The heat flux to the plate is:

f = 6 5 . 7 ^ - - - > 0 . 1 4 2 - ^ L = 0 . 5 9 5 ^ = 1 8 8 5 - ^ = 3.83 ^

A s 2 15.22 cm2 cm2 s cm2 ft2 h m2

The ideal heat flux from this source temperature is obtained from:

= o T * = 0.164 - ^ i - = 0.685 ^ 5 = 2172 ^ = 4 . 4 2 ^A cm2 s cm2 ft2 h m2

The radiant heater efficiency is given as:

Efficiency = 100 = 87%

The energy conversion efficiency is given by the ratio of the energy actuallyemitted by the heater to the rated heater efficiency. For the heater rating of8 W/in2, the energy conversion efficiency is:

Energy Conversion Efficiency = 100 - ^ - = 47.9%8

3.8 Real HeatersEfficiencies

As noted earlier, only a fraction of the energy supplied by utility companies to thethermoforming machine is converted to radiant energy to heat the sheet (Fig. 3.22)[9]. Efficiencies of actual radiant heating sources are given in Table 3.8. Theefficiencies of various types of heating sources for various polymers are given inTable 3.9. These values represent net efficiencies. The energy conversion from powersource to radiant thermal energy at the heater surface is relatively efficient (Example3.13). Quartz heaters are more efficient at higher temperatures (Fig. 3.23). About50% of the electrical power input is converted to radiant energy at 316C or 6000F.Essentially all is converted at 9000C or 16500F. As seen in Table 3.8, tubular andspiral wire heaters have similar efficiencies at about 50% when new. Gas combustionefficiency at 9000C or 16500F for one type of surface infrared burner is reported tobe 82% to 84% [10], with an average heat flux at this temperature of 236.5 kW/m2 or75,000 Btu/ft2 h. The ideal black body energy emitted at this temperature is 107.4kW/m2 or 33,970 Btu/ft2 h.

Other types of surface burners show efficiencies somewhat lower than this. Notein Table 3.8 that the effective surface heat fluxes for most gas-fired burners operatingat very high temperatures are substantially greater than the values predicted by blackbody radiation. Convective energy transfer is apparently a major factor with theseburners. Since all radiant heaters operate in an air environment, convection lossesfrom heater surfaces reduce heater efficiency, sometimes by as much as 30% to 50%.

Previous Page

Energy Absorbed by Sheet

Figure 3.22 Schematic of heat transfer energy distribution in thermoforming operation [9]. Figureused by permission of Society of Plastics Engineers, Inc.

An estimate of the convection heat loss is detailed below. Heaters radiant energy toall visible surfaces, including: Plastic sheet, Reflectors, Other heaters, Rails, Sheet clamping devices, Heater guards, Objects outside the oven edges, Oven sidewalls, and Shields and baffles.

As much as 20% to 30% of the energy emitted by radiant heaters is lost to theenvironment in this way. Further, a fraction of net radiant energy absorbed by theplastic sheet is convected to the cooler air environment from the hot sheet itself.Thus, only about 20% to 50% of the power supplied by the utility is converted intoincreasing the enthalpy of the sheet. The actual efficiency depends on:

Matching source temperature with plastic radiation absorption range, Minimizing all thermal sinks other than the plastic sheet, and Controlling the convective energy losses from heater and sheet surfaces.

Energy Supplied to Heaters

Energy Loss During Conversion to Radiant Heat

Energy Convected From HeatersRadiation Loss to Surroundings

Reradiation From Surroundings

Reradiation From Heaters

Radiation From Sheet to Heaters

Convection Heat Loss to Surroundings

Table 3.8 Efficiencies of Commercial Radiant Heating Sources1

Comments

Spot output, colorsensitive

Needs reflector, colorsensitive

Spot outputNeeds reflector, color

sensitive, seals mayneed cooling, must bekept clean

Needs reflector, sur-face exposed, airflowcauses large heatloses, resists shock,vibration

Needs reflector

Very even heating, lowtemperature leads toairflow losses

Even heating, can bezoned

See comments onmetal sheathed tube

Available with soft orhard face

Response time

Cooling

10s

10s

5-10 min10s

5 min

20s

2-4 min

5 min

5 min

5 min

5-10 min

Heating

3s

3s

5-10 min3s

5 min

1 min

2-4 min

5 min

5 min

5 min

5-10 min

Maximumefficienty

(%)

Thermally shocksensitive

Shock hazard, con-vective heat lossescan be high

Efficiency decreaseincreased output

Same as aboveSame as above,screen maintenancecan be a problem

Emitter can be dam-aged by force orfluids

Low temperature,emitter can bedamaged by forceor fluids

5-20 min

10s

2-4 min

2-3 min1 min

4-8s

30 min

5-20 min

4s

1-3 min

2-3 min1 min

4-8s

30 min

44

74

3-53

20-55320-553

33-652-3

20-553

44,700

25,300

99,400

37,30030,900

34,000

3,100

141

79.7

314

117.7

97.7

107.7

9.8

1800

1500

2300

17001600

1650

700

980

815

1260

930

870

900

370

19660

18770

600000

800004000

47000

5000

62

59.2

1890

252126

148

15.8

Quartz or hardceramicExposed foil

GaS-IR impingment:Ceramic plate

Gas-IR surface burn:Ceramic plateScreen

Ceramic fiber

Catalytic

1 Adapted from [42], with copyright permission

2 Greater than 100%

3 Listed efficiency, but greater than 100% of black body efficiency for given temperature

Radiative Heat Transfer Coefficient

Convective energy losses are determined with an energy balance around all solidsurfaces, including the heater and plastic sheet. The effect of radiation heat transfermust be included as well. This is done by examining the surface boundary conditionfor the transient one-dimensional heat conduction (Equation 3.9):

~= - k ^ =frr,T,,F,Fg)h,eJ (3.27)^

0 X (6,L)

For radiation absorption (only) on a solid surface, x = L, the proper form for f[---]is:

ITT5T009F9F^eJ = {aFFg}[T* 4 - T*4] (3.28)As noted, the radiation boundary condition is nonlinear, unlike the convectionboundary condition that is linear with temperature, f[---] = h(Too T). For certaincases, the radiation nonlinearity can be dealt with by letting T* = a TJ,, where a isa proportionality. Then {[-] becomes:

IjT5T005F9F89G^eJ Ji1(T00 - T) (3.29)where hr9 a radiation heat transfer coefficient, is given as:

hr = {aFFg}T* 3(a + l)(a2 + 1) = FFg R (3.30)where R is the radiation factor (Fig. 3.24). Note that the proportionality a is notconstant but varies with the absolute value of the sheet temperature. If a is verysmall, or TJ3T* throughout the heating cycle, R is approximately constant.Further if a does not vary much throughout the heating cycle, values of R aredetermined at the beginning and end of the heating cycle and an average value or R

