Boundaries on the Heating Process
In my Rotomolding Tech Minute I, I commented in general on model building for the heating and cooling process in rotational molding. And further, I remarked on the six technical papers presented at the 1997 SPE ANTEC that focused on this aspect of rotational molding. According to the 1998 SPE ANTEC advance program, there will be another spate of model building programs in Atlanta. While this activity brings much-needed technical interest to a process that has suffered a dearth of attention, I believe that there are some simple schemes that can aid the engineer to understand the heating process. The following is a thought problem to help bring some order to the subject.
Forget the Powder!
Really! At least insofar as energy uptake is concerned. Andy Rao
and I determined a long time ago that, insofar as heat sinks in
the process, the mold absorbs ten to 50 times more energy than
the plastic. So, as the mold is heating through to the final
plastic temperature, its temperature is essentially unaffected
by the small amount of thermal heat sink offered by the
sticking, densifying plastic. Thus, modelers should feel
reasonably safe in assuming, for all intents, that when the mold
is inserted in an isothermal oven, the outer mold surface will
heat in a first-order fashion
according to
where Ta is the isothermal oven temperature,
To is the initial
mold temperature, h is the heat transfer coefficient for forced
air convection,
is the thermal
diffusivity of the metal,=k/
cp,
L is the thickness of the
mold, k is the thermal
conductivity of the metal,
is its density, cp is its heat
capacity, and 
is time.
For more information on the equations above and below, check out the chapter on Conduction in the Handbook of Heat Transfer, McGraw-Hill.
Note that the inside mold cavity temperature always lags the outside mold cavity temperature. In fact, the time at which the inside mold cavity temperature first begins to increase from To is given by:
Really! This little equation incorporates the type of mold
material, the rate of heating, the wall thickness, everything!
It should be apparent that the mold cavity temperature lag time
is weakly dependent on the energy input from the oven [h, the
heat transfer coefficient], but extremely dependent on the mold
wall thickness, L [to the almost square] and the mold thermal
diffusivity, 
.
Although not theoretically correct, the inside mold wall
temperature, TL, can be assumed to lag the outside mold wall
temperature, Tw, according to:
The reader should note that this equation is correct for constant heat flux. The heat flux in rotational molding slowly decreases as the mold temperature increases. And further, this equation assumes that the inside mold cavity surface is insulated, which it obviously is not! However, since this is just a thought problem and not an academic exercise, this assumption is okay, at least for the moment.
Again, note that the temperature offset is proportional to the convection air heat transfer coefficient and the thickness and thermal properties of the mold material. The asymptotic temperature difference increases with more rapid air motion, thicker molds and lower mold thermal conductivity.
Furthermore, it can be shown that the inner and outer mold
surface temperatures become parallel when
So what, you say? Well, this approach adequately defines the inner mold cavity temperature throughout the heating [and cooling] cycle! The inner and outer mold temperatures look something like curves A and B of Figure 1.
Figure 1.
What About the Mold Cavity Air Temperature?
Devices such as Rotolog and Rotolog II contain thermocouples that monitor internal temperatures. Mold cavity air temperature is one of the temperatures that these devices measure. Consider energy transfer between the mold and the mold cavity air temperature for a moment 4. And further, assume that [horrors!], we forgot to put plastic powder in the mold cavity. The energy uptake by the air 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 that of the inner mold cavity surface. Since the volume of air in a given mold cavity is known, the exact air temperature can be calculated by solving the traditional transient heat conduction equation with an appropriate adiabatic inner mold cavity surface boundary condition. For now, we'll leave this to the academics. Suffice it to say that the time-dependent mold cavity air temperature will quickly parallel that of the inner mold cavity surface, as Crawford et al have experimentally verified.This is shown in our thought Figure 1 as curve C.
So, What Happens When Powder Starts to Stick to the Mold?
The first question is " When does powder start to stick to the mold? " or more correctly, "At what temperature does powder start to stick to the mold? " In earlier writings, I've defined that temperature as the tack temperature. It's most certainly a temperature around the melting temperature of a crystalline polymer, although its exact value depends on subtle things such as particle shape and particle surface characteristics. For the purposes of this thought problem, consider polymer sticking when its melting temperature is reached. This is given as the horizontal line D in Figure 1.
Okay, but now a most difficult problem is posed. How do we model the sticking rate, the thermal properties of the initially fluffy, now stuck-together powder, the heat of melting, and so on? Aha! We don't!
Instead, we move to the other end of the time-dependent
temperature scheme and consider what happens after all the
powder is stuck to the mold and melted and densified. At this
condition, the polymer is molten and has uniformly coated 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. And once again, the air temperature should
be paralleling the outer mold surface temperature. The offset
temperature between the inner liquid polymer surface and the
outer mold surface temperature is given as
5:
where now
is the thickness of the polymer layer and kp is the
thermal conductivity of the liquid polymer. In other words, the
air temperature now lags the outer mold temperature by both the
thermal resistance through the metal, L/k, and that through the
polymer,
/kp.
It is immediately apparent that the thicker the
polymer layer becomes, the greater the thermal lag will become.
Thus, in Figure 1,
the air temperature is shifted away from its
original curve C to a new curve E by the amount of the thermal
resistance through the polymer.
Okay, So What?
Well, in Figure 1, we have now defined our thought problem in a series of curves and lines [A through E]. We know that as soon as the inner mold cavity temperature reaches curve D, the air temperature will begin to deviate away from curve C toward curve E. The rate at which it asymptotically reaches E will depend on those fussy little things like bed configuration and rates of coalescence and densification and air migration from the polymer and latent heat of melting. But, given sufficient time in the oven, the inner air temperature will asymptotically reach curve E.
And it seems that we've developed is a molding diagram of sorts for rotational molding, all based on a thought problem!
An Aside
Consider the following variant to this problem. Instead of charging the mold with polymer powder, leave a part in the mold and let it be reheated and remelted [horrors II!]. Should the time-dependent air temperature curve look like Curve E right from the beginning of the heating time?
Jim Throne, 13 March 1998
Thermoforming - Thermoplastic Foams – Powder Processing
Dr. Jim Throne, consultant to the Plastics Industry Since 1985"Thermoforming is not an easy process. It just looks easy." Keep informed about the newest technology...