Heating Seminar 09 06

8/10/2019 Heating Seminar 09 06

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Mini-Seminar

Dr. James Throne, Instructor

8:00-8:50 - Technology of SheetHeating

9:00-9:50 - Constitutive EquationsApplied to Sheet Stretching

10:00-10:50 - Trimming asMechanical Fracture


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Mini-Seminar

Advanced Topics in Thermoforming

Part 1: 8:00-8:50

Technology of Sheet Heating


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Lets

begin!


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Mini-SeminarAdvanced Topics in Thermoforming

All materials contained herein are the intellectualproperty of Sherwood Technologies, Inc., copyright1999-2006

No material may be copied or referred to in anymanner without express written consent of thecopyright holder

All materials, written or oral, are the opinions ofSherwood Technologies, Inc., and James L. Throne,PhD

Neither Sherwood Technologies, Inc. nor James L.Throne, PhD are compensated in any way bycompanies cited in materials presented herein

Neither Sherwood Technologies, Inc., nor James L.Throne, PhD are to be held responsible for anymisuse of these materials that result in injury ordamage to persons or property


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Mini-SeminarAdvanced Topics in Thermoforming

This mini-seminar requires you to have aworking engineering knowledge of heattransfer and stress-strain mechanics

Dont attend if you cant handle theoryand equations

Each mini-seminar will last 50 minutes,followed by a 10- minute bio break

Please turn off cell phones PowerPoint presentations are available at

the end of the seminar for downloadingto your memory stick


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Part 1: Technology of Sheet Heating

Outline Fundamentals

Dimensionless Groups - Definitions Radiation Explained Arithmetic Energy Dome Radiant Heat Transfer Equation Ditto - Where is the Problem? Problem Solved! No?


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Fundamentals Conduction

Convection Radiation


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CONDUCTION Solid-solid energy interchangeMoving thermal energy through sheetMoving thermal energy, sheet to mold

In very thin sheet, conduction is used to movethermal energy from heater to sheet throughdirect contactImportant Parameters

Thermal conductivityHeat capacity or Enthalpy



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CONVECTION Fluid-solid energy interchangeSheet heating (or cooling) during sheetheating in oven

Free surface cooling in mold cavityPrimary way of heating very thick sheetImportant Parameters

Convective heat transfer coefficientAir velocity, temperatureRecrystallization, recrystallization rate



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Convection heat transfer coefficient values

Fluid h(Btu ft2 hr

oF)

Quiescent air 0.8 - 2Air with fan 2 - 5Air with blower 5 - 20Air/water mist 50 - 100Fog 50 - 100Water spray 50 - 150


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Dimensionless Groups - DefinitionsBiot number Bi = hL/kFourier number Fo = aq /L2

h convective heat transfer coefficientL half-thickness (heating on both sides)k thermal conductivitya thermal diffusivity, k/ r cpq timer densitycp specific heat


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RADIATION Electromagnetic energy interchange at adistanceMethod commonly used to heat all but verythin or very thick sheet

Important parametersHeater temperatureHeater and sheet emissivitiesThin-gauge sheet transmissivitySheet, heater geometry, spacing



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Radiation Fundamentals Black body, gray body, real body Emissivity and absorptivity

Energy interchange Wavelength-dependent energy transmission in

the far-infrared region Semi-transparency in thin-gauge sheet


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Dimensionless Groups - DefinitionsRadiation Biot number Bi r = h rL/kRadiation number Rad = LFF gs /k

hr radiative heat transfer coefficientL half-thickness (heating on both sides)k thermal conductivityF View factorFg Emissivity factors Stefan-Boltzmann constant


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Radiation concepts - Explained Emissivity, e, absorptivity, a , considered

similar Emissivity = 1, black body Emissivity between 0 and 1 and wavelength-

independent, gray body

Typical polymer emissivity, 0.90 < e < 0.95 Typical polymer emissivity, e=e l),

absorptivity, a=a l) Thin films semi-transparent, t = t l)


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The Black Body EnergyCurve, ShowingIncreasing EnergyOutput (logarithmic

scale, left axis),Shifting of PeakWavelength toShorter Values WithIncreasing HeaterTemperature



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So What is Wavelength?

Visible Wavelength of light: 0.4 - 0.7 mmNear-infrared: 0.7 - 2.5 mmFar-infrared (most TFing): 2.5 - 15 mm (+)



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So What is Wavelength?

