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Mini-Seminar Dr. James Throne, Instructor 8:00-8:50 - Technology of Sheet Heating 9:00-9:50 - Constitutive Equations Applied to Sheet Stretching 10:00-10:50 - Trimming as Mechanical 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-Seminar Advanced Topics in Thermoforming All materials contained herein are the intellectual property of Sherwood Technologies, Inc., copyright 1999-2006 No material may be copied or referred to in any manner without express written consent of the copyright holder All materials, written or oral, are the opinions of Sherwood Technologies, Inc., and James L. Throne, PhD Neither Sherwood Technologies, Inc. nor James L. Throne, PhD are compensated in any way by companies cited in materials presented herein Neither Sherwood Technologies, Inc., nor James L. Throne, PhD are to be held responsible for any misuse of these materials that result in injury or damage to persons or property

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Mini-Seminar Advanced Topics in Thermoforming This mini-seminar requires you to have a working engineering knowledge of heat transfer and stress-strain mechanics Dont attend if you cant handle theory and 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 downloading to 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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Part 1: Technology of Sheet Heating Fundamentals –Conduction –Convection –Radiation

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CONDUCTION mSolid-solid energy interchange mMoving thermal energy through sheet mMoving thermal energy, sheet to mold rIn very thin sheet, conduction is used to move thermal energy from heater to sheet through direct contact rImportant Parameters mThermal conductivity mHeat capacity or Enthalpy Part 1: Technology of Sheet Heating

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CONVECTION mFluid-solid energy interchange mSheet heating (or cooling) during sheet heating in oven mFree surface cooling in mold cavity rPrimary way of heating very thick sheet rImportant Parameters mConvective heat transfer coefficient mAir velocity, temperature mRecrystallization, recrystallization rate Part 1: Technology of Sheet Heating

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Convection heat transfer coefficient values Fluidh(Btu ft 2 hr o F) Quiescent air0.8 - 2 Air with fan2 - 5 Air with blower5 - 20 Air/water mist50 - 100 Fog50 - 100 Water spray50 - 150

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Part 1: Technology of Sheet Heating Dimensionless Groups - Definitions Biot numberBi = hL/k Fourier numberFo = /L 2 hconvective heat transfer coefficient Lhalf-thickness (heating on both sides) kthermal conductivity thermal diffusivity, k/ c p time density c p specific heat

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RADIATION mElectromagnetic energy interchange at a distance mMethod commonly used to heat all but very thin or very thick sheet rImportant parameters mHeater temperature mHeater and sheet emissivities mThin-gauge sheet transmissivity Sheet, heater geometry, spacing Part 1: Technology of Sheet Heating

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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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Part 1: Technology of Sheet Heating Dimensionless Groups - Definitions Radiation Biot numberBi r = h r L/k Radiation numberRad = LFF g /k h r radiative heat transfer coefficient Lhalf-thickness (heating on both sides) kthermal conductivity FView factor F g Emissivity factor Stefan-Boltzmann constant

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Part 1: Technology of Sheet Heating Radiation concepts - Explained Emissivity,, absorptivity,, considered similar Emissivity = 1, black body Emissivity between 0 and 1 and wavelength- independent, gray body Typical polymer emissivity, 0.90 < < 0.95 Typical polymer emissivity, absorptivity, Thin films semi-transparent,

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The Black Body Energy Curve, Showing Increasing Energy Output (logarithmic scale, left axis), Shifting of Peak Wavelength to Shorter Values With Increasing Heater Temperature Part 1: Technology of Sheet Heating

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So What is Wavelength? Visible Wavelength of light: 0.4 - 0.7 m Near-infrared:0.7 - 2.5 m Far-infrared (most TFing): 2.5 - 15 m (+) Part 1: Technology of Sheet Heating

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So What is Wavelength? Visible Wavelength of light: 0.4 - 0.7 m Near-infrared:0.7 - 2.5 m Far-infrared (most TFing): 2.5 - 15 m (+) Part 1: Technology of Sheet Heating

