Air Dilution, under Calm, of Spreading Dense Vapor from an Instantaneous Spill of Volatile Liquid

Abstract

We adopt depth-averaged, shallow-water gravity-wave theory to furnish a simplistic, tractable description of the lateral axisymmetric spreading of liquid and vapor in the aftermath of an accidental, instantaneous spill on level ground of a large quantity of volatile liquid. Our attention is focused on the spill of a flammable and/or toxic liquid (e.g., liquid petroleum gas, or liquid chlorine), such that the phase transition, facilitated primarily by heat transfer from the ground to the liquid, yields a vapor which, even upon equilibration to atmospheric temperature, remains appreciably denser than the surrounding air. Thus, the vapor remains confined to a ground-contiguous layer as the vapor laterally disperses (in the absence of obstacles, in the modeling). Our objective is to estimate the time after release until dilution with air of a combustible vapor results in the local concentration being everywhere below the fuel-lean flammability limit (or until the dilution of a toxic vapor results in the local concentration being everywhere tolerable for human occupation). We limit attention to no-wind conditions, for which the dilution is protracted (because the dispersion remains limited to the proximity of the spill site). For an instantaneous release of sufficient liquid volume, we anticipate (and can justify a posteriori) that a balance of inertial and buoyant forces constitutes an excellent approximation to the conservation of momentum for the gravity-current dynamics of the spilled liquid and evolved vapor [until the (cylindrical-)radial spread results in liquid and vapor layers so thin that a viscous–buoyant balance is more appropriate]. Whereas virtually all previous analyses of wind-free dense-fluid dispersion take the initial condition to be a uniform-fluid spill in the configuration of a right-circular cylinder, and eventually a selfsimilar behavior, based parametrically only on the fixed spill-occupied volume and the effective gravitational acceleration, evolves, we adopt an initial spilled-fluid configuration in the form of a finite-radius mound, with the layer thickness monotonically decreasing with increasing radial distance and smoothly vanishing at finite radius. We find that no selfsimilar behavior evolves: vestiges of the initial conditions persist and, asymptotically, the radial position of the spilled-fluid edge would increase linearly with time in the absence of vaporization; furthermore, we find that at all times the thickness of the spilled-fluid layer would decrease monotonically with radius, from the axis of symmetry to the spilled-fluid-layer front. Also, whereas virtually all simple gas-cloud models are of box type, and take the contents to be spatially homogeneous, we investigate the spatial inhomogeneity of the vapor content of a cloud formed by evaporation, and show that distribution of the ambient-air-diluted vapor is well approximated as decreasing with increasing height as a Gaussian function, with peak concentration on the axis at all times.