Visible Wavelength of light: 0.4 - 0.7 mmNear-infrared: 0.7 - 2.5 mmFar-infrared (most TFing): 2.5 - 15 mm (+)



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Radiation concepts - Explained Energy interchange between heater [source]

and sheet [sink]

What you see is what you heat! Wavelength symbol is l , units are microns

[mm] For l < 0.38 mm, ultraviolet

For 0.38 < l < 0.7 mm, visible For 0.7 < l < 2.5 mm, near infrared For 2.5 < l < 100 mm, far infrared Most thermoforming, l between 3 and 15 mm


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IR scans for thin-gage polymers

Note key 3.5 mm and 8 mm wavelengths


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Wavelength-DependentTransmissionThrough ThinFilms ofPS

PE

and PVC

[Thank you, Ircon!]


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IR scans for thin-gage polymers


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Thick - or Heavy-Gauge Heating, ShowingTemperature Profile Through Sheet



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Thin-Gauge Heating ,Showing RadiantEnergy Absorbing andTransmitting ThroughFlat TemperatureProfile Sheet



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Radiation is extremely complex Diffuse v. specular surface (textured v.

polished sheet) Planar v. curved surface (effect of sag) Effect of pigment (differential energy

absorption at surface)

Radio-opaque v. volumetric absorbing Internal reflection v. transmission Reflection, absorption at multilayer interface


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Radiation is extremely complex This morning we consider only the simplest of

radiant energy transfer, viz, diffuse radiationfrom gray body heaters to planar, radio-opaque, unpigmented, gray body, single layer,amorphous sheet

If time, we look at a slightly more complexsituation


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Emissivity - What isit anyway? Incoming radiation iseither:

Absorbed by theplasticReflected from theplasticTransmittedthrough the plastic



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An ideal material

absorbs all incomingradiation ( a =1)An ideal material emitsall of its energy ( e=1)

That ideal material iscalled a black bodyMost real materialshave absorptivities,emissivities less than

one [1]Plastics, rusty,oxidized metals haveemissivities of 0.9-

0.95



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Conduction Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent)

Initial sheet temperature independent of sheetthickness Step change in surface temperature Surface temperature independent of sheet

thickness Thermal properties independent of temperature Surface temperature same on both sides of

sheet


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Conduction Heat Transfer - Equation

T(x, q) is instant temperature, q is time, a is thermaldiffusivity, x is distance into sheet

BC1: T(x,0) = T i BC2: T(0, q) = T oBC3: (symmetry about the centerline

of the sheet)

2

2

xT T =a q or

0== L x x

T k

= xT k

xT c p q

r


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Conduction Heat Transfer -Dimensionless Equation

Y = (T-To)/(Ti-To),Fo = aq /L2, x = x/L

2

2

x = Y

FoY


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Convection Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent)

Initial sheet temperature independent of sheetthickness Surface temperature dependent on thermal

gradient, convection heat transfer coefficient

Environmental temperature independent of sheetthickness Thermal properties independent of temperature Convection condition same on both sides of

sheet


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Convection Heat Transfer - Equation

T(x, q) is instant temperature, q is time, a is thermal diffusivity, xis distance into sheet

BC1; T(x,0) = T i

BC2:

BC3: (symmetry about the centerlineof the sheet)

2

2

x

T T =a q

)(0 a x T T h xT

k ==

0== L x x

T k


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Convection Heat Transfer -Dimensionless Equations

Y = (T-Ta)/(Ti-Ta),Fo = aq /L2, x = x/L, Bi=hL/k

2

2

x = Y

FoY

Y BiY

==0x x

01

==x x

Y


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Convection Heat Transfer - Graph

Surface

Centerline


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Radiant Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent)

Initial sheet temperature independent of sheetthickness Surface temperature dependent on radiant

heat flux, fourth-power of temperature

differences


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Radiant Heat Transfer - Assumptions

Geometric factors independent of sheet, heater

shapes Radiant properties independent of temperature Diffuse surface absorption only (radio-opaque) Radiant boundary conditions same on both sides

of sheet


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Radiant Heat Transfer - Equation

T(x, q) is instant temperature, q is time, a isthermal diffusivity, x is distance into sheet

BC1: T(x,0) = T i

BC2:

BC3: (symmetry about the centerlineof the sheet)

2

2

xT T =a

q

0== L x x

T k

( )4*4*

0 sh g

xT T FF x

T == s


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Radiant Heat Transfer - EquationWhat is F g?

Fg is gray body correction factor for non-blackbody radiationIf sheet, heater emissivities ( es, eh) are


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Radiant Heat Transfer - Equation What is F?