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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, units are microns [ m] For < 0.38 m, ultraviolet For 0.38 < < 0.7 m, visible For 0.7 < < 2.5 m, near infrared For 2.5 < < 100 m, far infrared Most thermoforming, between 3 and 15 m

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Part 1: Technology of Sheet Heating IR scans for thin-gage polymers Note key 3.5 m and 8 m wavelengths

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Part 1: Technology of Sheet Heating Wavelength- Dependent Transmission Through Thin Films of PS PE and PVC [Thank you, Ircon!]

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Part 1: Technology of Sheet Heating IR scans for thin-gage polymers

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rThick - or Heavy-Gauge Heating, Showing Temperature Profile Through Sheet Part 1: Technology of Sheet Heating

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rThin-Gauge Heating, Showing Radiant Energy Absorbing and Transmitting Through Flat Temperature Profile Sheet Part 1: Technology of Sheet Heating

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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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Part 1: Technology of Sheet Heating Reflectivity and transmissivity of a thick semi- transparent sheet of plastic Properties are determined through ray tracing What goes on inside the sheet

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Part 1: Technology of Sheet Heating Radiation is extremely complex This morning we consider only the simplest of radiant energy transfer, viz, diffuse radiation from gray body heaters to planar, radio- opaque, unpigmented, gray body, single layer, amorphous sheet If time, we look at a slightly more complex situation

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rEmissivity - What is it anyway? rIncoming radiation is either: mAbsorbed by the plastic mReflected from the plastic mTransmitted through the plastic Part 1: Technology of Sheet Heating

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An ideal material absorbs all incoming radiation ( =1) An ideal material emits all of its energy ( =1) rThat ideal material is called a black body rMost real materials have absorptivities, emissivities less than one [1] rPlastics, rusty, oxidized metals have emissivities of 0.9- 0.95 Part 1: Technology of Sheet Heating

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Conduction Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent) Initial sheet temperature independent of sheet thickness 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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Part 1: Technology of Sheet Heating Conduction Heat Transfer - Equation T(x, ) is instant temperature, is time, is thermal diffusivity, x is distance into sheet BC1:T(x,0) = T i BC2:T(0, ) = T o BC3:(symmetry about the centerline of the sheet) or

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Part 1: Technology of Sheet Heating Conduction Heat Transfer - Dimensionless Equation Y = (T-To)/(Ti-To), Fo = /L 2, = x/L

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Part 1: Technology of Sheet Heating Convection Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent) Initial sheet temperature independent of sheet thickness Surface temperature dependent on thermal gradient, convection heat transfer coefficient Environmental temperature independent of sheet thickness Thermal properties independent of temperature Convection condition same on both sides of sheet

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Part 1: Technology of Sheet Heating Convection Heat Transfer - Equation T(x, ) is instant temperature, is time, is thermal diffusivity, x is distance into sheet BC1;T(x,0) = T i BC2: BC3:(symmetry about the centerline of the sheet)

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Part 1: Technology of Sheet Heating Convection Heat Transfer - Dimensionless Equations Y = (T-Ta)/(Ti-Ta), Fo = /L 2, x = x/L, Bi=hL/k

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Part 1: Technology of Sheet Heating Convection Heat Transfer - Graph Surface Centerline

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Assumptions One-dimensional (thickness) Transient (time-dependent) Initial sheet temperature independent of sheet thickness Surface temperature dependent on radiant heat flux, fourth-power of temperature differences

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Part 1: Technology of Sheet Heating 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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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation T(x, ) is instant temperature, is time, is thermal diffusivity, x is distance into sheet BC1: T(x,0) = T i BC2: BC3: (symmetry about the centerline of the sheet)

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation What is F g ? F g is gray body correction factor for non-black body radiation If sheet, heater emissivities ( s, h ) are <1 and independent of wavelength, then If h = s = 0.9, F g =0.82

