out and fresh air entering the room. This result can be achieved through the combination of the equations mentioned as follows:

$T_{g a s}-T_{i n}=T_{e x}-T_{i n}$ dan $\dot{m}_{e x}=\dot{m}_{i n}=\rho_{i n} V . n$           (20)

$\dot{Q}=\left(c_p \dot{m}_{ {ex }}+h_T A_T\right)\left(T_g-T_{ {in }}\right)$            (21)

Zhang et al. [82] discovered the mechanical ventilation heat transfer coefficient (h_T).

$h_T=0.0035 \dot{Q}^{1 / 3}$            (22)

$h_T=1 / A_T\left[\dot{Q} /\left(T_{\text {gas }}-T_{i n}\right)-c_p \rho_{\text {in }} V_n\right]$          (23)

$h_T=k_{T n}=6.93 \times 10^{-4} n$           (24)

The average fire temperature in the ship's engine room under quasi-steady state conditions can be calculated as follows:

$T_{g a s}=T_{i n}+\dot{Q} /\left(c_p \dot{m}_{e x}+h_T A_T\right)=T_{i n}+\frac{\chi A_f \Delta H_c}{c_p \rho_{i n} V+k_T A_T}\left(\frac{\dot{m}_0^{\prime \prime}}{n}+R_n\right)$           (25)

$\begin{gathered}T_{\text {gas }}=35^o+46.57\left[\frac{19.75}{n}+\frac{428.99+e^{(115-n / 12.34)}}{3600}\right] \\ 38 h^{-1} \leq n \leq 126 h^{-1}\end{gathered}$           (26)

5.3 Smoke and flame characteristics in the engine room air vents of marine vessels

Numerical investigations conducted by Lan et al. [83] explored the impact of ventilation location and airflow velocity on fire and smoke distribution within ship engine compartments. The study's findings reveal that increased ventilation airflow velocity influences the cooling rate within the ship's engine room. Method 3, which involves placing the intake at the middle of the top side and the exhaust at the bottom, is identified as a more effective ventilation configuration for enhancing the safety of the ship's engine and associated equipment. This research is a valuable resource for optimizing ship ventilation systems and ensuring the safety of ship equipment through simulation techniques.

The heat release rate per unit volume can be calculated based on the mass production rate of components and the mass of the heat source, as expressed by the following equation:

$q^{\prime \prime \prime}=\sum_a m_a^{\prime \prime \prime} \Delta h_{f, a}$           (27)

where, ∆h_f,a is the heat generator component, $m_a^{\prime \prime \prime}$ is the average chemical mass value, and q''' is the average heat released per unit volume.

The general equation for determining a hydrocarbon fuel fire's heat transfer rate involves calculating the ignition source and the air parameters within the cabin, as described in the following equation.

$Q=m_{f, \infty}\left(1-e^{k D}\right) \Delta H X_{\text {chem }} \pi D^2 / 4$           (28)

where, X_chem is the combustion efficiency rate, ∆H is the net heat of combustion, D is the pool diameter, k is the absorption rate coefficient, m_f,∞ is the mass rate of asymptomatic combustion, and Q is the release rate of the chemical heat.

The equation for the flame height model is as follows:

$\begin{gathered}\frac{H_f}{D}=3.7 Q^{* 2 / 5}-1.02 \\ \text { with, } Q^*=Q / \rho_o C_p T_{\infty}\left(g D^5\right)^{1 / 2}\end{gathered}$          (29)

where, Q is the average heat release from the fire, ρ_o is the environmental air density, C_p is the air heat capacity, T_∞ is the environmental temperature, and g is the gravitational acceleration.

It is essential to calculate the flame source diameter to mesh size ratio, thereby evaluating the mesh resolution [84].

$D^*=\left(\dot{Q} / \rho_0 c_p T_{\infty \sqrt{g}}\right)^{2 / 5}$           (30)

where, $\dot{Q}$ is the average total heat released by the fire, and $D^*$ is the characteristic diameter of the fire.