Table 3.9 Radiant Heater Efficiencies for Several Polymers1

Polymer

LDPEHPDEPSPVCPMMAPA-6Cellulose acetateFor the typical graybody thermoformingwavelength range,1.4 to 3.6 um

1 Adapted from [33]

Heater type

Ceramic5100C(9500F)4.0 um

13%13%13%5%0%

30%18%28%

Metal rod5500C(10220F)3.8 um

15%15%15%5%2%

28%28%33%

Quartz680C(1256F)3.0 jim17%17%.17%22%50%24%48%70%

Quartz7600C(14000F)2.8 um

20%20%20%25%65%28%56%77%

Peak Wavelength, um

Figure 3.23 Temperature-dependent peak wavelength and quartz heater output

is used. Example 3.16 illustrates the use of the radiation factor. As is apparent, thevalue of R increases with increasing heater temperature and increasing sheet temper-ature. In the example, a 2000F increase in heater temperature results in a 29%increase in the rate of heating. Similarly increasing the sheet temperature to theforming temperature results in a 20% to 30% increase in heating rate. Typically, theaverage value for R is accurate to within 15% to 20% of the actual value. To obtaina value for the radiation heat transfer coefficient, hr, the average value of R must becorrected for the gray-body interchange factor, Fg and the view factor, F. As a result,the actual value for the radiation heat transfer coefficient, hr, can be substantially lessthan the value for R. Typical values for hr are 1 to 10 times those for moving airconvection heat transfer coefficients in Table 3.2. The use of the artificial radiationheat transfer coefficient should be restricted to problems where rapid solutions andapproximate answers are acceptable.

Convection and the Heat Transfer Coefficient

Air trapped between the heater banks and the plastic sheet surfaces is very slowmoving or quiescent. It therefore attains a nearly isothermal temperature having a

Heat

er Te

mpe

ratur

e, 0 F

Typical Quartz Heater Output W/in2

Human Body

Radia

tion Fa

ctor

, R,

Bt

u/ft2

-h-

F

Absorber SurfaceTemperature, 1000F

Emitting Surface Temperature, 100 0F

Figure 3.24 Source and sink temperature-dependent radiation heat transfer coefficient

value somewhere between that of the sheet surface value and that of the radiant source.The nature of energy transfer is by rising, buoyant warm air and settling cool air. Thisis natural convection. The natural convection heat transfer coefficient is obtained from:

(3.31)

Heater

Convection Film

Sheet

HeaterFigure 3.25 Location of various convection heat transfer coefficients between sheet and top andbottom heaters

G is the length of the plate heater or the diameter of a rod heater. G T is thetemperature difference between the hot surface and the air. The proportionalityconstant K depends on the heater geometry G and whether the heater faces up ordown (Table 3.10). Example 3.14 illustrates the method of calculation for theconvection heat transfer coefficient. The range of 0.5 to 2 Btu/ft2 h 0F or 2.8 x 10~3to 11.3 x 10 ~3 kW/m2 0C is typical of natural convection heat transfer coefficientsfor quiescent air (Table 3.2). The range is a factor of 10 or so less than the typicalrange for forced air convection heat transfer coefficients and 20 times lessthan those for radiation heat transfer coefficients. Note that if the air is hotter thanthe plastic sheet, energy is convected to the sheet. If the sheet is hotter than the air,as in Example 3.15, energy is convected from the sheet. A combined convection andradiation heat transfer coefficient is written as:

heffective = h + hr (3.32)

Example 3.14 The Radiation FactorConsider heating a plastic sheet initially at 800F to 4000F using a heating source at8000F. Determine the initial and final values of the radiation factor. Obtain anaverage value. Increase heater temperature to 10000F and recalculate values.

From Fig. 3.24, at 8000F, the initial value of R1 = 6.0 Btu/ft2 h 0F. Thefinal value of Rf = 8.1. The average value of Ra = 7.05 Btu/ft2 h 0F.

From Fig. 3.24, at 10000F, R1 = 8.15, R f = 10.05 and Ra = 9.1 Btu/ft2 h 0F.

Example 3.15 Convection Heat Transfer CoefficientConsider 2000F air trapped between a 3000F sheet and a 8000F heater. The sheetis sandwiched between two heaters. Determine the heat transfer coefficients betweenthe air and the sheet and the air and the heaters. G=I.

Table 3.10 Convection Heat Transfer Coefficients for Natural Convec-7AT\l/4

tion from Flat Plates and Rods h = K \ G /

Geometry/attitude

Heat plate (G = L)Facing upwardFacing downward

Rod (G = D)

1 From reference [43]

xvmetric(h in kW/m2 0C)(AT in 0C)(G, D, L in m)

0.001490.0007460.0015330.0028391

^English(h in Btu/ft2 h 0F)(AT in 0F)(G, D, L, in ft)

0. 2630.1310.270.501

There are actually four heat transfer coefficients to consider, as shown in Fig.3.25:

Ji1 (heater, facing down) = 0.131 - (800 - 200)1/4 = 0.65 Btu/ft2 h 0Fh2 (heater, facing up) = 0.263 (800 - 200)1/4 = 1.30 Btu/ft2 h 0Fh3 (sheet, facing down) =0.131 - (300 - 200)1/4 = 0.41 Btu/ft2 h 0Fh4 (sheet, facing up) = 0.263 (300 - 200)1/4 = 0.83 Btu/ft2 h 0F

The range is 0.4 to 1.3 Btu/ft2 h 0F.

Example 3.16 shows how the effective heat transfer coefficient changes in value as thesheet is heated. In this idealized case, the radiation contribution to the overall heattransfer coefficient overwhelms the convection contribution. In practical thermo-forming, the radiation contribution is diminished by values of F and Fg that are lessthan unity. Nevertheless, in most cases, radiation heat transfer dominates the overallheat transfer coefficient.

Example 3.16 Combined Heat Transfer CoefficientGiven the conditions of Examples 3.14 and 3.15, determine the effective heattransfer coefficient. Assume that F= Fg= 1.

The initial sheet temperature is 800F with a 2000F air temperature and aradiant heater temperature of 8000F. The final sheet temperature is 4000F with a2000F air temperature and the same radiant heater temperature.

