Chapter 2
Controlled Resistive Heating of
Carbon Fiber Composites
Temperature control is the key to effective resistive heating for composite rigidization. The
ability to monitor and even control the temperature provides an active control strategy in
prescribing and predicting the stiffness of the hardened composite. Further, understanding
how the material responds to an electrical input determines both the type of control to be
used as well as the level of control that can be attained.
This chapter investigates how temperature control of carbon-fiber reinforced polymer
composites is implemented. The material was investigated experimentally and modeled
analytically for use in the final control scheme. A PID feedback controller was applied to
the resistive heating process and the control gains were then experimentally tuned. Infrared
imaging of the composite during heating was also performed as a method for visualizing the
heating process.
2.1
Introduction to Resistive Heating
When Georg S. Ohm (1787-1854) published Die galvanische Kette mathematisch bearbeitet
in 1827, he described the theory and applications of electric current. For his achievements,
the German scientist’s name has been forever attributed to electrical science terminology:
Ohm’s Law (2.1) states the proportionality of current and voltage in a resistor and the SI
18
unit of resistance is the Ohm (Ω) [32].
V = IR.
(2.1)
Ohm’s work provides a direct correlation between the potential voltage drop, V , across a
resistive (or Ohmic) material, R, and the electric current, I, flowing through the medium.
In doing so, Ohm’s Law is a fundamental concept that helped establish the basis of modern
electrical theory. James Joule (1818-1889) coupled Ohm’s Law with his own endeavors
in relating heat to mechanical work. Joule worked with Lord Kelvin in developing the
absolute temperature scale (Kelvin, K) and was also acknowledged for his contributions
that eventually led to the First Law of Thermodynamics. His experiments initiated the
concept of the mechanical equivalence of heat, which relates the energy required to raise
the temperature of water 1◦ F. Accordingly, the SI unit of work was named the Joule (J).
However, Joule might be best remembered for his discovery of the relationship, appropriately
named Joule’s Law, between current flow and heat dissapation in a resistive element (2.2).
Specifically, Joule found that the rate of thermal energy generated, Ė, within a resistive
material,
Ė = I 2 R,
(2.2)
is proportional to the square of the current, I, and directly proportional to the resistance,
R [33, 34].
Together, Georg Ohm’s and James Joule’s accomplishments close the gap between
electrical energy input (either as a voltage potential or an electric current) and thermal
energy output. As a result, resistive heating can provide both undesired and desired heat
generation. Because of Joule’s Law, resistive heating occurs whenever an electric current
passes through a resistive material. The design of electronic circuitry must account for this
phenomena or the risk of overheating, even melting, the hardware becomes a reality. On the
other hand, Joule’s law sets forth the notion that the amount of thermal energy generated
can be controlled by the inherent resistance of the heating element as well as the applied
electric current. The selection of the input, i.e. current, voltage, or even resistance, to obtain
a desired temperature rise is the most common use for resistive heating. Appliances like
electric stove tops, hairdryers, curling irons, and electric blankets employ resistive heating
to raise the temperature of heating elements.
Internal resistive heating of carbon-fiber reinforced materials provides a way to a
19
new method of rigidization. If current passing through the fibers can generate sufficient
heat, then it is thought that the localized temperature increase can be used to cure the
adjacent thermosetting resin. As Joule demonstrated, this process is an active one; the
temperature of the composite can be controlled via the supplied electrical input. Therefore,
the requirement for a given temperature to be obtained during the rigidization process is
merely a function of the electrical energy supplied to the material.
2.1.1
The Resistive Nature of CFRP Materials
In order for carbon fiber reinforced polymers to be rigidized by resistive heating, the composite must have the appropriate electrical properties. When an electric field is applied
to a material, the motion of electric charges within the material generates current flow.
However, all materials exhibit some resistance, R, to this charge motion. Resistance, then,
depends on both the resistive nature of a given material, called electrical resistivity, and
the geometry of the material
l
R=ρ .