5.4 Effect of air ventilation in the marine engine room on flame and hot-surface ignition (HSI)

Wang et al. [85] conducted a study on the detection of fuel leaks in the engine room and investigated the behavior of flames on hot surfaces under different ventilation airflow velocities. The experimental approach involves varying the airspeed from 1 m/s to 5 m/s. It was found that the impact of flammable fuels on ignition and flame propagation intensifies with an increase in airflow velocity. The position of the installed air vent influences the leak's location in the ship's engine room. Additionally, the airflow conditions in the engine room decrease as the surface heat temperatures rise. In the steady state, the heat transfer coefficient equation, denoted as h_n, can be expressed as follows [86].

$h_n=0.425\left[\frac{k_{v f}^3 H_{f g} \rho_v \rho_L^{3 / 2} g}{\left(T_w-T_{s a t}\right) \mu_{v f} \sigma^{1 / 2}}\right]^{1 / 4}$            (31)

where, h_n (W/m².K) is the heat transfer coefficient at the nuclear point, K_vf (W/m.K) is the conductivity of hot vapor in the film, σ (mN/m) is the surface tension of liquids, μ_vf (Pa.s) is the vapor viscosity between walls and liquid, H_fg (J/kg.K) is the latent heat of vaporization, ρ_L (kg/m³) is the liquid fuel density, and ρ_v (kg/m³) is the fuel vapor density.

$m_F=h_F c_{p, g}^{-1} A \ln (B+1)$            (32)

$b=h_{f g}^{-1} c_{p, g}\left(T_w-T_b\right)$        (33)

where, m_F (gm/s) is the average fuel evaporation mass, and B is the mass transfer number. The relationship between the height of heat impact surface (HIS) and the surface temperature can be determined using the following equation:

$H=H_{max }+\frac{T_m}{\sqrt{0.0716 \pi T_s}} \exp \left[-\frac{\ln \left(\frac{T_s}{T_{min }}\right)^2}{46.875 H_{ {min }}}\right]$              (34)

where, H_max (m) is the initial flame height, T_m (℃) is the average change in surface heat temperature, T_s (℃) is the surface heat temperature, T_min (℃) is the initial surface temperature, and H_min (m) is the height of the ship's fuel during the stationary period.

5.5 Smoke height in the ship engine room with experimental and theoretical approaches

Wu et al. [70] conducted a study to assess the height of the smoke layer in a ship's engine room equipped with forced air ventilation. Both experimental and theoretical approaches were employed to investigate the impact of air volume and air height on the ship's engine room. The findings of the tests indicate that an increase in air volume within the ventilation system influences the fire source's mass rate and the smoke layer's height in the engine room. A comparison between the theoretical predictions and experimental results reveals a relative error of approximately 11% in estimating the height of the smoke layer during fires in both the engine room and cabin.

The conservation concept utilized to formulate an equation for predicting the height of the smoke layer in the engine room is as follows:

$\dot{m}_p+\dot{m}_b-\dot{m}_e=\frac{d \rho_s V_u}{d t}$             (35)

where, $\dot{m}_p$ represents the mass flow rate of smoke entering the top control from the fire source, $\dot{m}_b$ represents the mass flow rate of air entering the smoke layer, $\dot{m}_e$ represents the mass flow rate of smoke coming out of the mechanical exhaust, ρ_s represents the density value of smoke, and V_u represents the volume of the smoke layer in the ship's space. The thickness of the smoke layer at any time $\frac{d Z_u}{d t}$ can be calculated using the following equation:

$\frac{d Z_u}{d t}=\frac{\dot{m}_p+\dot{m}_b-\dot{m}_e}{A_z \rho_s}$                (36)

The plume model adheres to the law of conservation of mass, which is encapsulated by the following equation:

$\dot{m}_p=0.21\left(\frac{\rho_0 g}{C_p T_0}\right)^{1 / 3} \dot{Q}_C^{1 / 3} Z^{5 / 3}=C \dot{Q}_C^{1 / 3} Z^{5 / 3}$              (37)

where, ρ_o represents the air density, g denotes the acceleration of gravity, C_p represents the heat capacity at constant pressure, T_o represents the ambient temperature, $\dot{Q}$ represents the ignition power convection, and Z represents the height of the smoke from the surface of the fire. The mass loss rate of a fire source is expressed as follows:

$\dot{m}=f\left(V_s\right)$            (38)

$\dot{Q}_c=\approx 0.7 \dot{Q}=0.7 \chi \cdot f\left(V_s\right) \cdot \Delta H_c$           (39)

where, $\Delta H_c$ is the heat of combustion, $\chi$ is the combustion efficiency, and $\dot{Q}_c$ denotes the total average of the heat output from the fire source.

