Fuel Surface Cooling by Aqueous Foam: A Pool Fire Suppression Mechanism

Abstract

Aqueous foams are generally thought to suppress pool fires by forming a transport barrier (either an aqueous film or the foam itself) that prevents fuel vapor transport from the hot pool surface into the fire above. The present work is aimed at evaluating a different potential suppression mechanism wherein the fuel vapor pressure is reduced due to pool surface cooling that occurs when a room-temperature foam is brought in direct contact with the hot pool surface. We present a model to predict the sudden decrease in pool surface temperature when aqueous foam is applied instantaneously and uniformly onto a shallow, burning, heptane fuel pool. Conduction is assumed to dominate heat transfer at short time scales (a few seconds) due to the steep temperature gradient at the interface. The model describes the time evolution of the temperature profile by numerically solving a transient, one-dimensional, heat-conduction equation in the liquid pool and in the foam layer. We also obtained an analytical solution that is valid immediately after contact between fuel and foam. Model predictions show a significant decrease in fuel surface temperature in less than a second after the foam layer is placed on top of a hot liquid pool surface, causing a decrease in the fuel vapor pressure of over 75%. The predictions indicate that surface cooling could be an important mechanism of fire suppression by aqueous foams.

Appendix: Analytical Solution for Surface Cooling

We derive below an analytical solution for heat transfer between the fuel and foam, as described in Sect. 2.2. The key advantage of the solution is that it involves an interface with zero thickness, which is more realistic than the interface of finite thickness used as an approximation in the numerical model.

The idealized problem includes two semi-infinite layers (fuel and foam), both with uniform initial temperatures that are brought into direct contact at t = 0 (see Figure 4 of the main paper); we solve the unsteady heat conduction equations in the two layers. The interface between the materials is at x = 0. The fuel layer corresponds to x < 0, and the foam layer corresponds to x > 0. We use subscripts 1 and 2 to denote the properties of the fuel and foam layers, respectively. The initial temperatures are T ₁(t = 0) = T ⁰₁ and T ₂(t = 0) = T ⁰₂ , respectively, and T ⁰₁  > T ⁰₂ . The thermal properties k, ρ, and c are different in the two layers. We solve for the temperatures in each layer (T ₁ and T ₂) as functions of time, with particular interest on the interface temperature.

The heat conduction equation for each layer is

$$ \frac{{\partial \Theta_{i} }}{\partial t} = \alpha_{i} \frac{{\partial^{2} \Theta_{i} }}{{\partial x^{2} }}, $$

(14)

where i is an index for the layers (i = 1, 2), α _i is the thermal diffusivity (k _i /(ρ _i c _i )), and Θ _i is the non-dimensionalized temperature, given by

$$ \Theta_{i} = \frac{{T_{i} - T_{2}^{0} }}{{T_{1}^{0} - T_{2}^{0} }}. $$

(15)

The conduction equation requires one initial condition and two boundary conditions for each layer. The initial conditions correspond to uniform temperatures T ⁰₁ and T ⁰₂ in layers 1 (fuel) and 2 (foam), respectively:

$$ \Theta_{1} (x < 0,t = 0) = 1 $$

(16)

$$ \Theta_{2} (x > 0,t = 0) = 0. $$

(17)

Very far from the interface, the temperatures remain at the initial temperatures, corresponding to the boundary conditions

$$ \Theta_{1} ( - \infty ,t) = 1 $$

(18)

$$ \Theta_{2} (\infty ,t) = 0. $$

(19)

The heat flux entering the interface from layer 1 (fuel) is equal to the flux exiting the interface into layer 2 (foam), corresponding to the boundary condition

$$ k_{1} \left. {\frac{{\partial \Theta_{1} }}{\partial x}} \right|_{x = 0} = k_{2} \left. {\frac{{\partial \Theta_{2} }}{\partial x}} \right|_{x = 0}. $$

(20)

Upon contact, the temperature becomes continuous at the interface, corresponding to the boundary condition

$$ \Theta_{1} (0,t) = \Theta_{2} (0,t). $$

(21)

