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
Mini-Seminar
Advanced Topics in Thermoforming
Part 1: 8:00-8:50
Technology of Sheet Heating
Let’s
begin!
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
Mini-Seminar
Advanced Topics in Thermoforming
• This mini-seminar requires you to have a
working engineering knowledge of heat
transfer and stress-strain mechanics
• Don’t attend if you can’t 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
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?
Part 1: Technology of Sheet Heating
Fundamentals
– Conduction
– Convection
– Radiation
Part 1: Technology of Sheet Heating
CONDUCTION
Solid-solid energy interchange
Moving thermal energy through sheet
Moving thermal energy, sheet to mold
 In very thin sheet, conduction is used to move
thermal energy from heater to sheet through
direct contact
 Important Parameters
Thermal conductivity
Heat capacity or Enthalpy
Part 1: Technology of Sheet Heating
CONVECTION
Fluid-solid energy interchange
Sheet heating (or cooling) during sheet
heating in oven
Free surface cooling in mold cavity
 Primary way of heating very thick sheet
 Important Parameters
Convective heat transfer coefficient
Air velocity, temperature
Recrystallization, recrystallization rate
Part 1: Technology of Sheet Heating
Convection heat transfer coefficient values
Fluid
Quiescent air
Air with fan
Air with blower
Air/water mist
Fog
Water spray
h(Btu ft2 hr oF)
0.8 - 2
2 - 5
5 - 20
50 - 100
50 - 100
50 - 150
Part 1: Technology of Sheet Heating
Dimensionless Groups - Definitions
Biot number
Bi = hL/k
Fourier number
Fo = aq/L2
h
convective heat transfer coefficient
L
half-thickness (heating on both sides)
k
thermal conductivity
a
thermal diffusivity, k/rcp
q
time
r
density
cp
specific heat
Part 1: Technology of Sheet Heating
RADIATION
Electromagnetic energy interchange at a
distance
Method commonly used to heat all but very
thin or very thick sheet
 Important parameters
Heater temperature
Heater and sheet emissivities
Thin-gauge sheet transmissivity
Sheet, heater geometry, spacing
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
Dimensionless Groups - Definitions
Radiation Biot number Bir = hrL/k
Radiation number
Rad = LFFgs/k
hr
radiative heat transfer coefficient
L
half-thickness (heating on both sides)
k
thermal conductivity
F
View factor
Fg
Emissivity factor
s
Stefan-Boltzmann constant
Part 1: Technology of Sheet Heating
Radiation concepts - Explained
• Emissivity, e, absorptivity, a, considered
similar
• Emissivity = 1, black body
• Emissivity between 0 and 1 and wavelengthindependent, 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)
Part 1: Technology of Sheet Heating
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
So What is Wavelength?
Visible Wavelength of light:
0.4 - 0.7 mm
Near-infrared:
0.7 - 2.5 mm
Far-infrared (most TFing): 2.5 - 15 mm (+)
Part 1: Technology of Sheet Heating
So What is Wavelength?
Visible Wavelength of light:
0.4 - 0.7 mm
Near-infrared:
0.7 - 2.5 mm
Far-infrared (most TFing): 2.5 - 15 mm (+)
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
IR scans for thin-gage polymers
Note key 3.5 mm and 8 mm wavelengths
Part 1: Technology of Sheet Heating
WavelengthDependent
Transmission
Through Thin
Films of
PS
PE
and PVC
[Thank you, Ircon!]
Part 1: Technology of Sheet Heating
IR scans for thin-gage polymers
Part 1: Technology of Sheet Heating
 Thick - or Heavy-Gauge Heating, Showing
Temperature Profile Through Sheet
Part 1: Technology of Sheet Heating
 Thin-Gauge Heating,
Showing Radiant
Energy Absorbing and
Transmitting Through
Flat Temperature
Profile Sheet
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
What goes on inside the sheet
• Reflectivity and
transmissivity of a
thick semitransparent sheet
of plastic
• Properties are
determined through
ray tracing
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, radioopaque, unpigmented, gray body, single layer,
amorphous sheet
• If time, we look at a slightly more complex
situation
Part 1: Technology of Sheet Heating
 Emissivity - What is
it anyway?
 Incoming radiation is
either:
Absorbed by the
plastic
Reflected from the
plastic
Transmitted
through the plastic
Part 1: Technology of Sheet Heating
 An ideal material
absorbs all incoming
radiation (a=1)
 An ideal material emits
all of its energy (e=1)
 That ideal material is
called a black body
 Most real materials
have absorptivities,
emissivities less than
one [1]
 Plastics, rusty,
oxidized metals have
emissivities of 0.90.95
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
Conduction Heat Transfer - Equation
T   T 
cp r
= k

