## Background

Part I detailed the arithmetic used to predict energy transfer to plastic sheet during the heating portion of the thermoforming cycle. In this part, we’ll take a look at heating of thin-gage sheet.

In Part I, we considered the importance of the three modes of heat transfer – conduction, convection and radiation – on the heating of a plastic sheet of nominal thickness. As noted, conduction describes energy transmission from the sheet surface toward its centerline. Convection deals with air motion around the sheet, and radiation is the dominant way in which energy reaches the sheet surface from the heaters. It is well-known that plastics have relatively poor thermal conductivities. As a result, energy transmission into heavy-gage sheet is strongly dependent on conduction. In fact, it is recommended that very heavy-gage sheet be heated in a forced air convection oven, to minimize overheating of the sheet surface before the sheet centerline temperature is within the forming range.

Conduction is relatively less important as the sheet thickness decreases. Typically, the temperature difference across a 60 mil or 1.5 mm sheet may be less than 10^{O}C at the end of the heating cycle, and for 40 mil or 1 mm sheet, it may be less than 5^{O}C. This allows us to simplify the transient heat transfer mathematics, as noted below.

However, another, perhaps more serious concern needs to be addressed for thin-gage sheet, that of sheet semi-transparency to incident radiation.

We explore both of these factors in the Technical Note.

## The Thin-Gage Arithmetic

In Part I, we presented the traditional one-dimensional transient heat conduction equation, together with conduction and radiation boundary conditions. For thin-gage sheet, this arithmetic can be substantially modified and simplified. We only need to assume that the sheet has uniform temperature throughout its cross-section. In other words, the center of the sheet has always the same temperature as the surface. Correctly, the arithmetic is referred to as the “lumped parameter method” or LPM, and is found in detail elsewhere [J.L. Throne, Technology of Thermoforming, Hanser, 1966, p. 164+.]. Technically, LPM is applicable when Bi [Biot Number] = h_{apparent}L/k, < 0.1, where h_{apparent} is the combined radiation and convection heat transfer coefficient, L is the total thickness of the sheet, and k is the thermal conductivity of the plastic. The Biot Number represents the relative importance of energy input to the surface [convection and radiation] to energy transfer into the bulk of the plastic [conduction]. Obviously the thinner the sheet, the smaller the Biot Number.

If this condition is satisfied, the LPM equation becomes a very simple first-order ordinary differential equation:

where q _{convection} = h _{air} (T _{air} – T _{sheet}).

The terms given here were, for the most part, defined in Part I. As above, L is the total thickness of the sheet.

Recall in Part I that we defined F_{g} as the geometric factor or *view factor*. In Part III, we will address the relevance of geometry on the heating of sheet. We also defined “F” as a *non-black body correction factor*. We also noted that this correction factor was defined as:

where ε_{h} is the emissivity of the heater, assumed to be 0 < ε_{h} < 1 and independent of wavelength. And where ε_{s} is the emissivity of the plastic sheet, also assumed to be 0 < ε_{s} < 1 and independent of wavelength. As noted, for most heaters ε_{h} = 0.9 to 0.95 and ε_{s} = 0.9 to 0.95.

In a paper (almost) presented at the 1997 SPE ANTEC, the issue of wavelength-independent plastic sheet emissivities was discussed and examples were (almost) given to illustrate the inappropriateness of this assumption. These results will be given here.

## Model Building

The finite difference version of the LPM is much simpler to write, unlike that for the distributed-parameter model given in Part I:

Only one boundary condition, the initial sheet temperature, T [θ = 0] = T_{0} , is needed. This equation is very easy to solve on even the smallest of computers.

## Wavelength-Dependent Emissivity

We rarely are able to measure emissivity. Instead, by means of an infrared scanner, sometimes known as an FTIR device, we measure wavelength-dependent transmission through a thin film of polymer. Now the sum of the transmissivity, τ, absorptivity, α, and reflectivity, ρ, of a material must add to unity: ρ + α + τ = 1.