$\dot{m}_b=\psi \dot{m}_s=g(h) \dot{m}_s=g(h) \rho_s V_s$             (40)

The source of fire under different air volume conditions for determining the smoke layer can be determined as follows:

$\begin{gathered}\frac{d Z_t}{d t}=-\frac{C\left[0.7 \chi\left(-204.54 e^{-\frac{V_S}{0.37}}+7.11\right) \Delta H_c\right]^{1 / 3} z^{5 / 3}+\psi V_s \rho_a-V_e \rho_s}{(k z+b) \rho_s} \\ \text{when} \,t=0, Z_t=1.9\end{gathered}$            (41)

In the case of C, the Zukoski model was employed with a specific value of C=0.076432. As for k, an independent compartment structure was adopted with a value of k=6.7. Similarly, b also follows an independent compartment structure with a value of b=51. In addition, $\psi$ represents the average level of air mixing, and $g(h)=-0.25 e^{-\frac{h}{0.64}}+0.76$.

5.6 Smoke filling and fire entry of the enclosed ship engine room

Wang et al. [87] investigated the behavior of smoke plumes as they entered and filled a ship's engine room in a closed environment. Both theoretical and experimental methods were utilized to examine the smoke inlet channel and the process of smoke filling within the ship's engine room. The findings of this study were corroborated by earlier research, indicating that sealed fires are more likely to result in smoke infiltration compared to open fires. The change in pressure ∆P=P-P₀ within a closed ship's engine room due to fire [87] is expressed as follows:

$\frac{\Delta P}{P_0}=\frac{\int_0^t\left(1-\chi_l \dot{Q}(t)\right) d t}{\rho_0 V_{c v} T_0}$              (42)

$\rho_u A(H-z)+\rho_l A_z=\rho_0 A H$             (43)

where, $\dot{Q}(t)$ represents the fire's heat release rate, $\chi_l$ denotes the overall heat loss fraction, ρ₀ stands for the initial density of the air, P₀ signifies the environmental pressure of the air, ρ_u indicates the average density of the upper layer of smoke, and ρ_l represents the average density of the lower layer of smoke.

The following equation can be used for determining the smoke height [15]:

$\frac{d z}{d t}=\frac{-R_M\left(\dot{m}_p c_p T_l+\frac{Z}{H}\left(1-\chi_l\right) \dot{Q}(t)\right)}{P A_{C v}}$            (44)

The equation for the mass flow rate of a plume of smoke at a certain height can be written as follows:

$\dot{m}_p=\rho_0 \pi b_u^2 u_0$           (45)

The equation for the mass flow rate of the smoke plume can be determined in the following manner [16]:

$\dot{m}_p=\frac{1}{C_T \beta}\left[\frac{g \rho_1^2}{c_p T_l}\right]^{1 / 3} Q_c^{1 / 3}\left(z-z_0\right)^{5 / 3}\left[1+C_T(1-\right. \left.\beta)\left(g^{1 / 2} c_p \rho_l T_l\right)^{-2 / 3} Q_c^{2 / 3}\left(z-z_0\right)^{-5 / 3}\right]$             (46)

The Heskestad model equation for Z₀ can be calculated as follows [88]:

$\frac{Z_0}{D}=-1.02+0.083 \dot{Q}^{* 2 / 5}$              (47)

where, D represents the diameter of the flame source or the diameter of the non-circular case. The dimensional heat release rate, denoted as $\dot{Q}^*$, can be calculated using the following equation:

$\dot{Q}^*=\frac{\dot{Q}}{\rho_l c_p T_l g^{1 / 2} D^{5 / 2}}$             (48)