The similarity method (e.g., [14] ) is used to obtain an analytical solution for the problem. The relationship between the physical heat conduction length scale and time scale are used to define a similarity variable as

$$ \eta_{i} = \frac{x}{{2\sqrt {\alpha_{i} t} }}. $$

(22)

Using the similarity variable, the heat conduction equation that depends on x and t (Eq. 14) is reduced to an ordinary differential equation that depends only on η and is given by

$$ \frac{{\partial^{2} \Theta_{i} }}{{\partial \eta_{i}^{2} }} + 2\eta_{i} \frac{\partial \Theta }{\partial \eta } = 0. $$

(23)

The solution to Eq. (23) is given by [14]

$$ \Theta_{i} = \phi_{i1} {\text{erf}}(\eta_{i} ) + \phi_{i2}, $$

(24)

where ϕ _i1 and ϕ _i2 are constants determined by the boundary conditions. Two “boundary” conditions for η in each layer are needed to solve for the constants, which are determined below.

Using the similarity variable, the initial condition in Eq. (16) and boundary condition in Eq. (18) can be included in a single boundary condition

$$ \Theta_{1} (\eta_{1} = - \infty ) = 1. $$

(25)

Similarly, in layer 2 (foam), Eqs. (17) and (19) correspond to

$$ \Theta_{2} (\eta_{2} = \infty ) = 0. $$

(26)

The heat flux boundary condition at the interface in Eq. (20) corresponds to

$$ \sqrt {k_{1} \rho_{1} c_{1} } \left. {\frac{{\partial \Theta_{1} }}{{\partial \eta_{1} }}} \right|_{{\eta_{1} = 0}} = \sqrt {k_{2} \rho_{2} c_{2} } \left. {\frac{{\partial \Theta_{2} }}{{\partial \eta_{2} }}} \right|_{{\eta_{2} = 0}}. $$

(27)

The continuous temperature at the interface (Eq. 21) corresponds to

$$ \Theta_{1} (\eta_{1} = 0) = \Theta_{2} (\eta_{2} = 0). $$

(28)

Equations (25)–(28) are used to solve for the constants ϕ _i1 and ϕ _i2 in Eq. (24), which gives

$$ \phi_{12} = \frac{1}{{1 + \sqrt {\frac{{k_{2} \rho_{2} c_{2} }}{{k_{1} \rho_{1} c_{1} }}} }} = \phi_{22} = - \phi_{21} = \phi_{11} + 1. $$

(29)

The solutions for Θ ₁ and Θ ₂ are given by

$$ \Theta_{1} = \left( {\frac{1}{{1 + \sqrt {\frac{{k_{2} \rho_{2} c_{2} }}{{k_{1} \rho_{1} c_{1} }}} }}} \right) - \left( {\frac{1}{{1 + \sqrt {\frac{{k_{1} \rho_{1} c_{1} }}{{k_{2} \rho_{2} c_{2} }}} }}} \right){\text{erf}}(\eta_{1} ), $$

(30)

$$ \Theta_{2} = \left( {\frac{1}{{1 + \sqrt {\frac{{k_{2} \rho_{2} c_{2} }}{{k_{1} \rho_{1} c_{1} }}} }}} \right)\left[ {1 - {\text{erf}}(\eta_{2} )} \right]. $$

(31)

The interface temperature corresponds to η ₁ = 0 in Eq. (30) (or, equivalently, η ₂ = 0 in Eq. 31), given by

$$ \Theta_{i} (\eta_{i} = 0) = \left( {\frac{1}{{1 + \sqrt {\frac{{k_{2} \rho_{2} c_{2} }}{{k_{1} \rho_{1} c_{1} }}} }}} \right). $$

(32)

This analysis shows that the interface temperature decreases instantaneously when the foam layer makes direct contact with the pool surface and thermal equilibrium is established at the interface. The interface cools because the two layers are at different temperatures, and the cooling by conduction occurs instantaneously because the interface has no volume and no thermal mass. The cooling of liquid fuel underneath the interface does take time and depends on the rate of heat absorption by the foam, as described in the main body of the paper.