According to Kirchhoff’s law, absorptivity and emissivity are equal for systems in thermal equilibrium. And usually we can ignore reflectivity, which can be related to the difference in indices of refraction between the plastic and air, and which represents no more than 5% of the total incident energy. Therefore, if we measure transmission, we can assume that absorption is given as 100% – percent transmission. In other words, the chemists’ IR measuring device yields important information about the heating characteristics of thin-gage polymers, as we shall see.

## Figure 1: Far-Infrared Spectra for Two Thicknesses of Polystyrene

Figure 1 shows a typical generic IR curve for polyethylene. Along the x-axis is the logarithmic wavelength, in microns or μm. Percent transmission is given along the y-axis. Before we use this graph for heat transfer, consider certain characteristics on it. First, typical infrared spectra range from 2.5 microns to 10 microns or more. A heater at 1600^{O}F or 870^{O}C produces its peak energy at 2.5 microns. A heater at 1000^{O}F or 540^{O}C produces its peak energy at 3.5 microns. A heater at 700^{O}F or 370^{O}C produces its peak energy at 4.5 microns. And you and I at 98.6^{O}F or 37^{O}C produce our peak energy at about 9.7 microns. Note that polyethylene is completely radiantly opaque in the range of about 3.5 microns. Technically, all carbon-hydrogen organics, including most polymers, absorb all incident radiation at this wavelength, owing to C-H stretching. Other bumps and wiggles along the wavelength path, particularly around 8 microns, are also indicators of molecular absorption characteristics of given polymer types.

Then note that even at 10 mils, 0.010 inches or 0.25 mm, polyethylene is relatively transparent to incident radiation, particularly in the 4 to 7 micron range, where its wavelength-dependent transmission is 60 percent or more.

And finally note that, as expected, thicker films have less infrared transmission and hence greater absorption than thinner films. Most IR scans display spectra for at least two film thicknesses. This allows us to obtain thickness-dependent infrared information.The appropriate model is the Bouguer-Lambert-Beer law which states that wavelength-dependent energy decreases exponentially with increasing thickness, with absorptivity, α (λ), as the proportionality:

## Obtaining α(λ) and Hence α _{average}

The average absorptivity is given as:

If you are in control of obtaining transmission curves, set the machine to subtract the transmission value from unity, then integrate the result over the wavelength of the machine. To obtain average absorptivity, correct the data for the appropriate film thickness. The result will be (almost) the appropriate average absorptivity [and through Kirchhoff’s law, average emissivity], which is then applicable for any sheet thickness. This emissivity then becomes the proper one to be used in the F-equation, given earlier.

Now, what happens if the only IR transmission curve you have is one that is published in a book or article? You will need to descritize the transmission curve into dozens of narrow wavelength bands, then subtract each transmission value from unity, then obtain the descritized absorption values. The average absorptivity is obtained from the above equation, where the integrals are replaced with summations:

While this is a messy job, it works. Then, the average absorptivity is set equal to the average emissivity and the arithmetic proceeds apace, as before

## Results

The descritization method was used to obtain heating efficiencies for both PET and PVC. Tables 1 and 2 show the error generated when the absorptivity [=emissivity] is assumed to be 0.95, compared with the thickness-dependent values obtained from generic IR curves, using the second method described above. This technique has been written into a QBasic computer program.

### Computed Data For PVC Heated To 280^{O}F Using Heater at 640^{O}F

Table 1

Sheet Thickness [mils] |
Calc’d Time When [sec] |
Calc’d | Calc’d Time When [sec] | Percent Error |

1 | 1.05 | 0.334 | >30 | * |

3 | 3.13 | 0.521 | 7.77 | 148 |

5 | 5.21 | 0.601 | 9.81 | 88 |

10 | 10.52 | 0.716 | 15.02 | 43 |

### Computed Data For PET Heated To 280^{O}F Using Heater at 640^{O}F

Table 2

Sheet Thickness [mils] |
Calc’d Time When [sec] |
Calc’d | Calc’d Time When [sec] | Percent Error |

1 | 1.32 | 0.513 | 3.28 | 148 |

3 | 3.83 | 0.753 | 5.14 | 34 |

5 | 6.38 | 0.834 | 7.49 | 17 |

10 | 12.75 | 0.906 | 13.51 | 6 |