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bubble dissolution. Both effects increase dramatically as the bubble collapses to zero radius. The role of air diffusion from the collapsing bubble is important to the mechanics of bubble collapse. When diffusion is very rapid, small bubbles in a viscoelastic polymer collapse catastrophically and larger bubbles oscillate only a few times before collapse. When diffusion is very slow, bubbles always oscillate, regardless of the bubble dimension or viscoelastic nature of the polymer. Furthermore, if diffusion controls, bubbles do not collapse to zero radius, regardless of their initial size or the viscoelastic character of the polymer melt. The level of saturation of gas in the bulk polymer melt also influences the extent of bubble collapse. For example, if the polymer is initially saturated with air and the bubbles contain air, the diffusional concentration gradient will be small and the bubbles may not collapse to zero radius. Further, if there are many bubbles, the regions around these bubbles may be quickly saturated and the bubble collapse may be retarded or even stop. Figures 6.21 and 6.22 show excellent agreement between theory and experiment for air bubbles in HDPE at various isothermal mold surface temperatures.
Figure 6.22 Time-dependent bubble extinction model and Spence's experimental data46
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In practical rotational molding, air buoyancy in the polymer melt is not a factor. For static tests such as that shown in Figure 6.19, on the other hand, air buoyancy could be a factor, albeit a very slight one.43
It is apparent that the three mechanisms described above all act to den-sify the polymer structure. Both capillary action and air diffusion and solution show that the rate of densification is proportional to. And all three show that the rate of densification increases rapidly, probably exponentially, with increasing polymer temperature.
Although these mechanisms yield comparable results for static tests, the vagaries of the actual process make comparisons questionable. Keep in mind that the powder bed contacts only a portion of the mold surface at any instant. In-mold videography54 shows that as the depleting powder bed flows across the powder already affixed to the mold surface, only a portion adheres to the tacky powder. In many cases, by the time the flowing powder returns, that portion that had adhered previously is tacky and may be almost fully coalesced into a discrete powder-free surface. This observed event would be best simulated in a static fashion by periodically applying thin layers of powder atop previously applied layers which are in contact with a hot plate that is increasing in temperature. Of course, the uncertainty of the process is that both the time and frequency of contact between the flowing powder and the affixed powder are unknown for most mold designs. Further, these aspects undoubtedly vary with location across the mold surface, with continuing depletion of the free powder bed, and with the changing nature of the temperature-dependent interparticle adhesion.
Having said that, it is apparent that the time of contact between the free powder bed and the fixed substrate is greatest when the powder first begins to stick to the mold surface. This implies that the thickest layer of powder affixed to the surface occurs in the beginning of the powder laydown. If the periodicity at any point is fixed by the rotation of the mold and if the rates of coalescence and densification do not dramatically increase with increasing temperature between periods of bed flow, then the greatest amount of porosity should occur at the beginning of powder laydown onto the mold, or in the polymer layer nearest the inner mold surface. Particle size segregation is an additional factor.
Finer particles should fluidize more than coarser particles. As a result, coarser particles should be preferentially at the bottom of the rotating
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powder bed and should therefore contact the hot mold surface more frequently than finer particles. However, certain experiments prove the contrary. In the 1960s, decorator acrylic globes were manufactured using a mixture of powder and pellets. The powder coated the mold first, with the pellets adhering to the molten polymer. The product had a smooth exterior surface and a roughened interior surface. Recently, this experiment has been repeated with fine black polyethylene powder and coarse natural polyethylene powder of the same molecular weight. When a small amount of fine powder was used, the powder only partially coated the mold surface prior to coalescence of the coarser powder.* When the ratio of black fine powder to coarse natural powder was increased, the final part showed a distinct black polymer layer at the outer part surface and a distinct natural polymer layer at the inner part surface. In another study in a double-cone blender,112 at a fill level of, say, 25%, the larger particles segregated to the center and the finer particles to the outsides. At a slightly lower fill level, the finer particles segregated to the center. And at a fill level in between, the finer particles migrated to one side and the coarser particles to the other. Once one of these patterns is established, it requires heroic measures to disturb it.
