At this point, all of the bubbles that have burst during this timestep have been removed from the sample, and the radial locations and sizes of all of the remaining bubbles are known. The finite element geometry is now reimposed upon the problem in preparation for the determination of bulk material properties for each element.
Given the known quantity of polymer within an element, the task is to determine which bubbles extend into the element and how much of the gas from each of these bubbles lies within the element. As bubbles grow and migrate toward the surface of the spherical sample, many will extend beyond the dimensions of a single element. For the proper determination of the physical properties of these elements, the gases in these bubbles are apportioned by volume over all elements through which they extend. The location of each finite element node is determined as part of this calculation, which progresses from the center of the sample outward.
The basic geometry of a spherical bubble sliced by a sphere delineating the edge of a spherical shell element is shown in Figure 8. Here, the fraction of a small sphere of radius Rb that protrudes above a larger sphere of radius R can be calculated as the difference of sectors of the two spheres cut by a common plane passing through the intersection of the sphere surfaces. The height of the sectors is hb for the small sphere and h for the larger one. With a representing the distance from the centerline to the surface intersection of the two spheres, and with their centers separated by a distance d, the volume of the small sphere outside of the larger sphere is given by:
|
| (39) |
The surface area of this portion of the small sphere is equal to the zone of the sphere sliced by a plane,
| S = 2πRbhb | (40) |
These equations hold for a large sphere slicing through any portion of a small sphere. Setting d = rk, Rb = Rk, and R = ri, equation (39) gives the volume of the portion of bubble k contained within element i when the outermost extent of the bubble, rk + Rk, is located within this element. For a bubble k whose innermost point at rk − Rk is located within element i, the bubble volume contained within the element is equal to the total volume of the bubble minus the portion cut by the sphere defining the outer extent of the element at ri+1:
| (41) |
where again d = rk and Rb = Rk, but in this case R = ri+1 in equations for a2, hb, and h.
Figure 8: Bubble Volume extending beyond a sphere of radius R.
A straightforward but more complex equation can be written for the volume of the midsection of a bubble that extends completely through an element.
Given the radial location of the inner node of an element, the quantity of gas within the element, and therefore the location of the outer node, is determined by iteration. The calculation thus proceeds from the innermost to the outermost element. The gases in portions of unburst bubbles that extend beyond the surface of the sample are assigned to the outermost element.