From Example 3.14, the radiation heat transfer coefficients are R1 = 6.0Btu/ft2 h 0F and Rf = 8.1. There are four initial convection heat transfercoefficients and four final ones. Only the values between the sheet and the airare important here:

h u (heater, facing down) = 0.131 (800 - 200)1/4 = 0.65 Btu/ft2 h 0Fh2'i (heater, facing up) = 0.263 (800 - 200)1/4 = 1.30 Btu/ft2 h 0Fh3,'i (sheet, facing down) =0.131 - (200 - 80)1/4 = 0.43 Btu/ft2 h 0FV 1 (sheet, facing up) = 0.263 (200 - 80)1/4 = 0.87 Btu/ft2 h 0F1I1 f (heater, facing down) = 0.131 (800 - 200)1/4 = 0.65 Btu/ft2 h 0Fh2f (heater, facing up) = 0.263 (800 - 20O)1/4 = 1.30 Btu/ft2 h 0Fh3 / (sheet, facing down) =0.131 (200 - 400)1/4 = -0.49 Btu/ft2 h 0Fh4 / (sheet, facing up) = 0.263 (200 - 400)1/4 = -0.99 Btu/ft2 h 0F

Note that the signs on the convection coefficients indicate the way in whichenergy is being transferred.

The initial and final effective heat transfer coefficients, he { and he f, are:hej (sheet, facing up) = R1 + h4, = 6.0 + 0.87 = 6.87 Btu/ft2 h 0Fh^ (sheet, facing down) = R1 + h3'1 = 6.0 + 0.43 = 6.43 Btu/ft2 h 0Fhe / (sheet, facing up) = Rf + h4,f = 8 . 1 - 0.49 = 7.61 Btu/ft2 h 0Fhef (sheet, facing down) = Rf + h3f = 8.1 - 0.99 = 7.11 Btu/ft2 h 0F

An effective heat transfer coefficient can also be obtained from an overall heatbalance on a given plastic sheet. Effective values in Table 3.11 are obtained from

Sheet Thickness, mm

Figure 3.26 Two-sided quartz heating of sheet. Heat flux = 40 kW/m2 or 12,700 Btu/ft2 h 0F, peakwavelength = 2.8 um, heater temperature = 7600C or 14000F. Solid points obtained at heat flux = 43kW/mm2 or 12,700 Btu/ft2 h 0F, peak wavelength = 3.7 urn, heater temperature = 5100C or 9500F

thin-gage heating rate data of Fig. 3.26 and typical forming conditions. Values rangefrom about 4.5 to 9.7 Btu/ft2 h 0F or 0.0255 to 0.0548 kW/m2 0C. If theconvection contribution is essentially constant at about 1 Btu/ft2 h 0F or 0.005kW/m2 0C, the radiation contribution is about 4 to 9 times that of the convection

Heat

ing Tim

e, s

Table 3.11 Rod Heater Reflector EfficienciesEffective Heat Transfer Coefficients1

Material

Gold, newGold, agedStainless steel, newStainless steel, agedAluminum, newAluminum, aged

Emissivity

0.92

0.60

0.30

Heatertemperature(0C)

690683686668719693

Reflectortemperature(0C)

320323304352274287

Convectionheat transfer(kW/m2 0C)

0.02440.01760.01250.01420.01990.0199

1 Adapted from [11], with permission

Foam PS

HIPS

FPVC

ABS/PVC

CABRPVCHDPE

PMMA

Figure 3.27 Radiation between metal rod heaters and planar sheet [41]. Figure used by permissionof McGraw-Hill Book Co., Inc.

contribution1. Further, if the average black-body net radiant interchange yields aneffective radiation heat transfer coefficient of about 10 to 15 Btu/ft2 h 0F or 0.05 to0.075 kW/m2 0C, the radiant interchange efficiency is about 40% to 60%. Thisefficiency is the product of the gray-body factor, Fg, and the view factor, F. Thisefficiency agrees reasonably well with values that are discussed below. These effectiveheat transfer coefficient values are typical of experimental data obtained in otherways [H].

Rod Heaters

Rod heaters, with or without reflectors, are used to heat sheet in many thermoform-ers. The energy emitted from rod heaters is related to that emitted by a heated plane.Figure 3.27 assumes that the surface behind the rod heaters is nonconducting.Example 3.17 illustrates how to determine the relative energy efficiency of rodheaters. As is apparent, the closer the heaters are to one another, the more efficientthe energy transfer becomes. The gray-body correction factor Fg for gray surfaceradiation between a plane and a tube bank is:

F 8 = G00-G8 (3.33)

Example 3.17 Rod Heater EfficiencyConsider a single row of rod heaters 0.5 in or 12.7 mm in diameter, spaced 3 in or76 mm apart. Determine the relative energy efficiency as compared with a flat plate.Change the spacing to 1.5 in or 38 mm and recompute.

1 Note that this assumes that the convection energy transfer is from the air to the sheet, with the

air temperature hotter than the sheet temperature. Obviously if the convection contribution isnegative, the radiation effect is 6 to 11 times greater.

Ener

gy Fr

actio

n Co

mpa

red

With

In

finite

Pl

ane

Rod Spacingto Rod Diameter, d/D

Two Row Total

One Row Total

First Row of Two

Second Row of Two Planar Sheet

Non-Conducting Absorber

Fig. 3.27 requires the determination of R, the ratio of center-to-centerdistance to the diameter.

R = 3/0.5 = 6

From Curve B of Fig. 3.27, F = 0.46 or the heating is 46% as efficient asfrom a flat plate.

For R = 1.5/0.5 = 3, F = 0.73 or 73% as efficient.

Example 3.18 compares the gray-body correction factors for rod and plate heaters.Usually the gray-body correction factor values are quite comparable. However, therod heater efficiency is low when compared with the flat plate, as also shown inExample 3.18. Radiant energy loss from the back of rod heaters is minimized byreflectors. New aluminum and gold-fired porcelain enamel give the greatest reflectorefficiencies. However efficiencies deteriorate with age. Stainless steel appears toprovide the best long-term efficiency (Table 3.11). The effective heat transfercoefficient from the top of the reflector is essentially independent of reflectormaterial and reflector temperature (Fig. 3.28). The range in heat transfer coefficientis about 2 to 4 Btu/ft2 h 0F or 0.01 to 0.02 kW/m2 0C. Essentially all of this isreradiation from the reflectors.

Example 3.18 Gray-Body Correction Factor for Rod HeaterCompare the values for Fg for flat plates and rod heaters if ^ = 0.9 andzs= 0.85. Then determine the relative gray-body efficiencies.