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation What is F? F is view factor, a geometric parameter F is the measure of average fraction of energy transferred from the heater to the sheet surface In other words, what you see is what you heat

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Part 1: Technology of Sheet Heating What do heaters heat? 3All visible surfaces 3Plastic sheet 3Heater reflectors 3Other heaters (esp. with transparent sheet) 3Pin-chain rails 3Heater guards 3Objects outside the oven edges 3Oven sidewalls, shields, baffles Sag 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/Z 2 Oven efficiency depends on minimizing energy transfer to non-sheet - rails, oven walls, etc. Part 1: Technology of Sheet Heating

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Radiant Heat Transfer - Equation What is F? Often the view factor is given as a number (0<F<1) This is incorrect We return to an examination of view factor shortly

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Part 1: Technology of Sheet Heating Radiative Heat Transfer - Dimensionless Equations Y = (T-T h )/(T i -T h ), Fo = /L 2, = x/L, Bi r =h r L/k h r = radiative heat transfer coefficient : :

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Part 1: Technology of Sheet Heating Combined Conduction, Convection, Radiation Equation with Boundary Conditions : : Keep in mind that these equations assume temperature-independent physical properties and equal boundary conditions on both sides of the sheet

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Part 1: Technology of Sheet Heating Combined Conduction, Convection, Radiation Equation with Boundary Conditions Parabolic equation with nonlinear boundary conditions Two methods of solution Finite Element Analysis [FEA] Finite Difference Equations [FDE]

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Part 1: Technology of Sheet Heating Combined Conduction, Convection, Radiation Finite Difference Equation with Boundary Conditions where 0 and N represent the top and bottom surfaces of the sheet and where...

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Part 1: Technology of Sheet Heating Combined Conduction, Convection, Radiation Finite Difference Equation with Boundary Conditions where Average temp: F 12 is the average view factor, s is the sheet emissivity, h is the heater emissivity

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Part 1: Technology of Sheet Heating Combined Conduction, Convection, Radiation Finite Difference Equation with Boundary Conditions This equation is easily solved using Fortran or Q-Basic The calculated time-dependent sheet temperature is based on an average value for the energy transferred between the heater and the surface, viz, a fixed value for the view factor, F This result may not apply to many practical cases

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Part 1: Technology of Sheet Heating 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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Part 1: Technology of Sheet Heating The Energy Dome Concept When a sheet of finite dimensions is heated uniformly with a heater of similarly finite dimensions The center of the sheet is hotter than the edges The edges of the sheet are hotter than the corners

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? The use of a constant value for F, the view factor

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? Consider the energy interchange between differential elements on the sheet and on the heater Here, direction cosines and the solid angle radius, r, are defined

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? The energy interchange between a heater element dA 1 and a sheet surface element dA 2 is given as where cos i are the direction cosines defined earlier, and r is the distance between the two elements

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? The equation assumes all heater elements have the same temperature and all sheet surface elements have the same temperature The double integral correctly represents the view factor integrated over all heater elements and all sheet surface elements

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? Now replace the differential elements with discrete elements, A 1 and A 2

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? The terms cos are direction cosines, and r is the solid angle radius between each pair of the difference elements, as defined earlier For each of the planar heater and sheet elements, the direction cosines are cos 1 = cos 2 = z/r z is the line distance between the two elements

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? For each of the planar heater and sheet elements, r, the spherical radius, is given as:

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? The direction cosines and the term for r are now 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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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? For the energy transfer from one heater element to all sheet elements For the energy transfer from all heater elements to one sheet element

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? These are the new radiation boundary condition equations They are now combined with the traditional convection boundary conditions And the finite difference equation is solved for each heater element and each sheet element

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? As an example, a 7 x 7 matrix heater interchanges 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 iteration is necessary

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Part 1: Technology of Sheet Heating Radiant Heat Transfer - Equation Where is the problem? If all 49 heater elements have the same temperature, the result is the energy dome, as predicted