6.14 Phase Change During Heating
As noted, crystalline polymers such as polyethylenes, nylons, and polypropy-lenes, represent the majority of rotational ly molded polymers. As seen in Figure 6.9,** crystalline polymers require substantially more energy to heat to fusion temperatures than do amorphous polymers such as styrenics and vinyls. Thermal traces during heating rarely show abrupt changes in the polymer heating rates. There are two reasons for this. First, crystalline polymers typically melt over a relatively wide temperature range. And the powder flows periodically across the polymer affixed to the mold surface. As a result, the effect of melting is diffused over a relatively wide time frame, with the result being an extended time to fusion. Figure 6.23 clearly illustrates this for time-dependent mold cavity air temperature profiles for crystalline polyethylene and amorphous polyvinyl chloride.55
*This experiment demonstrated local hot spots on the mold inner surface, since the black
powder fused first to the hotter regions.
**This figure is discussed in detail in the oven cycle time section.
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Figure 6.23 Comparison of the heating characteristics of crystalline (PE) and amorphous (PVC) polymers,55 redrawn
6.15 The Role of Pressure and Vacuum
Commercially, the application of pressure during the densification portion of the process yields parts with fewer, finer bubbles. Technically, pressure acts to increase air solubility in the bulk polymer. Increasing bulk polymer pressure also acts to decrease bubble dimension and internal air pressure in the bubble, which in turn increases the concentration gradient. The overarching effect is one of accelerating bubble extinction. It has also been shown that vacuum or partial vacuum is also beneficial in promoting void-free densification prior to the bubble formation stage.
Note that there are competing effects. Low pressure inside the mold is important as the gas pockets are being formed into bubbles. If vacuum is applied when the bubbles are fully formed, they will get larger. However, the concentration of air in the bulk polymer will drop dramatically, implying that the bubbles should disappear even quicker. A hard vacuum is not required. The vacuum does not need to be applied throughout the heating process. In fact, there is strong evidence that vacuum applied during the early heating stages of the process may be detrimental to uniform powder flow across the mold surface.
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6.16 Mathematical Modeling of the Heating Process
It is apparent from the discussion above that the mechanics of powder heating, coalescence, and densification are quite complex and certainly not fully understood. Nevertheless, a general, holistic view of the process is possible. Figure 6.24 is a schematic of the typical heating process.56 First, it is well known that the mold absorbs substantially more energy than the plastic. As the mold is heating in a nearly constant temperature air environment, its rate of heating is essentially unaffected by the small amounts of thermal heat sink offered either by the sticking, densifying plastic or the air in the mold cavity. As a result, the mold should heat as a lumped parameter first-order response to a step change in temperature, as described above. For all intents, the inside mold surface sees the outside mold surface energy in less than one second. Once the inner mold surface begins to heat, its temperature TLlags behind the outside mold surface temperature TV by approximately:*
(6.40)
The temperature offset is about proportional to the convection heat transfer coefficient and the thickness and thermal properties of the mold material. High oven air flow, thicker molds, and molds of low thermal conductivity act to increase the temperature difference across the mold thickness. The rate of heating of both mold surfaces become equal when the heating time is approximately:
(6.41)
The thermal offset across the mold thickness is shown in schematic as curves A and Вin Figure 6.24. For most rotational molding materials, the thermal offset may be only a few degrees at best.**
Consider the case where there is no polymer in the mold cavity. The energy uptake by the air in the cavity depends on convection through a relatively stagnant air layer at the interface between the mold cavity air and the inner mold cavity surface. Thus the air temperature will lag behind that of the inner mold cavity surface. Since the volume of air in a given mold cavity is
*This equation is technically correct for constant heat flux to the surface. The heat (lux in rotational molding slowly decreases as the mold temperature increases. For this approximate analysis, it can be considered constant.
**Again, as given in the discussion about Figure 6.1, temperature differences of as much as ЗОoChave been measured. The anomaly between the predicted and measured temperature differences is not understood.
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known, the air temperature can be approximated at any time by solving the transient heat conduction equation with an appropriate adiabatic inner mold cavity surface boundary condition. However, for this heuristic analysis, the time-dependent mold cavity air temperature quickly parallels that of the inner mold cavity surface, as described earlier in this chapter. This is shown as curve D in Figure 6.24.