From Equation 3.33, F g r o d = 0.9 0.85 = 0.765From Equation 3.23, Fg 'plate = [1/0.85 + 1/0.9 - I ] " 1 = 0.777The two factors are essentially the same.From Example 3.17, at 1.5-in spacing of 0.50-in diameter rods, the rodefficiency is 0.73. As a result,

Rod efficiency = 0.73 0.765 = 0.558Plate efficiency = 1 0.777 = 0.777

Or the hot plate transfers nearly 40% more energy than the rod heaters.

3.9 Long-Term Radiant Heater Efficiencies

Radiant heater efficiency decreases with time as seen in Table 3.12. The valuesrepresent overall efficiencies or effective energy conversion for several commercialheaters. Efficiency is thought to decrease exponentially with time as a first-ordersystem response:

(3.34)

Figure 3.28 Convection heat transfer coefficients for metal rod heaters with reflectors

where a is the time constant of the heater, in month"1. The expected efficiencies ofheaters at various times are shown in Table 3.121.

Since heater efficiency is directly related to the radiant heat transfer coefficient,any decrease in heater efficiency at constant heater temperature increases the time toachieve sheet forming temperature. Since heater efficiency loss is gradual, cycle timescan lengthen imperceptibly over weeks. Usually power input or heater temperature isgradually increased to compensate for the decrease in efficiency. An increase inheater temperature results in a reduction in the peak wavelength and this effect mightresult in heating in the less efficient regions of the infrared spectrum. Since efficientsheet heating is a key to optimum economic performance, all heater manufacturersnow recommend strict, scheduled periodic replacement of all elements, regardless oftheir apparent performance.

3.10 Edge LossesView Factor

Net radiant energy interchange between ideal infinite parallel heat sources and sinksdoes not depend on the distance between them. This is not the case for finitedimensions of heaters and sheets. The spacing between the plane of the heater andthat of the sheet surface affects the efficiency of energy transfer. So long as the sheetwidth dimension is much larger than the sheet-to-heater spacing dimension, radiationlosses to machinery elements are small. The relative amount of energy actuallyreceived by the sheet depends on the ability of the heater to "see" the sheet. In simpleterms:

What the heater sees is what it heats

1 These values assume that the heaters are still functioning at these times.

Conv

ectio

n He

at Tr

ansfe

rCo

effic

ient

Btu/

ft2-rr

F

Reflector Temperature, 0F

Newly Gold-Plated New Aluminum

Oxidized Gold Stainless Steel

Table 3.12 Commercial Radiant Heater Overall Efficiencies1

Expected efficiency3

24 mo

1-1.1

2.638324

18 mo

2-2.3

5.34337

7

12 mo

4-4.5

10.5494213

Efficiencyat end oflife

11-13

1931-3633-3611-12

Time constanta (month-1)

0.0926-0.1155

0.11550.020.02270.0926-0.104

Averagelife (h)

1500

300012,000-15,0008,000-10,0005,000-6,000

r|0, efficiencyafter 6 months2

8-10

21554825

new

16-18

42625540-45

Heater type

Coiled wire,nichrome

Tubular rod4Ceramic panelQuartz heaterGas-fired IRpanel

1 Adapted from [6], with copyright permission

2 One month = 440 h, assumed for time constant only

3 After 6 months use, 4-8% efficiency can be gained by replacing all reflectors

4 Sanding, polishing increases efficiency by 10-15%

Diameter or Side Dimensionto Distance Between Surfaces

Figure 3.29 Radiation view factor for radiant interchange between parallel surfaces

The radiant energy interchange between black bodies of equal finite dimensionconnected by reradiating walls is given as Fig. 3.29. The factor F is called a radiationfactor or "view factor" and typically has a value less than one. Furthermore, F variesacross the sheet surface. Example 3.19 illustrates the effect of sheet-to-heater spacingon the view factor. To obtain the proper net energy interchange value between graysurfaces, this view factor must then be multiplied by the gray-body correction factor,Fg. Example 3.19 includes the relative effect. The energy that is not transmitted to thesheet is lost to the surroundings and is called "edge losses". In Example 3.19, edgelosses amount to 36% for the wide spacing and 23% for the narrow spacing. The edgeloss is reduced if the side walls reradiate or reflect. Although spacing is used tocontrol the heating characteristics of the sheet without changing the heater tempera-ture, it is now recognized that this is an inefficient use of energy. Heater spacing isusually governed by sheet sag and minimization of sheet "striping" or local overheat-ing beneath rod and quartz heaters.

Example 3.19 View Factor and Edge LossesConsider a 600 mm x 600 mm sheet being heated with a 600 x 600 mm plate heater.Ignore edges. What is the view factor F, from Fig. 3.29, for sheet-to-heater spacingfor 150mm? For 75mm? What are the equivalent values if the sides reradiate?

Then consider a gray-body correction factor for ^ = 0.9 and ev = 0.85.R = side/spacing = 600/150 = 4. From Fig. 3.29, F = 0.64. For R = 600/75 = 8, F = 0.77.

For reradiating sides, FR==4 = 0.765. FR = 8 = 0.86. An oven with reradi-ating sides is 19% more efficient at R = 4 than one that has no reradiatingsides. It is 12% more efficient at R = 8.

View

Fa

ctor,

F

Total Radiation - Surfaces Connected byNon-Conducting Reradiating Surfaces

SquaresCircular Disks

The gray-body correction factor, Fg - [1/0.9 + 1/0.85 - I ] " 1 = 0.777. Theadjusted efficiencies, r\ FFg, are now:

rjR = 4 = 100 0.64 0.777 = 49.7% rjR = 8 = 100 0.77 0.777 = 59.8%

Local Energy Input

The view factor obtained from Fig. 3.29 yields an average radiant energy transferefficiency. The specific local energy transfer rate is also important. As seen in Fig. 3.30[12] for uniform energy output from the radiant heaters, the edges of the sheet receivesubstantially less energy than the center. This is because the heaters in the centersee substantially more sheet than those at the edges. In other words, the heatersat the edge radiate to a greater amount of non-sheet than those in the center.Figure 3.31 illustrates this. An accurate estimate of the energy of Fig. 3.30 is

Heater Elements

h/b=0.2

bbFigure 3.30 Energy received by finite sheet from

uniform energy output by heaters [12]

Radiation Overlap

Figure 3.31 Schematic of radiation overlap from heaters to sheet

Sheet

Heater Heater

Figure 3.32 Radiation ray tracing between finite parallel plane elements [13]

obtained from radiant heat transfer theory. Consider energy interchange betweentwo differential surface elements (Fig. 3.32) [13]. The intensity of the energy emittedfrom surface element dAx is constant in a hemisphere of radius r from the surface.Any element that intersects this hemisphere receives an amount of energy propor-tional to its projected area relative to the area of the hemisphere. The projected areadepends on the attitude of that element to the source plane. In differential form, thetotal energy interchange between these elements is:

Qi ^ 2 = crFg(T*e4ater - Ts*h4eet) (IA1 dA2 (3.35)JA2 JA1 ^r

The double-integral term on the right side represents the view factor, F . The terms,cos 4>, are direction cosines and r is the solid angle radius between the elements.Figure 3.29 is obtained through proper integration of the double integral of Equation3.35.

