## Background

In this technical note, we examine the arithmetic used to predict energy transfer to plastic sheet during the heating portion of the thermoforming cycle. Our discussion is restricted to radiantly opaque sheet. There are three modes of heat transfer:

** Conduction**, where energy is moved through a solid or non-flowing substance. The key to conduction energy transfer is the thermal conductivity and its sister, thermal diffusivity, of the substance. In effect, conduction energy transfer decreases as the thermal conductivity of the substance decreases.

** Convection**, where energy is transferred by a flowing substance. The key to convection energy transfer is the flow rate of the fluid, with fluid viscosity and thermal properties important secondary influences. A measure of convection energy transfer is the heat transfer coefficient, being the proportional factor between the thermal driving force and the amount of energy transferred.

** Radiation**, where energy is interchanged between two substances that do not touch. The key to radiation energy transfer is that the two substances have different temperatures. The radiation wavelength-dependent energy absorption characteristics are important secondary influences.

All three modes of heat transfer are relevant in thermoforming. Energy reaches the radiantly opaque surface from the heater source primarily by radiation. Heat may be added to or removed from the sheet surface by convection to the air surrounding the sheet. And energy is transferred from the sheet surface to the sheet interior by conduction.

## The Arithmetic

The general approach to modeling the heating process begins with the basic assumption that the thermoforming sheet is arithmetically and thermodynamically thin when compared with its planar dimensions. This allows us to consider the sheet as one-dimensional in the thickness direction only. Since we are considering the sheet to be opaque to incident radiant energy, energy is moved throughout the sheet only by conduction. Convection and radiation energy input is consigned to the sheet surface only, or as will be noted, to the boundary condition of the mathematical model. As a result, the standard textbook transient or time-dependent heat conduction equation is valid:

Here, T is the local temperature, T = T(x,), where x is the distance into the sheet and f is the elapsed time. It is common to assume that the polymer thermal conductivity, k, is neither temperature- nor position-dependent. Thus thermal diffusivity, a = k/rcp, becomes the common polymer thermal property, where r is polymer density and c is thermal heat capacity. The equation then becomes:

## Boundary Conditions

If the energy input is equal to both sides of the sheet, the thermal gradient in the sheet is symmetrical about the mid-plane. The plane of symmetry is given as:

This means that the heat fluxes from each surface are equal.

The energy input at the sheet surface is given as:

The first term on the right is the convective energy transferred between the sheet, at temperature **T**, and the environmental air, at temperature **T _{a}**. Note that if the air temperature is less than the sheet temperature, energy is transferred from the sheet surface to the environmental air. The second term on the right is the radiative energy transferred between the sheet and the heater, at temperature

**T**. The asterisk indicates that the temperatures used here are absolute. That is, for Celsius or Centigrade,

_{h}***T**=

^{*}**T**+

**273**, and for Fahrenheit,

**T**=

^{*}**T**+

**460**. There are several terms ahead of the bracketed temperatures. σ is the Stefan-Boltzmann constant.

**F**is a

*non-black body correction factor*, discussed below,

**0**<

**F**<

**1**.

**F**is a geometric factor, or

_{g}*view factor*, also discussed below,

**0**<

**F**<

_{g}**1**.

## Non-Black Body Correction Factor

All radiation theory is based on the maximum amount of energy interchanged between two bodies. The higher temperature body is commonly called the *source*. The lower temperature body is called the *sink*. One radiation physical characteristic of a body is its *emissivity*, ε . The body is called a *black body* when ε = 1. When ε << 1, the body is called a non-black body. the emissivity can be wavelength-dependent, ε = ε (λ). If the emissivity is not wavelength-dependent, the body is called a *gray body*. Typically, as a first approximation, radiant heaters used in thermoforming can be assumed to be gray bodies, where ε_{h} = 0.90 to 0.95. And most polymer sheet can also be assumed to be gray bodies, where ε_{s} = 0.90 to 0.95. As a result, for planar surfaces, F is given as:

## The Geometric or View Factor

When the source and sink are infinite planes, radiation interchange occurs without any loss due to geometry. When the source and sink are finite however, a portion of the radiation emitted by the source does not impinge on the sink and vice versa. Instead, this radiation is lost from the interchange. In thermoforming, energy interchange is maximum in the center of the sheet and is lowest at the corners of the sheet. This is sometimes referred to as the *energy dome effect*, Figure 1. The relative effect of sheet dimensions and heater-to-sheet spacing is given in terms of the geometric factor, F.

For most thermoforming applications, F is usually greater than about 0.7. In other words, at least 70% of the energy transfer is between the sink and the source. On the other hand, up to 30% of the energy transfer goes into heating non-sheet, viz, rails, clamp frames, oven sidewalls, the outside world, etc.

It should be noted that the value of F is an average

Arithmetically, 0.818 < f < 0.905. in other words, for most thermoforming operations, the energy transmission rate is 80 to 90% of the theoretical maximum rate. d value. For local heating on the sheet surface, it is necessary to devise a more complex arithmetic. This is treated elsewhere {J.L. Throne, Technology of Thermoforming, Hanser, 1996, pp. 155-159}].

## Model Building

There are two general approaches to the arithmetic needed to model the heating process. The older technique is called finite difference equation or FDE. The newer technique is called finite element analysis or FEA. For conduction heat transfer, FDE is simpler to understand and to implement than FEA. While there are sophisticated FDE models for minimizing error generation, the speed of modern computers allows the use of the simplest model, that of explicit FDE. In this model, the partial differential equation is written as:

T _{Θ+1} = T _{Θ} + αƒ [ T _{Θ} X _{i} , T _{Θ} ( X _{ i – 1} ), T _{Θ} ( X _{ i + 1} ) ]

for i = 1 to N

The boundary conditions become quite simple, as well. In order to maintain computational stability, the time step, ΔΘ, must be less than:

Δθ ≤ [ Δx² / 2α ]

where Δx is the differential slice of sheet parallel to the sheet surface. As is apparent, if the temperature profile must be very accurate or if the sheet is very thin, Δx must be small and ΔΘ is reduced in proportion to the square of Δx. In other words, the differential time step must be very small and so the number of iterations needed to complete the computation must be very large. Again, however, with the speed of modern microcomputers, the actual computation time is quite short. For example, TF202, a computer program written in QBasic, takes approximately 4 seconds to run the example below, on a Pentium 166MHz computer.

## Example

A 20 inch x 30 inch x 0.060 inch thick polystyrene sheet is to be heated to its forming temperature using the following environmental settings:

Top and bottom heater temperature, T_{h}= 600F

Top and bottom air temperature, T_{a} = 175F

Top and bottom heat transfer coefficient, h = 2

Top and bottom sheet-to-heater spacing = 6 in

Sheet and heater emissivity, ε_{s} = ε_{h}= 0.9

The sheet thickness is divided into 10 elements [ Δx = 0.060 in/10.] The calculated time when the average sheet temperature equals the lower forming temperature of 260F is **43.4 s**. The calculated time when the average sheet temperature equals the normal forming temperature of 300F is **56.6 s**. The calculated time when the average sheet temperature equals the upper forming temperature of 360F is **82.4 s**.