## Introduction

Often, when a sheet of plastic is heated prior to forming it into a part, there are planes of heaters above and below the sheet. The heaters often have many segments, with each segment temperature controllable. As a result, the energy input to the sheet can be “shaped” to provide uniform heating of the sheet. The raison d’etre for this work can be found in [1].

## Differential and Finite Element Interchange

As a first assumption, we consider both the sheet and the radiant heaters to be very simple finite black body parallel planes. The differential element energy interchange yields the following well-understood form:

(1)

For parallel differential elements, the direction cosines are just z/r, yielding:

(2)

If we replace the differential elements with finite elements, we can easily generate the standard double integral form:

(3)

The extended version of this equation yields a quadruple integral form where dA_{1}=dx_{1}dy_{1} and dA_{2}=dx_{2}dy_{2}:

(4)

## Stokes’ Theorem

Stokes showed that an areal integral can be replaced with a integral that follows the contour of the area in a counterclockwise fashion [2]. This is written as:

(5)

We can replace surface integration with contour integration and carry out the integration.

(6)

If all the heater and sheet temperatures are equal, the energy interchange can be written as:

(7)

s referred to as the view factor or configuration factor.

When the double integral is parsed in finite elements, desired geometric elements can be algebraically related back to the Stokes’ integrated form. Consider the case where there is only one heater element and one sheet element, Figure 1 [3].

The contour integral for this case yields the following equation for the view factor:

(8)

Where X=x/z and Y=y/z.

## Extension of the Stokesian Model to Many Elements

Although integrating the contour integrals can be done for a few parsed elements on each surface, the practical problem involves, say, m x n heater elements and x x y sheet surface elements. When the parsed elements become many, the arithmetic becomes tedious at best and very prone to “juggling” errors. There does not appear to be a Stokesian-driven algorithm that allows the sum of all the energy interchanges between, say, element xy on the sheet and the m x n heater elements, where xy is any position on the sheet surface.

## The Differential Form of the Double Areal Integral

In earlier publications, the differential area elements were replaced with differential area elements of the form [4].

(9)

To determine the sum of energies interchanged between all the m x n heaters and sheet element ab, say, the areal integral has been replaced with summation of the form:

(10)

While this equation has been used to generate the “energy dome” for equal-sized square heater and sheet elements, it is difficult to apply when heater and sheet elements are neither of equal size nor square.

## The Algorithm for the Stokesian Contour Integral

While searching for an appropriate computer algorithm, we found a reference to a 1982 catalog of Radiation Configuration Factors [5,6] The coordinates associated with the algorithm are shown in Figure 2.

= a and b in the arithmetic that follows. Again, following tradition, A_{1} is the area of the sheet element and A_{2} is the area of the heater element. The view factor for heater element ab and sheet element xy is given as:

(11)

Note that there is no restriction to the sizes or locations of the various elements in this equation. It is, in fact, the general solution of the parallel plate radiant energy interchange between m x n heater elements and any or all of the x x y sheet elements.

## Extension to Typical Thermoforming Heat Transfer Issues

For the black body interchange between heater element mn and sheet element xy:

(12)

For the black body interchange between all heater elements M x N and sheet element xy:

(13)

For the black body interchange between a heater element mn and all sheet elements X x Y:

(14)

And the total black body interchange between all heater elements M x N and all sheet elements X x Y:

(15)

For gray-body energy interchange, where _{mn} is the emissivity of the heater and _{xy} is the emissivity of the sheet (assuming these values represent diffuse energy transfer and are independent of wavelength, sheet and heater temperatures), the total radiant interchange between MxN heaters and X x Y sheet is given as:

(16)

## Checking the Arithmetic

Case 1. In [7] is given a plot of the view factor for radiative energy transfer between identical, parallel, directly opposed rectangles. For X=Y=Z=1, the view factor F_{1-2} = 0.200. When the appropriate values for the differential form for the view factor equation are used, F_{1-2} = 0.218.

Case 2. Consider the view factor between heater element 3 x 3 and sheet element 1 x 1. From equation (11), the view factor is 0.00448. When the appropriate values for the differential form for the view factor equation are used, F_{1-2} = 0.00393.

At this point, this author is unsure why these two methods do not yield exactly the same values. Although the “exact” solution, equation (11) is perhaps slightly more complicated than the differential form, both involve multiple summations and both involve relatively simple arithmetic.