Quartz and ceramic heating elements are discrete and isothermal 'bricks'. As aresult, the differential form of the view factor that yields the double integral can bereplaced with the difference form:

_

/ rT^4 _ . T ^ ^ cos (J)1 cos 4>2 1Qi-2 = ^Fg(T*e4ater - Ts*h4eet) X E 2 AA1 AA2 (3.36)

LA1 A2 nv

Jwhere the " 1 " element is the heater and the " 2 " element the sheet1. Consider a gridof heater and sheet elements in the X-Y direction separated by a distance z in the Zdirection. For parallel surfaces z units apart:

1 This assumes that the sheet is made of elements as well. In fact, the sheet should be considered

as an infinite number of infinitesimal elements and the double integral replaced with a inte-grodifferential form. This is not done in this discussion.

Direction CosineDirection Cosine

ZY

X

A1

A2

r

COS(J)1 = COSCt)2 = - (3.37)

The spherical radius between any two heater and sheet element is given as:r = y x 2 + y2 + z2 (3.38)

The amount of energy emitted from a single heater element to all plastic elements is:Qi - Z 2 = a F g [ x rc(x2 + y2 + z2)2 (Th*e4ater - TJKL)AA1 A A 2 ] (3.39)

Th is the single heater element temperature and Ts represents one of the many sheetsurface element temperatures. Likewise, the amount of energy received by a singleplastic element from all heater elements is:

q,~a = F g [ l 7t(x2 + y2 + z2)2 (Ti ter - TSU1)AA1 AA2] (3.40)Note that the individual element temperatures are now incorporated within thesummation. Individual heater element and sheet element temperatures vary and thisexpression accommodates these variations. Furthermore note that the summation inEquation 3.39 implies that the [XY] position of the heater element is fixed and the[XY] position of each sheet element is computed relative to that [XY] position.Although Equations 3.35 through 3.40 appear formidable, they are rapidly solved ona computer. Figure 3.33 gives the computer solution for energy input to a sheetcontaining 49 elements from a heater bank containing 49 elements. The energy

Figure 3.33 Local heat flux distribution from 7 x 7 uniform 5400F heaters. Values based on 100%at element [4,4]. Relative heater-to-sheet spacing, Z = I [14]

1160.9%

2174.0%

3176. 8%

4177.3%

5176. 8%

6174.0%

7160. 9%

1274.0%

2290.8%

3294. 4%

4295.1%

5294. 4%

62

yu, cvo72

74. 0%

1376.8%

23

94.4%33

98.4%43

99. 2%53

98. 4%

63

94. 4%73

76.8%

1477.3%

24

95.1%34

99.2%44

100%54

99.2%

6495.1%

7477. 3%

1576.8%25

94.4%35

98.4%45

99.2%55

98. 4%

6594. 4%75

76. 8%

1674.0%26

90.8%36

94.4%46

95.1%56

94.4%

6690.8%

7674. 0%

1760.9%

2774.0%

3776.8%

4777.3%57

76. 8%

6774.0%

7760.9%

output is the same from each heater element. The elemental values represent theamount of energy received by a given sheet element relative to that received by thecenter sheet element [14]. As is apparent, the values of Fig. 3.33 support theproposed scheme of Fig. 3.30. The energy flux from each heater element can bevaried to achieve near-uniform energy input to the plastic sheet. Figure 3.34 is oneproposed scheme of an optimized heating system where the energy flux is the sameto each element [15]. Figure 3.35 is the computer solution obtained by varying theindividual heater element temperatures. As is apparent from Equation 3.40 andearlier discussion, small changes in absolute heater element temperatures yield

Figure 3.34 Energy received by finite sheet from zonal en-ergy output by heaters [15]b

b

h/b=0.2

Heater Element

11185%706F

21130%608F

31135%618F

41135%618F

51135%618F

61130%608F

71185%706F

12130%608F

2280%486F

3290%514F

4290%514F52

90%514F

6280%486F

72130%608F

13135%618F23

90%514F

3395%527F43

90%514F5395%

527F63

90%514F

73135%618F

14135%618F

2490%514F34

90%514F44

92.5%521F5490%514F64

90%514F74

135%618F

15135%618F

2590%514F3595%527F4590%

514F5595%527F65

90%514F75

135%618F

16130%608F26

80%486F^

3690%514F4690%514F56

90%514F

6680%486F76

130%608F

17185%706F27

130%608F

37135%618F47

135%618F

57135%618F

67130%608F77

185%706F

Figure 3.35 Uniform heat flux everywhere [+1.5%]. Relative heater temperature in 0F. Relativeheater-to-sheet spacing, Z = I

substantial changes in emitted energy. This is apparent in Fig. 3.35 for the 7 x 7heater by 7 x 7 sheet configuration. The heater temperature profile predicted in Fig.3.35 mirrors current forming practice, with corner heaters running hotter than edgeheaters and center heaters running substantially cooler than peripheral heaters.

Pattern Heating

Pattern heating is the placing of welded wire screens between the sheet and the heaterin strategic locations to partially block the radiant energy. Radiant screens arefrequently used to achieve uniform wall thickness in odd-shaped parts [16-18] whenthe heater output is fixed, as with plate and rod heaters. Fine welded stainless steelwire mesh is cut to an approximate shape of the blocking region and is placedbetween the heater plane and the sheet surface (Fig. 3.36). The screens are frequentlylaid on the wire screen protecting the lower heaters from sheet drop. They are wiredin position below the upper heaters. If fs is the fraction of open area in the screen andT*. is its absolute temperature, the energy interchanged between the heater and thesheet beneath the screen is given as:

5= {aF^F^.J-f, -[T -^Tf] (3.41)

The energy interchanged between the heater and the screen is:

(3.42)

Heater

Hanger

Welded Wire

Sheet

Welded Wire

Support

Heater

Figure 3.36 Examples of attaching welded wire screen for pattern heating on rod heaters

And that interchanged between the screen and the sheet is:

= I G F F I f [T*4 T*41 (3 43)A \ w x s c s A g , s c s i 1 S L x s c A s J y~y.-T~>j

Note that there are three view factors and three gray-body correction factors. Thesheet-to-heater distance, the sheet-to-screen distance and the screen-to-heater dis-tance have different values and the respective view factors will therefore be different.Furthermore, the emissivities of the screen, sheet and heater are different. Example3.20 illustrates the extent of reduction in energy interchange. The fraction of openarea in the screen is the primary method of controlling energy interchange in patternheating. Multiple screens are used if necessary (Fig. 3.37).