q x  x 
or
T
 2T
=a 2
q
x
T(x,q) is instant temperature, q is time, a is thermal
diffusivity, x is distance into sheet
BC1: T(x,0) = Ti
BC2: T(0,q) = To
BC3:
(symmetry about the centerline
T
k
= 0 of the sheet)
x
x =L
Part 1: Technology of Sheet Heating
Conduction Heat Transfer Dimensionless Equation
Y  2Y
= 2
Fo x
Y = (T-To)/(Ti-To),
Fo = aq/L2, x = x/L
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
Part 1: Technology of Sheet Heating
Convection Heat Transfer - Equation
T
 2T
=a 2
q
x
T(x,q) is instant temperature, q is time, a is thermal diffusivity, x
is distance into sheet
BC1;
T(x,0) = Ti
BC2:
BC3:
k
T
= h(T  Ta )
x x =0
T
k
=0
x x =L
(symmetry about the centerline
of the sheet)
Part 1: Technology of Sheet Heating
Convection Heat Transfer Dimensionless Equations
Y  2Y Y = 0
= 2
Fo x x x =1
Y
= Bi  Y
x x =0
Y = (T-Ta)/(Ti-Ta),
Fo = aq/L2, x = x/L, Bi=hL/k
Part 1: Technology of Sheet Heating
Convection Heat Transfer - Graph
Centerline
Surface
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
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
Part 1: Technology of Sheet Heating
Radiant Heat Transfer - Equation
T
 2T
=a 2
q
x
T(x,q) is instant temperature, q is time, a is
thermal diffusivity, x is distance into sheet
BC1: T(x,0) = Ti
(
)
* 4
* 4
BC2:  T
= sFFg T h  T s
x x =0
BC3:
(symmetry about the centerline
T
k
= 0 of the sheet)
x x =L
Part 1: Technology of Sheet Heating
Radiant Heat Transfer - Equation
What is Fg?
Fg is gray body correction factor for non-black
body radiation
If sheet, heater emissivities (es, eh) are <1 and
independent of wavelength, then
1 1 
Fg = 1 /    1
e h e s 
If eh = es = 0.9, Fg =0.82
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”
Part 1: Technology of Sheet Heating
What do heaters heat?
 All visible surfaces
 Plastic sheet
 Heater reflectors
 Other heaters (esp. with transparent sheet)
 Pin-chain rails
 Heater guards
 Objects outside the oven edges
 Oven sidewalls, shields, baffles
 Sag bands and other sheet supports
Part 1: Technology of Sheet Heating
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 energy
transfer to non-sheet - rails, oven walls, etc.
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
Radiative Heat Transfer Dimensionless Equations
Y
=0
x x =1
Y  2Y
= 2 :
Fo x
:
Y
= Bir  Y
x x =0
Y = (T-Th)/(Ti-Th), Fo = aq/L2, x = x/L, Bir=hrL/k
hr = radiative heat transfer coefficient
(
)(
hr = sFFg Th*  Ts* Th*2  Ts*2
)
Part 1: Technology of Sheet Heating
Combined Conduction, Convection, Radiation
Equation with Boundary Conditions
Y
Y  2Y
Y
= [ Bi  Bir ]  Y
= 2
=0 :
x x =0
Fo x : x x =1
Keep in mind that these equations assume
temperature-independent physical
properties and equal boundary conditions
on both sides of the sheet
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]
Part 1: Technology of Sheet Heating
Combined Conduction, Convection, Radiation
Finite Difference Equation with Boundary
Conditions
Ti ' = Fo(Ti 1  Ti 1 )  (1  Fo)Ti
(
)
(
)
T0' = T0  2Fo(T1  T0 )  2Fo(Ta,0  Bi0T0 )  2FoRad0 Th*,40  T0*4
TN' = TN  2Fo(TN 1  TN )  2Fo(Ta, N  BiNTN )  2FoRadN Th*,4N  TN*4
where “0” and “N” represent the top and bottom
surfaces of the sheet and where...
Part 1: Technology of Sheet Heating
Combined Conduction, Convection, Radiation
Finite Difference Equation with Boundary
Conditions
where
 1  e s ,i
1 1  e h ,i 