F is view factor, a geometric parameterF is the measure of average fraction of energy

transferred from the heater to the sheetsurface

In other words, what you see is what you heat


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What do heaters heat?All visible surfacesPlastic sheetHeater reflectorsOther heaters (esp. with transparent sheet)Pin-chain rails

Heater guardsObjects outside the oven edgesOven sidewalls, shields, bafflesSag bands and other sheet supports


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The View Factor Heaters heat what they see! Heating efficiency - From a given heater

decreases in proportion to the square of the

distance to the sheet, E prop.to 1/Z2

Oven efficiency depends on minimizing energytransfer to non-sheet - rails, oven walls, etc.



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Radiant Heat Transfer - EquationWhat is F?

Often the view factor is given as a number(0


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Radiative Heat Transfer -Dimensionless Equations

Y = (T-T h)/(T i-T h), Fo = aq /L2, x = x/L, Bi r=hrL/khr = radiative heat transfer coefficient

01

==x x

Y 2

2

x = Y

Fo

Y :

:

( )( )2*2*** sh sh g r T T T T FF h = s

Y BiY

r ==0x x


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Combined Conduction, Convection, RadiationEquation with Boundary Conditions

2

2

x = Y

FoY

01

==x x

Y Y Bi BiY

r =

=][

0x x : :

Keep in mind that these equations assumetemperature-independent physicalproperties and equal boundary conditionson both sides of the sheet


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Combined Conduction, Convection, RadiationEquation with Boundary Conditions

Parabolic equation with nonlinear boundaryconditions Two methods of solution

Finite Element Analysis [FEA]

Finite Difference Equations [FDE]


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Combined Conduction, Convection, RadiationFinite Difference Equation with Boundary

Conditions

where 0 and N represent the top and bottomsurfaces of the sheet and where...

( ) iiii T FoT T FoT )1(11' = ( ) ( ) ( )4*04*0,0000,010'0 222 T T FoRad T BiT FoT T FoT T ha =

( ) ( ) 4*4*,,1' 222 N N h N N N N a N N N N T T FoRad T BiT FoT T FoT T =


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Conditions

k xFF Rad i g i /, s =where

= =

1

1

0

21 N

ii

N avg T

T T N

T Average temp:

=

ih

ih

i s

i si g F

FF ,

,

12,

,,

111/1

e

e

e

e F12 is the average viewfactor, es is the sheetemissivity, e

h is the heater

emissivity


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Conditions

This equation is easily solved using Fortran orQ-Basic The calculated time-dependent sheet

temperature is based on an average value for

the energy transferred between the heater andthe surface, viz, a fixed value for the viewfactor, F

This result may not apply to many practicalcases


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The Energy Dome Cannot be predicted using average value for

view factor, F Needs first principles in radiation

[What is the Energy Dome?]

[More importantly, how do you calculate it?]


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The Energy DomeConcept When a sheet of finite

dimensions is heateduniformly with a heater

of similarly finitedimensions The center of the sheet

is hotter than the edges The edges of the sheet

are hotter than thecorners


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Radiant Heat Transfer - EquationWhere is the problem?

The use of a constant value for F, the view factor


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Consider the energyinterchange betweendifferential elementson the sheet and onthe heater

Here, directioncosines and the solidangle radius, r, aredefined


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The energy interchange between a heater elementdA1 and a sheet surface element dA 2 is given as

where cos f i are the direction cosines definedearlier, and r is the distance between the twoelements

( ) =2 1

212214*4*

21coscos

A A sh g dAdA

r T T F q

f f

s


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The equation assumes all heater elements havethe same temperature and all sheet surfaceelements have the same temperature

The double integral correctly represents theview factor integrated over all heater elementsand all sheet surface elements


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Now replace the differential elements withdiscrete elements, A1 and A2

( ) = 212 214*4*21 coscos21

A Ar

T T F q A A

sh g

f f s


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For each of the planar heater and sheetelements, r, the spherical radius, is given as:

222 z y xr =


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The direction cosines and the term for r arenow combined in the double summation equation. When the local sheet and heater temperatures

are also moved inside the double summation, the

following equations obtain...