Example 3.20 Pattern HeatingEfficienciesConsider a screen having a 0.030-in wire with a square 0.060-in center-to-centerdistance. The screen is positioned halfway between a 3Ox 30 in sheet and a 3Ox 30in heater, spaced 6 inches apart. The heater emissivity is 0.9, the sheet emissivity is0.85 and the stainless steel screen emissivity is 0.3. The heater temperature is8000C, the emitter temperature is 5000C and the sheet temperature is 2000C.Determine the efficiency of heat transfer relative to the unscreened sheet.

The area of a single square is 0.060 x 0.060 = 0.36 x 10~2 in2. The projectedarea of the wire in the square is 2 x 0.06 x 0.015 + 2 x (0.06 - 2 0.015) x0.15 = 0.027 x 10~2 in2. Thus, the wire covers 75% of the surface area.fs = 0.25.

For the heater-to-screen interchange, Fg = 0.9 0.3 = 0.27. The view fac-tor is obtained from Fig. 3.27 for R = 0.060/0.030 = 2, and is F = 0.86. Thusthe heater-to-screen efficiency is:

r|sc_ ^ = FFg(l - fs) = 0.27 0.86 0.75 = 0.174For the screen-to-sheet interchange, Fg = 0.3 0.85 = 0.255. The view factoris obtained from Fig. 3.29 and is F = 0.86. The screen-to-sheet efficiency is:

nsc_s = FFg(l - fs) = 0.255 0.86 0.75 = 0.164For the heater-to-sheet interchange, Fg = [1/0.9 + 1/0.85 - I ] " 1 = 0.777. Theview factor is obtained from Fig. 3.29 for R = 30/6 = 5 and is F = 0.7. Theheater-to-sheet efficiency is:

Ti00 _s = FFg fs = 0.777 0.7 0.25 = 0.136

The energy interchange equation is:

^={aFFg}fs[Ts*o4urce-Ts*4k]

For the heater-to-screen interchange:

5 = 56>74 . Q 1 7 4 . [ L 0 73 4 - 0.7734] = 9.56 kW/m2

For the screen-to-sheet interchange:

^ = 56.74 0.164 [0.7734 - 0.4734] = 2.86 kW/m2

For the heater-to-sheet interchange:

^ = 56.74 0.136 [1.0734 - 0.4734] = 9.84 kW/m2

The total energy transfer is:

Y = 9.56 + 2.86 + 9.84 = 22.26 kW/m2

This compares with the unscreened energy transfer:

5 = 5 6 J 4 . o.7O 0.777 [1.0734 - 0.4734] = 31.57 kW/m2

The screen provides a 29.5% reduction in the amount of radiant energyinterchange between the heater and the sheet.

Zone, Zoned or Zonal Heating

With the advent of discrete heating elements, the effect of shielding or screeningcertain areas of the sheet has been, for the most part, replaced with local heatingelement energy output control. The earliest heating stations employed manually setproportional controllers on every heating element. Computer-aided controllers arenow used. In certain circumstances, the energy output from every heating element iscontrollable. For very large ovens and very many heating elements, individualcontrol is impractical. Regional banks of heating elements have a single controllerand thermocouple. Thus, for an oven with 100 x 100 elements, top and bottom,requiring 20,000 controlling elements, the oven may have 40 zones, top and bottom.In certain circumstances, individual elements may be transferred from one zone toanother electronically. In other cases, hard rewiring is necessary. Usually, zonalconditions are displayed on a CRT screen. As noted in the equipment section, mostceramic and metal plate heaters use PID-based controls and thermocouple tempera-ture is the indicating readout variable. Quartz heaters operate on percentage of thetime on and percentage is the indicating readout variable. Technically, of course,these variables are simply measures of intrinsic energy output of the heater or bankof heaters. Zone heating or zonal heating is used to change local energy input to thesheet in much the same way as pattern heating. With pattern heating, the patternmust be some distance from the sheet surface to minimize a sharp edge effect,shadowing or spotlighting where the pattern ends. In zone heating, the heaters aresome distance from the sheet surface to begin with. As a result, energy change in alocal heater or heater bank not only affects the sheet directly below it but alsochanges the energy input to the sheet elements in the vicinity. This is seen in the

Time, sHeating Pattern, Shown in Insert

Figure 3.37 Effect of patterning on thermoforming part wall thickness and temperature forpolystyrene, PS [16,17]. Initial sheet thickness = 2.1 mm. In lower figure, up to four layers of tissuepaper are used as screening. In lower figure, thickness ratio, t/t0 = 0.29 to 0.32 over entire part.Figure used by permission of Krieger Co.

computer-generated energy input scheme of Fig. 3.38 [14]. Increasing a specific heaterelement energy output 14% results in a 6% increase in energy input to the immediatesheet element neighbors and lesser amounts elsewhere even though energy outputsfrom neighboring heater elements have not changed. If this is an undesirable effect orthe effect sought requires greater focus, the bank of heaters making up the specificzone must be reduced in number.

Heater to Sheet Distance

As stated earlier, radiation does not depend on fluid or solid medium. Relativeheater-to-sheet spacing does affect radiant energy interchange however. This wasdemonstrated in Fig. 3.29 and is apparent in Equation 3.40. Figure 3.39 shows theeffect of a 50% increase in heater-to-sheet spacing relative to the optimum energy

Tem

pera

ture,

0 F

Tem

pera

ture,

0 F

Uniform Heating Pattern,Local Thickness Shown in Insert

Top Surface

Bottom Surface

Time, s

Top CornerTop CenterTop Side

Bottom CenterBottom Side

Bottom Corner

11O0.1%210

0.1%310O4100510061007100

1200.2%220

0,2%320

0.1%420

0.1%5200.1%62007200

130.2%0.4%

230.2%0.5%

330.2%0.4%

430.1%0.3%530

0.1%630

0.1%7300

140.8%1.6%24

1.1%2.2%

340.7%1.5%

440.4%0.7%54

0.1%0,3%

640

0.1%7400

153.1%6.2%25

6.9%14.0%35

3.1%6.3%45 -

0.7%1.5%55

0.2%0.4%650

0.1%7500

167.0%

14.1%26

28.2%56.4%

366.9%

14.0%46

1.1%2.2%56

0.2%0.5%66

0.1%0.2%760

0.1%

173.1%6.3%

277.0%14.1%

373.1%6.2%47

0.8%1.6%57

0.2%0.4%67

0.1%0.2%770

0.1%

Figure 3.38 Spotlighting effect from two-fold and four-fold increases in heater output at [2,6].Percentage represents local increase in heat absorption [14]