Figure 6.24 Heating temperature profile schematic56
As indicated earlier, the sticking, coalescence, and densification processes are complex interactions of free powder flow and neck formation between irregular particles. Instead of immediately modeling these processes, consider the conditions when all the powder has stuck, melted, and densified. At this time, the polymer is molten and has uniformly coated
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the inner mold cavity wall surface. The energy transfer now is through the mold wall, through the liquid polymer layer and into the mold cavity air. The mold cavity air temperature should now be increasing at a rate parallel to the outer mold surface temperature. The offset temperature between the inner liquid polymer surface and the outer mold surface temperature is given approximately by:
(6.42)
whereis the thickness of the liquid polymer layer and Kpis the thermal conductivity of the liquid polymer. As is apparent from this approximation, the thicker the polymer layer becomes, the greater the thermal lag becomes. This is seen as a shift away from the original curve D in Figure 6.24 to a new curve E, the amount of shift being the amount of thermal resistance through the polymer.
As discussed earlier, the transition from curve D to curve E begins at about the time the inner mold surface reaches the tack temperature of the polymer. The air temperature asymptotically approaches curve E when the entire polymer is densified and molten. This temperature is greater than the melting temperature of the polymer and certainly depends on powder flow, mold geometry, and rate of heating, among other parameters discussed earlier.
This analysis has made some technically inaccurate assumptions. Nevertheless, it illustrates some of the general concepts connected with the rotational mold heating process.
With this overview in mind, now consider mathematical models for the early portion of the heating process. One approach is to consider the powder bed as an infinitely long stationary continuum of known thickness. The appropriate model is the simple one-dimensional transient heat conduction equation, with appropriate boundary conditions:58*
(6.43)
*This model was originally proposed as a simpler version of an earlier steady-state circulation model for powder flow.2 In reality, it represents a model for steady-state slip flow of the powder bed.57
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whereand.Here Tmis the mold tempera-
ture and rair is the mold cavity air temperature.
For the simplest version of this modelis considered con-
stant. Standard graphical solutions for this equation are available when Tm is a known function, such as constant or linear with respect to time.57 Computer models are easily generated when Tmis more complex or when powder thermal properties are temperature-dependent. As one example, the crystalline heat of melting is accommodated by assuming the powder bed specific heat to be temperature-dependent, or. Densifica-
tion can be approximated by assuming that the polymer density is also temperature-dependent, or. As a result, this model can be used
to approximate the entire heating process, from cold mold insertion into the isothermal oven environment to full densification of the molten polymer. Slip flow of the powder bed comes closest to being characterized by this model.
Recently, a more complex model has been developed. Here the mold is first opened to a flat surface. Then a two-dimensional transient heat conduction equation is applied to a static powder bed of length less than that of the mold.59 This model allows the mold and any affixed polymer to be mathematically separated from the static powder bed, thus allowing simulation of mold parameters such as contact time length and frequency.
Another approximate energy model has been used when the powder bed appears to circulate in a steady-state fashion.2 The first assumption is that while a portion of the powder bed is in contact with the mold surface, it is static or nonflowing, and is heated by conduction from the mold surface. The static contact is short-lived, however, as that powder releases from the mold and cascades across the newly-formed static bed. During cascading, the powder particles mix sufficiently well to produce powder of a uniform bulk temperature, which now form a new static bed.*
Energy is transmitted by conduction through the surface of the bed that is in contact with the mold surface. Essentially no energy is transmitted to the bed from the mold cavity air. Since the powder contacts the mold surface for a relatively short time, the powder bed is considered to be infinitely deep relative to the thermal wave entering the bed at the
*The reader should review Figure 6.3 to understand lliis model.
Processing 249 mold-bed interface.* The appropriate mathematical model is:
(6.44)
Here x is the distance into the powder bed, assumed to be essentially planar relative to the planar mold surface.is the thermal diffusivity of the
powder bed, as discussed below. The mold surface temperature is given by the exponential equation:
(6.45)
where, and Т* is called the offset temperature. Ifis the distance
into the powder bed beyond which the effect of the increasing mold temperature is not felt, then the temperature in the powder bed can be approximated by a cubic temperature profile60 as:
(6.46)
The solution to the partial differential equation yields the following expression for, the thermal penetration distance:
(6.47)
For a simple step change in surface temperature, the thermal penetration distance is given as:
(6.48)
This model is valid so long as the dimensionless time is at least:61
where 0.000 КBi < 1000 (6.49) And
For a linear change in surface temperature,, the thermal penetra-
tion distance is given as:
*In the discussion that follows, the powder bed is considered to be a continuum with uniform thermophysical properties such as bulk density and thermal diffusivity. If specific bed characteristics are known, the analysis can be modified to include variable thermophysieal properties.
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(6.50)
For linear heating of the mold, the temperature in the powder bed at any time and distance x is then given as:
(6.51)