Keywords

1 Introduction

A large amount of liquefied petroleum gases (LPG) is transported by rail in special tank cars. A number of accidents with fires and explosions that occurred with railway tank cars for the transportation of LPG indicate a high fire and explosion hazard of the processes of transportation of these substances. Often accidents of LPG tank cars proceed according to the scenario when the tank with LPG is exposed to the fire source, as a result of which an explosion of the tank is possible, with consequences that can be catastrophic. One of the methods of fire protection of LPG tank cars is to choose a safety valve of such a passage section, which, on the one hand, prevents the explosion of the tank car for a given period of time under various accident scenarios, on the other hand, provides a minimum discharge of LPG into the environment.

There are few studies of such emergency scenarios. Therefore, the purpose of this work is to develop a more accurate mathematical model of the behavior of LPG tank car located in the fire center and the calculated state of the LPG tank car in a model fire center during a full-scale experiment.

Often, accidents in LPG tank cars proceed according to the scenario when a tank with LPG is exposed to a fire source, as a result of which an explosion of the tank is possible, with consequences that are sometimes catastrophic. To prevent such accidents, various methods of fire protection of LPG tank cars are proposed:

  • Increasing the thickness of the bottom of the tanks;

  • Equipping tanks with protective screens to protect them from impacts;

  • Arrangement of safety arches on tank hatches;

  • Use of thermal insulation;

  • Application of a fire retardant coating on the outer surface of tanks;

  • Increase in flow area of safety devices.

These and other measures for fire protection of LPG tank cars can be developed on the basis of experimental and theoretical research.

In the process of calculating the critical parameters for the operation of the safety valves of tanks, it is necessary to obtain positive or negative results of the selected methods of fire protection of LPG tank cars.

2 Materials and Methods

In this mathematical model, a metal tank with LPG is considered, on the outer surface of which there can be a heat-insulating layer, consisting partly of a layer of porous non-combustible material (such as mineral wool), and the remaining part of the outer surface of the tank can be covered with foaming fire-retardant paint. It is assumed that at the moment of time τ = 0 the temperature of LPG and the temperature of the layers of the two-layer tank wall are equal to the temperature of the ambient air TV and on a certain part of the outer surface of the tank F0 (heat-insulating layer, if any), an external surface heat source with power q. If there is a fire-retardant coating on the walls of the tank, the heat flux causes its quasi-instant swelling with the formation of a heat-insulating layer with a thickness of Lfrc, which depends on the initial thickness of the coating layer and its properties. It is assumed that the temperatures of the tank walls, as well as the temperature and pressure of the LPG, do not depend on the coordinates; the ambient temperature is considered constant. The thermophysical properties of the metal of the tank walls, the heat-insulating layer and the layer of the intumescent fire-retardant coating are assumed constant. The convective and radiative heat exchange on the outer surface of the tank with the ambient air temperature TB is taken into account. In the tank boiler, depending on the combination of LPG operating parameters, the following heat transfer modes are possible: surface, bubble or film boiling of liquid; natural convection of a liquid or gas in a large volume. When the safety valve is opened, a critical outflow of the two-phase LPG medium into the surrounding space occurs.

Two-phase region:

$$ P < P_{{cr}} ;T < T_{{cr}} ; $$
(1)

Continuity equation

$$ V\frac{{d\left[ {\rho ^{\prime}\left( {1 - \beta } \right) + \rho ^{\prime\prime}\beta } \right]}}{{d\tau }} = - G, $$
(2)

where, V—tank volume; \(\rho ^{\prime},\rho ^{\prime\prime}\)—density of the liquid and gas phases at the saturation line, respectively;

β—volumetric gas content; G—critical flow rate of a gas–liquid mixture flowing out of a safety valve

$$ G/S = \left[ {\frac{{\left( {1 - \beta } \right)\rho ^{\prime}}}{k} + \beta \rho ^{\prime\prime}} \right]\sqrt {\gamma RT_{S} \left( {1 + \frac{{1 - \beta }}{\beta } \cdot \frac{{\rho ^{\prime\prime}}}{{\rho ^{\prime}}}k^{2} } \right)} ; $$
(3)

\(k = w^{\prime\prime}/w^{\prime}\)—slip coefficient;

$$ k = 0.17x^{{0.18}} \left( {\frac{{\rho ^{\prime}}}{{\rho ^{\prime\prime}}}} \right)^{{\left( {1/2} \right)}} - {\text{Fauske model}} $$
(4)
$$ k = \left( {\frac{{\rho ^{\prime}}}{{\rho ^{\prime\prime}}}} \right)^{{\left( {1/3} \right)}} - {\text{Moody model}}; $$
(5)
$$ \gamma = c_{P}^{\prime } /c_{P}^{{\prime \prime }} ; $$
(6)

R—universal gas constant; S—safety valve flow area.