11

73%

21

85%

31

85%

41

85%

51

85%

61

83%

71

73%

12

83%

22

95%

32

97%

42

98%

52

97%

62

95%

72

83%

13

85%

23

97%

33

99%

43

99. 5%

53

99%

63

97%

73

85%

14

85%

24

98%

34

99. 5%

44

100%

54

QQ R/W, O/o

64

98%

74

85%

15

85%

25

97%

35

99%

45

99.5%

55

99%

65

97%

75

85%

16QQO/

26

95%

36

97%

46

98%

56

97%

66

95%

76

83%

17

73%

27

83%

37

85%

47

85%

57

85%

67

83%

77

73%

Figure 3.39 Effect of heater-to-sheet spacing on energy received by sheet elements. Local percentageof initial energy input for Z = 1.5 as given in Figure 3.35 for Z = I [14]

input profile of Fig. 3.35 [14]. As expected, energy input to edges and corners aremost affected. But the overall energy input to the sheet also substantially decreases.The energy output from each of the heaters must be changed to compensate for thechange in gap distance. Again, the arithmetic in Equation 3.40 is a most useful aidin this process.

3.11 Thin-Gage SheetApproximate Heating Rates

For thin-gage sheet, especially roll-fed film for packaging and blister-pack applica-tions, the time-dependent heating model can be significantly simplified. The netenthalpic change in the sheet is simply equated to the rate at which energy in thesheet is interchanged with its environment. As a first approximation, the temperaturegradient through the plastic film thickness is assumed to be zero. There are twogeneral approaches to this lumped-parameter approximationconstant environmen-tal temperature and constant heat flux to the sheet surface.

Constant Environmental Temperature Approximation

Consider T00 to be the constant environmental temperature. The lumped-parameterapproximation then becomes:

d(VH) = V pcp dT = !1A(T00 - T) d9 (3.44)V is the sheet volume, V = At, A is the sheet surface area and t is its thickness. T isthe sheet temperature. T00 can be the radiant heater temperature, with h being theapproximate radiation heat transfer coefficient. Or T00 can be air temperature with hbeing the convection heat transfer coefficient. This ordinary differential equation iswritten as:

cffM)"If t0 = T(G = 0), and T00 is constant:

ln(I^I) = Z^ (3.46)VT00 - T0; tpCp

or:

This is a first-order response of a system to a change in boundary conditions. Thislumped-parameter transient heat transfer model is valid only where conductionthrough the sheet thickness is less significant than energy transmission from theenvironment to the sheet surface. There are two dimensionless groups that define the

Table 3.13 Lumped-Parameter MaximumSheet Thickness

Moving air heat transfer coefficient, Table 3.2:0.0014cal/cm2-s-C1 Btu/ft2 h 0F

Plastics thermal conductivity, Table 3.12:4.1 to 8.3 x 10-4cal/cm- s 0C0.1 to 0.2 Btu/ft2 h F/ft

Maximum thickness for Bi = 0.1:0.025 to 0.5 cm0.010 to 0.100 in

10 to 100 mils

limits of the lumped-parameter model. One is the Biot number, Bi = ht/k, which isthe ratio of internal to external heat transfer. The second is the Fourier number,Fo = k9/pcpL2 = oc0/L2, where a is the thermal diffusivity, a = k/pcp, and L is thehalf-thickness of the sheet when heated equally from both sides1. The lumped-parameter model should be applied only when Bi < 0.1, or when the internalresistance is low. For air moving over plastic sheet, the sheet thickness should be lessthan about 0.010 in or 0.3 mm or so, Table 3.13 [19], but can be more than this forhigher thermal conductivity and higher air velocity. Figure 3.40 [20] expands thelimits of Table 3.13 by demonstrating the relative sensitivity of the sheet thickness tothe assumed temperature difference from the sheet surface to its centerline. Example3.21 explores the use of this figure in determining the appropriateness of thelumped-parameter model for convectively heating one side and both sides of athin-gage sheet. Practical heating times for various thin-gage polymers over a widerange in sheet thickness are given in Fig. 3.26 [21]. The linear relationship isapparent. The energy source temperature and the sheet temperature at forming timeis not given for these data. A radiant heater at T00 = 7600C or 14000F produces anenergy spectrum with a peak wavelength of about 2.8 um. The energy source outputat this temperature is 40 kW/m2. This energy input produces a near-linear heatingrate. A lower source temperature, T00 = 5100C or 9500F, does not produce a formingtime that is linear with sheet thickness.

1 Note throughout the discussion on sheet heating that the half-thickness of the sheet is used if the

sheet is heated equally on both sides. If the sheet is heated on only one side, as is the case withtrapped sheet heating, contact heating, or single-side radiant heating, and if the free surface canbe considered as insulated or without appreciable energy transfer to the surroundings, then theproper value for L is the total sheet thickness. If the sheet is unevenly heated on both sides orif one side of the sheet is heated in one fashion, such as contact heating and the other side isheated in another fashion, such as forced convection heating, then the proper value for L is thetotal sheet thickness. More importantly, models describing non-symmetric heating or one-sideheating with an insulated free surface cannot be applied. The proper model requires appropriateboundary conditions on each surface of the sheet.

Biot Number, Bi

Figure 3.40 Sensitivity of sheet thickness to temperature difference between sheet surface andcenterline [20]. Dimensionless time, Fourier number = a6/L2 and relative surface resistance, Biotnumber = hL/k

Example 3.21 The Limits on the Lumped-Parameter ModelA 0.020-in (0.5 cm) PET sheet is radiantly heated equally on both sides, from roomtemperature, 800F to its forming temperature, 3800F. The combined convection andradiation heat transfer coefficient is 10Btu/ft2 h 0^F. The thermal dijfusivity ofthe sheet is 0.002ft2/h and its thermal conductivity is 0.08Btujft h 0F. Deter-mine the heating time for a 1% difference in temperature between the sheet surfaceand center. Repeat for a 10% difference. What is the heating time for onesidedheating and a 1% or 10% temperature difference? Comment on the relative times.

For Fig. 3.40, values for Bi and Fo are required.