A
(2.3)
Equation 2.3 shows that the resistance of a heating element is proportional to its length, l,
and inversely proportional to its cross-sectional area, A. Further, the electrical resistivity,
ρ, for a given material is a function of temperature. The electrical conductivity, σe , is a
measure of the material’s ability to allow electrical current to flow and is defined as the
reciprocal of resistivity [3]
σe =
1
.
ρ
(2.4)
Table 2.1: Room Temperature Electrical Conductivities of Common Engineering Materials [3]
Metals and Alloys
σe , (Ω· m)−1 Nonmetals
σe , (Ω· m)−1
Silver
6.3 x 107
Graphite
105 (average)
7
Copper, commercial purity
5.8 x 10
Germanium
2.2
Gold
4.2 x 107
Silicon
4.3 x 10−4
Aluminum, commercial purity
3.4 x 107
Polyethylene (PE) 10−14
Polystyrene (PS)
10−14
Diamond
10−14
Quantified as Ω-m−1 , electrical conductivity (and thus resisivity) is an inherent material property that does not depend on geometry. Both conductivity and resistivity are
functions are temperature, however. The difference in conductivity varies greatly between
20
materials and helps categorize materials as conductors, insulators, and semiconductors. Table 2.1 illustrates that pure metals have the highest conductivities, which explains their use
as electrical wiring materials. On the opposite end of the spectrum, the electrical insulators
such as polyethylene (PE), polystyrene (PS), and diamond have very small conductivity
values. Between these extremes, semiconductors such as graphite, germanium, and silicon
have conductivities less than metals but much greater than insulators. An in-depth look at
the physical properties of both the carbon fiber tow and thermosetting resin is discussed in
Chapter 3. For now, the primary focus is kept on the method of applying internal resistive
heating to resistive materials.
2.1.2
Resistive Heating Experimental Goals
The application of resistive heating to the carbon-fiber reinforce polymeric materials of
interest required an understanding of how the material heats as well as the electrical power
requirements needed to reach curing temperatures. Thermal control was implemented on
these materials through a two-step process. First, the material was subjected to open-loop
heating tests, in which a constant voltage was applied to the material for a given amount
of time. In doing so, techniques for measuring the temperature were established and the
power requirements to achieve certain temperatures was recorded. The heating and cooling
time constants were also measured as a way of experimentally modeling the heated material.
Closed-loop, or feedback, control was then applied to the heating process. In this strategy,
a measured temperature was compared to a desired temperature and the input voltage
(or current) was regulated in order to reduce the error between the two. Selected for its
effectiveness and simplicity, a proportional-integral-derivative (PID) was chosen as the type
of control strategy to use. The goals, and resulting success, of this testing were chosen in
order to produce an effective, repeatable resistive heating rigidization method for CFRP
materials. The prescribed stiffening that this method can produce, then, is very much
dependent on the ability to control the temperature of the material. The specific detailed
objectives and goals for this testing are summarized:
• Establish temperature measurement of CFRP samples during resistive heating.
• Measure temperatures achieved for selected input voltages during open-loop testing.
• Record heating and cooling time constants for use in system identification.
21
• Apply feedback control, by way of a PID controller, to the resistive heating process.
• Tune the controller to achieve accurate and precise temperature control.
2.2
Thermal Control Experimental Setup
To perform temperature observation during resistive heating, the samples were first fixed on
each end. Each sample consisted of a 15 − 20cm length of polymer resin-coated carbon fiber
tow. An applied voltage (generated in Simulink and dSpace and then amplified by a ±20V,
±2A HP 6825A Bipolar Power Supply/Amplifier) across the sample resulted in current flow
between the voltage leads. Omega J-type (Iron-Constantan) thermocouples were placed at
three positions along the sample to measure temperature. An additional thermocouple was
also used to record the ambient air temperature. Temperatures from the thermocouples
were then measured and recorded by an Omega OMK-DAQ-56 data acquisition module
and Personal DAQView software. Then, data from this device was transmitted via USB to
a personal computer.