Energy equation

(7)

where \(i^{\prime},i^{\prime\prime}\)—enthalpy of liquid and gas phases on the saturation line, respectively; ik—enthalpy of outflowing gas–liquid mixture; P—pressure in the tank boiler; αв—heat transfer coefficient on the inner surface of the boiler tank; Fв—internal surface area of the boiler tank; Tw—temperature of the inner surface of the tank boiler; TS—LPG temperature at saturation line.

$$ P = f\left( {T_{S} } \right) - {\text{saturation line}} $$
(8)
$$ \begin{aligned} & {\text{Single phase}} - P > P_{S} ;T < T_{{cr}} ;\;\;{\text{and}} \\ & {\text{supercritical region}} - P > P_{{cr}} ;T > T_{{cr}} ; \\ \end{aligned} $$
(9)

Continuity equation

$$ V\frac{{d\rho }}{{d\tau }} = - G, $$
(10)

where V—tank volume; \(\rho = f\left( {P,T} \right)\)—density of single-phase LPG; G—critical flow rate of boiling single-phase liquefied gas flowing out of the safety valve.

Energy equation:

(11)

where \(i = f\left( {P,T} \right)\)—enthalpy of LPG; TW—temperature of the inner surface of the tank boiler.

To calculate the critical pressure in the tank, I took the following parameters:

  • model of 4-axle LPG tank 15-1200-02;

  • safety valve SV(32–2.25);

  • LPG—propane;

  • design temperature 20 °C.

3 Results

To calculate (simulate) the development of the accident according to the BLEVE scenario, the «Fobot» software package was used.

The use of technical means of protecting LPG tanks in the fire site should at least prevent the development of an accident according to the BLEVE scenario within the time (4 h) required for the deployment of the fire department [1,2,3,4,5,6,7,8,9,10,11].

The following methods of fire protection are supposed to be applied in the fire center:

  • Equipping the tank with safety valves:

Standard valve—(SV(32–2.25)).

Response pressure—2.25 MPa.

Nominal diameter of the flow area dy = 32 mm;

  • Covering 95% of the outer surface of the tank with thermal insulation thickness δin = 20 mm.

Thermal properties of insulation (identical to mineral wool):

  • density—ρ = 150 kg/m3,

  • heat capacity - c = 1000 J/(kg K),

  • coefficient of thermal conductivity—λ = 0.047 W/(m K);

    • Application of a foaming fire retardant coating SGK-1 in thickness δfoa = 20 mm on the remaining outer surface of the tank. Thermophysical properties of SGK-1:

  • density—ρ = 10 kg/m3,

  • heat capacity—c = 1000 J/(kg K),

  • coefficient of thermal conductivity—λ = 0.07 W/(m K).

Enter data into «Fobot», start modeling the situation.

Detailed results are structured, summarized and presented calculations are presented in Tables 1 and 2.

Table 1 Results of calculating LPG—Propane. Tank 15-1200-02
Full size table
Table 2 Results of calculating LPG - Propane. Tank 15–1200-02
Full size table

4 Conclusion and Discussion

The four-axle tank car for LPG 15–1200-02 is equipped with an SV(32–2.25) safety valve with an opening pressure of 2.25 MPa and a closing pressure of 1.95 MPa. Figure 1 shows that the pressure graph in the tank is limited to 2.25 MPa. After the pressure (Fig. 1) and temperature (Fig. 2) in the tank have dropped, the safety valve SV(32–2.25) closes [12,13,14].

Fig. 1
figure 1

The pressure in the tank vs. time

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Fig. 2
figure 2

The temperature in the tank vs. time

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The «Fobot» program is a complex of databases that can contain the data necessary for the calculation for any types of railway tanks, liquefied gases, safety valves, heat-insulating materials and fire-protective coatings, as well as software interpolation and other auxiliary modules, modules for calculating heat transfer characteristics and numerical solution of a system of differential equations.

Calculations using the «Fobot» software package show that the selected fire protection means:

  • equipment of a propane tank (model 15-1200-02) with a safety valve (actuation pressure 2.25 MPa, nominal bore diameter dy = 32 mm);

  • covering 90% of the outer surface of the tank with thermal insulation δis = 20 mm;

  • foaming fire retardant coating thickness δis ≈ 20 mm on the remaining outer surface of the tank, ensure that the tank remains in the model fire source for 4 h without the development of an accident according to the BLEVE scenario [15].