A brief description of the mathematical model of Amon and Denson and its application to intumescent coatings A brief introduction to the Amon and Denson theory have proposed a mathematic model to describe how spherical bubbles in a polymer liquid will expand when the interior pressure inside the bubbles is increased. Refer to Figure 1, which shows uniformly distributed bubbles of radius R, within a viscous polymer liquid that has a viscosity µ. Each bubble has an envelope of radius “S”. Based on conservation of mass and momentum, have obtained the following equation to describe the expansion ratio of the bubbles: (1) where and are the radius of the bubble and the outer shell respectively at time t, with initial values of and respectively, η is the viscosity of the surrounding liquid and σ is the bubble surface tension. By conservation of volume, (2) The outer shell radius is calculated at each time step. The initial outer shell radius is calculated by using the nucleation density, i.e. the number of nucleation sites per volume, N/V. (3) All nucleated cells are uniformly distributed and each cell has of volume at . The initial outer shell radius, is significantly greater than the initial bubble radius . Figure 1 shows a schematic representation of the cell model.
(7) (8) where is the initial gas pressure inside the bubble before mass conversion and expansion. Surface Tension In order to use Equation (1), the surface tension of the intumescent coating melt is required. Since there has been no measurement of this property for intumescent coating melt, the values of for polyethylene are adopted. This is shown in Figure 4. Figure 4 Surface tension vs. temperature for polyethylene melt [] Fortunately, prediction of intumescent coating expansion is not sensitive to the value of surface tension. For example, Figure 5 indicates that when using the variable surface tension in Figure 4 and a constant value of 20 N/m2 for predicting one (C12L1) of the tests of , the results nearly coincide. Figure 5 Sensitivity of intumescent coating expansion ratios to surface tension Viscosity For the intumescent coating studied by and [], there was no measurement of viscosity. Therefore, for the purpose of this paper, the viscosity value has to be estimated. This estimate is based on the results of which provided viscosity data for a number of intumescent melts given by . These data are shown in Figure 6 and Figure 7. Comparing the mass conversion (loss) data shown in Figure 2 for the intumescent coatings studied by with the TGA data of
Figure 6, it can be seen that of the intumescent coatings of is similar to the curve for Therefore, in this paper, it is assumed that the viscosity of the intumescent coatings studied by is that of curve for PP-APP/PER. Figure 7 shows the recorded relationship for PP-APP/PER and the fitted polynomial curve for use in this paper. Figure 8 shows the measured viscosity – temperature relationships for the intumescent coatings of . Figure 6 TGA curves of different mixture of intumescent coatings [] Figure 7 Assumed viscosity – temperature relationship, based on heat-treated PP/PER/APP system of Figure 8 Viscosity-temperature relationships for the intumescent coatings of Muller [] Mathematical Model Results and Comparison Zhang et al. [9] and carried out two series of tests, one under furnace fire exposure and one under cone calorimeter. performed cone calorimeter tests using different intumescent coating formulations. Comparison will be made for these three series of tests. Comparison for furnace fire tests of The furnace fire tests of were conducted by placing intumescent coated steel plates, all measuring 100 mm by 100 mm in planar dimensions but different plates having different thicknesses, each with the same intumescent coating thickness on both sides, to fire exposure