Case 3. In [8] is the computer solution for energy input to a sheet containing 49 elements from a heater bank containing 49 identical elements. Here x=y=Z=1. Because the energy interchange is symmetric, only one-fourth (+one) of the elements need to be used. Given in Table 1 below are the view factors for the left upper quadrant, relative to 100% at the center element.

## Table 1

Element Number | |||

(11) | (12) | (13) | (14) |

60.9 | 74.0 | 76.8 | 77.3 |

90.8 | 94.4 | 95.1 | |

98.4 | 99.2 | ||

100 |

The relative view factors calculated from equation (11) are given in Table 2 below:

#### Table 2

Element Number | |||

(11) | (12) | (13) | (14) |

53.6 | 68.9 | 72.3 | 73.0 |

88.9 | 93.3 | 94.1 | |

98.1 | 99.0 | ||

100 |

The errors between these two methods are shown in Table 3 below. It is apparent that the differential method yields more generous results than that of the more exact equation (11), particularly at the two-dimensional corners. Nevertheless, the difference in the overall energy interchange between these two methods is only 1.2%. It is believed that this difference is primarily the result of the use of the centerpoint positions on the differential model and the edge positions on the method used to generate equation (11).

**Table 3**

Element Number | |||

(11) | (12) | (13) | (14) |

-13.6% | -7.4% | -6.2% | -5.9% |

-2.1% | -1.2% | -1.1% | |

-0.3% | -0.2% | ||

0% |

## Conclusion

View factors are extremely important values when determining energy interchange between various heater and sheet elements. Originally, the double areal integrals, or the equivalent quadruple position integrals, were solved by replacing the differential element equation with differential elements and simply summing the view factors.

Recently, a thorough study of the very extensive literature in this area revealed that the areal double integral equation could be replaced with a contour integral equation, said contour equation then being integrated. Furthermore, it was found that there was a general mathematical form for energy interchange where the sheet element was placed anywhere in the x-y plane and the heater element was placed anywhere in the a-b plane. Furthermore, there was no restriction on the relative dimensions of either the heating element or the sheet element.

The values of view factors generated from solving this equation for simple elements and 9×9 elements were similar but not exactly the same as those determined by differential summation. Because the general Stokesian-generated equation is more versatile, we are recommending its use rather than the differential equation.

## Future Work

This model will be used to analyze the following thin-gauge processes:

- Thin-gauge heating times
- Multiple ovens in thin-gauge heating (including oven temperature profiling)
- Effect of edge heating of thin-gauge sheet
- Effect of environment on edge of thin-gauge sheet just prior to exiting the oven

Then the model will be used to analyze combined convection and radiation on the heating of thick-gauge sheet.

## References

- J.L. Throne,
**Technology of Thermoforming**, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, Section 3.7,*Radiant Heating*, pg. 138+. - S.A. Schelkunoff,
**Applied Mathematics for Engineers and Scientists**, 2^{nd}Ed., D.Van Nostrand Company, Inc., Princeton, NJ, 1965, pg. 131. - J.R. Howell, https://www.me.utexas.edu/~howell/tablecon.html, C-11,
*Identical, parallel, directly opposed rectangles.* - J.L. Throne,
**Technology of Thermoforming**, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, pp. 155-159. - J.R. Howell,
**A Catalogue of Radiation Configuration Factors**, McGraw-Hill, New York, 1982. Dr. J.R. Howell, Professor, University of Texas at Austin kindly referred us to updates of the catalog on his website, https://www.me.utexas.edu/~howell/tablecon.html. His catalog reference C-13 is entitled*Rectangle to rectangle in a parallel plane; all boundaries are parallel or perpendicular to x and boundaries.* - J.R. Ehlert and T.F. Smith, T.F., “View Factors for Perpendicular and Parallel, Rectangular Plates,”
*J. Thermophys. Heat Trans.*, vol. 7, no. 1, pp. 173-174, 1993. - M.F. Modest,
**Radiative Heat Transfer**, McGraw-Hill, Inc., 1993, Figure D-2, page 794. - J.L. Throne,
**Technology of Thermoforming**, Hanser-Gardner Publicatins, Inc., Cincinnati OH, 1996, Figure 3.33, pg 157.