FFg ,i = 1 / 


F12
e h ,i 
 e s ,i
Average temp:
Radi = xFFg ,is / k
F12 is the average view
factor,es is the sheet
emissivity, eh is the heater
emissivity
Tavg
1 T0  TN N 1 
= 
 Ti 
N 2
i =1

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
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?]
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
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
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
Part 1: Technology of Sheet Heating
Radiant Heat Transfer - Equation
Where is the problem?
The energy interchange between a heater element
dA1 and a sheet surface element dA2 is given as
(
q1 2 = sFg T
*4
h
T
*4
s
cos f1 cos f 2
A2 A1 r 2 dA1dA2
)
where cos fi are the direction cosines defined
earlier, and r is the distance between the two
elements
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
Part 1: Technology of Sheet Heating
Radiant Heat Transfer - Equation
Where is the problem?
Now replace the differential elements with
discrete elements, A1 and A2
(
q1 2 = sFg T  T
*4
h
*4
s


 A1
)

A2

cosf1 cosf 2
A1A2 
2
r

Part 1: Technology of Sheet Heating
Radiant Heat Transfer - Equation
Where is the problem?
• The terms cos f 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 f1 = cos
f2 = z/r
• z is the line distance between the two elements
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:
r= x y z
2
2
2
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...
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

z2
q1 2 = sFg A1 
2
2
2

 A2  x  y  z
(
)
2
(T
*4
h
T
*4
s

A2 

)
• For the energy transfer from all heater
elements to one sheet element
q 12


z2
= sFg A2 
2
2
2
 A1  x  y  z
(
)
2
(T
*4
h
T
*4
s

A1 

)
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
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
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
Part 1: Technology of
Sheet Heating
The net %
energy
received by
each element
when heater
temps are
equal
everywhere
Part 1: Technology of Sheet Heating
Part 1: Technology of Sheet Heating
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
Part 1: Technology of Sheet Heating
The % values
represent the
local heater
flux output
change from
the original
100%
Part 1: Technology of Sheet Heating
Part 1: Technology of Sheet Heating
Problem Solved!
No?
What Assumptions Need to be
Relaxed?
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.]
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 rollfed sheet? (Roll-fed sheet clamped on two
sides, “endless”, multiple shots in oven)
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 (Semitransparent sheet is heated by volumetric
energy absorption)
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 (H2O, CO2) needs to be included
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)
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
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”
Beer’s law: Wavelength-dependent absorption is
exponentially dependent on depth into the
plastic
Primary assumptions: Sheet remains planar; only
diffuse absorption, reflection
Part 1: Technology of Sheet Heating
Okay, give us an example of how just one
of these assumptions can be relaxed!
• Reflectivity and
transmissivity of a
thick semitransparent sheet
of plastic
• Properties are
determined through
ray tracing
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
t =e
k 2d / cosq 2
=e
4k2d / l0 cosq 2
Where k2 is the Beer’s law absorption coefficient,
l0 is the wavelength of the incident energy and
d/cos q2 is the distance the radiant beam
travels through the plastic in one pass
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
 n1  n2 

r12 = 
 n1  n2 
2
 1  n plastic 

r air plastic = 

 1  n plastic 
2
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
Rsheet
 (1  r )2t 2 
= r 1 
2 2 
1 r t 

Where r is the reflectivity at both inner and
outer sheet surfaces and t is the transmissivity
of a single pass through the sheet. Note: these
values are wavelength-dependent!
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
sheet
(
1  r )t
=
2
1 r t
2 2
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
Asheet
(
1  r )(1  t )
=
1  rt
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:
T
  T 
rc p
= 
  ql (T ), x 
q x  x 
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 timedependent 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…
Part 1: Technology of Sheet Heating
End of
Part 1
Technology of Sheet Heating
Part 1: Technology of Sheet Heating
Part 2
Constitutive Equations Applied to Sheet
Stretching
Begins promptly at 9:00!