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For the energy transfer from one heaterelement to all sheet elements

For the energy transfer from all heaterelements to one sheet element

( ) ( )=

2

24*4*

2222

2

121 A

sh g AT T z y x

z A F q

s

( ) ( )= 14*4*2222

2

2211

AT T z y x

z A F q sh

A g

s


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These are the new radiation boundary conditionequations They are now combined with the traditional

convection boundary conditions

And the finite difference equation is solved foreach heater element and each sheet element


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As an example, a 7 x 7 matrix heaterinterchanges energy with a 7 x 7 matrix sheet The FDE is solved 49 x 49 times for each time

step

Because the equation is parabolic, no iterationis necessary


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If all 49 heater elements have the sametemperature, the result is the energy dome, aspredicted


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Part 1: Technology ofSheet Heating

The net %energyreceived byeach elementwhen heatertemps areequaleverywhere


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If the temperatures of the heater elements arenow changed by trial and error so that theentire sheet heats at the same rate, theenergy dome is flattened


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The % valuesrepresent thelocal heaterflux outputchange fromthe original100%


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Problem Solved!

No?

What Assumptions Need to beRelaxed?


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Problem Solved! No?What Assumptions Need to be Relaxed?

1. Most models assume planar sheet (sheet sagsduring heating. Does this affect local heating?Art Buckel says no. Is he right?)

[Note: cosine geometry gets brutal. The 2D modelhas been solved using catenary equations to

describe sheet surface. The 3D model usingparabolic hyperboloid equations to describesheet surface has not. Read 4Q06 TechArticle for additional information.]


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2. Models assume wavelength-independent (andtherefore, temperature-independent) sheet andheater emissivities

3. Model designed for cut sheet. What about roll-fed sheet? (Roll-fed sheet clamped on twosides, endless, multiple shots in oven)


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4. Convection boundary condition assumes constantair temperature, constant convection heattransfer coefficient (in thin-gauge, sheetstart-stop during transit through oven)

5. Model designed for opaque sheet (Semi-transparent sheet is heated by volumetricenergy absorption)


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6. Model does not include role of pigment, fillerin sheet heating (Solid particles change thermalproperties, absorption characteristics)

7. Model assumes amorphous polymers, so latentheat of fusion, phase boundary are not

included8. For combustion radiant heaters, the role of

the absorption and reradiation of combustiongases (H 2O, CO 2) needs to be included


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9. Diffuse absorption, reflection, reradiation fromoven walls, etc., need to be included

10. Most models assume uniform radiant heating,

air environment on both sides of sheet (Radiantheaters are usually at different temperatures,air is trapped against underside of sheet)


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11. Model assumes monolayer. Multilayerlaminates involve interfacial reflection andlocalized absorption, as well as interfacialconduction


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Okay, give us an example of how just oneof these assumptions can be relaxed!

Number 5Radio-opacity v. volumetric absorption or

diathermanousity Beers law: Wavelength -dependent absorption is

exponentially dependent on depth into the

plasticPrimary assumptions: Sheet remains planar; onlydiffuse absorption, reflection


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Reflectivity andtransmissivity of athick semi-transparent sheetof plastic

Properties aredetermined throughray tracing



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Transmission is given as

Where k2 is the Beers law absorption coefficient,l 0 is the wavelength of the incident energy andd/cos q2 is the distance the radiant beamtravels through the plastic in one pass

20222 cos/4cos/ q l q k t d k d ee ==


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Reflectivity at the interface of the sheet surfaceis usually given in terms of the relative indices

of refraction at the interface

2

2

21

2112

1

1

=

=

plastic

plastic plasticair n

nnn

nn

r

r


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Reflectivity from the sheet is given as

Where r is the reflectivity at both inner andouter sheet surfaces and t is the transmissivityof a single pass through the sheet. Note: thesevalues are wavelength-dependent!

( )=22

22

11

1t r t r

r sheet R


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Transmissivity through the sheet is given as

( )22

2

1

1

t r

t r =

sheet


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Absorptivity within the sheet is given as

( )( ) rt

t r =1

11 sheet

A


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The general heat conduction equation now includesa term, q, for volumetric energy absorption:

( ) xT q xT

xT c p ,l q

r

=


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Has this problem been solved? Yes, analytically, inthe 1970s [Progelhof, Quintiere and Throne]for coextruded clear PMMA-pigmented ABS.The general effect of increasing volumetricabsorption is a flattening of the time-dependent temperature profile.

Viz, the instant surface temperature does notincrease as rapidly as with radio-opaque sheetwhile the instant internal temperatures increasemore rapidly


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End of

Part 1

Technology of Sheet Heating


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Part 2

Constitutive Equations Applied to SheetStretching

Begins promptly at 9:00!

Documents

Heating Seminar 09 06