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Part 1: Technology of Sheet Heating The net % energy received by each element when heater temps are equal everywhere

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Part 1: Technology of Sheet Heating

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Radiant Heat Transfer - Equation Where is the problem? If the temperatures of the heater elements are now changed by trial and error so that the entire sheet heats at the same rate, the energy dome is flattened

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Part 1: Technology of Sheet Heating The % values represent the local heater flux output change from the original 100%

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Part 1: Technology of Sheet Heating

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

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 1.Most models assume planar sheet (sheet sags during heating. Does this affect local heating? Art Buckel says no. Is he right?) [Note: cosine geometry gets brutal. The 2D model has been solved using catenary equations to describe sheet surface. The 3D model using parabolic hyperboloid equations to describe sheet surface has not. Read 4Q06 Tech Article for additional information.]

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 2.Models assume wavelength-independent (and therefore, temperature-independent) sheet and heater emissivities 3.Model designed for cut sheet. What about roll- fed sheet? (Roll-fed sheet clamped on two sides, endless, multiple shots in oven)

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 4.Convection boundary condition assumes constant air temperature, constant convection heat transfer coefficient (in thin-gauge, sheet start-stop during transit through oven) 5.Model designed for opaque sheet (Semi- transparent sheet is heated by volumetric energy absorption)

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 6.Model does not include role of pigment, filler in sheet heating (Solid particles change thermal properties, absorption characteristics) 7.Model assumes amorphous polymers, so latent heat of fusion, phase boundary are not included 8.For combustion radiant heaters, the role of the absorption and reradiation of combustion gases (H 2 O, CO 2 ) needs to be included

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 9.Diffuse absorption, reflection, reradiation from oven walls, etc., need to be included 10. Most models assume uniform radiant heating, air environment on both sides of sheet (Radiant heaters are usually at different temperatures, air is trapped against underside of sheet)

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Part 1: Technology of Sheet Heating Problem Solved! No? What Assumptions Need to be Relaxed? 11. Model assumes monolayer. Multilayer laminates involve interfacial reflection and localized absorption, as well as interfacial conduction

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Number 5 Radio-opacity v. volumetric absorption or diathermanousity Beers law: Wavelength-dependent absorption is exponentially dependent on depth into the plastic Primary assumptions: Sheet remains planar; only diffuse absorption, reflection

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Part 1: Technology of Sheet Heating Reflectivity and transmissivity of a thick semi- transparent sheet of plastic Properties are determined through ray tracing Okay, give us an example of how just one of these assumptions can be relaxed!

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Transmission is given as Where 2 is the Beers law absorption coefficient, 0 is the wavelength of the incident energy and d/cos 2 is the distance the radiant beam travels through the plastic in one pass

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Reflectivity at the interface of the sheet surface is usually given in terms of the relative indices of refraction at the interface

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Reflectivity from the sheet is given as Where is the reflectivity at both inner and outer sheet surfaces and is the transmissivity of a single pass through the sheet. Note: these values are wavelength-dependent!

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Transmissivity through the sheet is given as

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Absorptivity within the sheet is given as

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! The general heat conduction equation now includes a term, q, for volumetric energy absorption:

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Part 1: Technology of Sheet Heating Okay, give us an example of how just one of these assumptions can be relaxed! Has this problem been solved? Yes, analytically, in the 1970s [Progelhof, Quintiere and Throne] for coextruded clear PMMA-pigmented ABS. The general effect of increasing volumetric absorption is a flattening of the time- dependent temperature profile. Viz, the instant surface temperature does not increase as rapidly as with radio-opaque sheet while the instant internal temperatures increase more rapidly…

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Part 1: Technology of Sheet Heating End of Part 1 Technology of Sheet Heating

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Part 1: Technology of Sheet Heating Part 2 Constitutive Equations Applied to Sheet Stretching Begins promptly at 9:00!

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