-
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