Fo - .6,L= - 0.002 . J L ) . J L ft-= = 0.8

From Fig. 3.40, Fo = 5.2 at 1% AT. Therefore 6 = 5.2/0.8 = 6.5s.From Fig. 3.40, Fo = 2.1 at 10% AT. Therefore 0 = 2.1/0.8 = 2.6s.In other words, to keep the centerline essentally at the surface tempera-

ture, the heating rate must be adjusted to achieve the forming temperaturein about 6.5 seconds. At the forming temperature of 3800F, the centerlinetemperature will be 0.99 (380 - 80) + 80 = 377F, or 3F below the surfacetemperature. If the heating rate is faster than this, the centerline temperature

Four

ier Nu

mbe

r, Fo

10% Difference

1% Difference

5% Difference

will lag the surface temperature by more than 1%. If the heating rate is suchthat the sheet reaches the forming temperature in about 2.6 seconds, thecenterline temperature will lag the surface temperature by about 10%.At the forming temperature of 3800F, the centerline temperature will be0.90 (280 - 80) + 80 - 3500F, or 300F below the surface temperature.

For one-sided heating, L = 0.020 in.

ft2 6 144Fo = aB/U = 0.002 _ _

(s) _ ft- = 0.29 (s)From Fig. 3.40, Fo = 4.05 at 1% AT. Therefore 0 = 4.05/0.2 = 20.25s.

From Fig. 3.40, Fo = 1.8 at 10% AT. Therefore 0 = 1.8/0.2 = 9.0s.It takes 20.25/6.5 = 3.1 times longer to heat the one-sided sheet to 1%

temperature difference and 3.5 times longer to heat it to 10% temperaturedifference.

Constant Heat Flux Approximation

If the heat flux to a thin sheet is constant, Q/A = constant, then:O dT

^ = constant = tpcp (3.48)Rearranging:

dT = ^ d 9 (3.49)Atpcp

Integrating this yields:T

-

T = x i ^ (3-50)

For a given set of processing conditions, the constant heat flux approximationindicates that the time to heat a very thin sheet of plastic to a given formingtemperature is proportional to the sheet thickness. The data of Fig. 3.26 indicate thislinearity, even though no values for forming temperature or heat flux are given.

Thin-Gage ApproximationsComments

The heating efficiencies for several polymers can be determined by using the normalforming temperatures from Tables 3.1 or 2.5. For a given polymer, the enthalpicchange between room temperature and the normal forming temperature is determinedfrom Fig. 2.17. The individual heating rate is determined from the slope of the curveof Fig. 3.26, for example. The net energy increase is then calculated. As seen in

Table 3.1, most thin-gage polymers absorb 40% to 60% of the energy supplied by theheating source. The relatively low efficiency of LDPE is unexplained. PP heatingefficiency is also reported to be low [22]. This indicates that the 7600C sourcetemperature used in the calculation may be improper for efficient heating of olefinmaterials. This is discussed below.

This analysis is restricted to one very specific processing areathin-gage poly-mersand to very stringent conditionslumped-parameter with linear approxima-tion of the logarithmic function. But it serves to illustrate that only a fraction of theenergy emitted by the source, about half in the cases examined, is actually taken upby the polymer sheet. The rest is lost to the environment or passes completelythrough the sheet unabsorbed.

3.12 Heavy-Gage SheetInternal Temperature Control

For thin-gage sheet and film, energy transmission to the sheet controls the heatingcycle time. Radiant heating is far more efficient than convection heating and so ispreferred for thin-gage thermoforming. For heavy-gage sheet however, energy ab-sorbed on the sheet surface must be conducted through the thermally insulatingplastic to its centerline1. For very thick sheets, the overall heating cycle time iscontrolled by the sheet centerline temperature and so the overall heating rate must becontrolled to prevent surface overheating. As with the thin-gage discussion earlier,there are two general cases to be consideredconstant environmental temperature,T00 = constant, and constant heat flux to the sheet surface, Q/A = constant.

Constant Environmental Temperature

Usually hot air is used as a heating medium for very heavy sheet. As a result, theT00 = constant case prevails. As with all transient heating problems, the centerlinetemperature lags the surface temperature. This is seen by reviewing the graphicalsolution to the one-dimensional time-dependent heat conduction equation with aconvection boundary condition (Figs. 3.41 and 3.42) [23,24]. Figure 3.41 gives theconditions at the sheet centerline. Figure 3.42 gives the equivalent conditions at thesheet surface. Similar figures for intermediate points throughout the thickness of thesheet are found in standard handbooks [25]. As with the thin-gage approximation,the dimensionless temperature dependency, Y, for heavy-gage sheet is a function oftwo dimensionless groups, the Biot number and the Fourier number:

1 Again, symmetric heating is assumed throughout this discussion. The general arithmetic de-

scribed herein must be modified if the sheet is heated in an unsymmetric fashion or if it is heatedonly on one side.

Next Page

Front MatterTable of Contents3. Heating the Sheet3.1 Introduction3.2 Energy Absorption by Sheet3.3 Heat Transfer Modes3.4 Incorporating Formability and Time-Dependent Heating3.5 Conduction3.6 Convection Heat Transfer CoefficientThe Biot NumberEffective Radiation Heat Transfer CoefficientConstant Heat Flux

3.7 Radiation HeatingBlack Body RadiationGray Body - EmissivityRadiant Heater Efficiency - Constant Heat Flux Application

3.8 Real Heaters - EfficienciesRadiative Heat Transfer CoefficientConvection and the Heat Transfer CoefficientRod Heaters

3.9 Long-Term Radiant Heater Efficiencies3.10 Edge Losses - View FactorLocal Energy InputPattern HeatingZone, Zoned or Zonal HeatingHeater to Sheet Distance

3.11 Thin-Gage Sheet - Approximate Heating RatesConstant Environmental Temperature ApproximationConstant Heat Flux ApproximationThin-Gage Approximations - Comments

3.12 Heavy-Gage Sheet - Internal Temperature ControlConstant Environmental TemperatureThe Constant Heat Flux CaseThe Thickness EffectSummary

3.13 EquilibrationConvection HeatingConstant Heat FluxComputed Equilibration TimesThe W-L-F EquationThe Arrhenius EquationRelating Shift Factors to Sheet Thickness

3.14 Infrared-Transparent Polymers3.15 Computer-Aided Prediction of Sheet TemperatureThe Radiant Boundary Condition

3.16 Guidelines for Determining Heating CyclesThe Biot NumberThin-Gage GuidelinesHeavy-Gage GuidelinesIntermediate-Gage Guidelines

3.17 References

Index