OMK-DAQ-56
Simulink
Th1
V+
Th4
Th6
V-
i
dSpace
Controller
Thamb
Data
V(t)
Personal
Computer
Power
Supply/Amp
Figure 2.1: Test setup schematic for open-loop resistive heating with temperature monitoring.
The temperature was measured at multiple positions in order to detect possible
temperature differences, or gradients, along the length of the material. Since the tow
consisted of such fine fibers, the placing of the thermocouples (much larger in contrast)
required multiple iterations. At first, untwisted fiber tow was position into the fixture
(2.3). However, the fibers were easily separated, allowing the thermocouple to pass through
the thickness of the material. Poorly placed thermocouples not only lead to imprecise
22
Th1
V+
Th2
Composite Sample
Th3
V-
8 ¼”
Figure 2.2: Resistive heating with temperature monitoring experimental setup.
temperature measurements, but in the presence of a feedback control algorithm can cause
incorrect amounts of energy to be supplied, resulting in an inaccurate heating method. One
solution to this problem was twisting the fiber tow samples. This technique kept the fibers
in a “tighter knit” configuration and in turn, caused the sample to have a more consistent
cross-sectional area.
Figure 2.3: Thermocouple positioning without twisting (left) and with twisting
(right). Notice how the tip of the thermocouple protrudes through the tow in
the first picture. Twisting the samples helped to hold the thermocouple in a
position to better measured internal temperature.
This experimental setup was also designed to measured the amount of current flowing
through the material. Knowing how much current flows allows for the electrical power to
be calculated as well as the resistance of the material to be monitored during heating.
To perform these tasks, a current-measuring circuit was designed and built (Figure 2.4).
Modeling the sample as a resistor with unknown resistance, Rstrands , the output voltage,
Vout , was measured directly during the tests. Knowing the voltage drop across a known
23
R2
R1
-
Vi
V+
+
OP177
+
Rstrands
+
Vout
+ ∆V -
-
R3
Vin
-
Figure 2.4: Current sensing circuit used to measure current flow and material
resistance.
resistance, R3 , allowed the current flowing through R3 (and also the sample, Rstrands ) to
be determined. Likewise, the voltage drop, ∆V , across the strands was measured and from
that, the resistance of the sample was obtained. Figure 2.5 illustrates a flow-chart type
representation of how each unknown value was computed.
Vin ( RR12 + 1)
Vin
Vout =
Vout
V + =V − =
R2/R1
R3
(1 + )
Rst
R3
V
⇒ Rst = R3  in
 Vout
R2
R1

+ 1  − R3

)
Rst
V+
Vout
(1 + RR12 )
∆V
∆V = Vin − V +
i=
(
V+
R3
i
Figure 2.5: Post-processing circuit and associated equations for indirect current
measurement.
This testing apparatus (Figure 2.1) was then used to perform open-loop resistive
heating on samples of the CFRP tow. However, when closed-loop control was performed
the experimental setup was changed to accomodate temperature as a feedback control signal.
Specifically, four signal conditioners (Omega #CCT-22-0/400C) with a range of 0 − 400o C
were used to provide the cold-junction reference point for the thermocouples and then
output 0 − 10V voltage signals proportional to the measured temperatures. These signal conditioners replaced the OMK-DAQ-56 temperature data acquisition module, which
could not output temperature signals. When incorporated into the previous experimental
24
Thermocouple
Signal Conditioners
Figure 2.6: Signal conditioners used to measure and output temperature signals
required for feedback control.
setup (Figure 2.1), the new experimental setup sends measured temperatures back into
Simulink/dSpace, where the control algorithm could process them.
*Temperature Feedback
LowPass RC
Filtering
*PID Controller
Algorithm
Simulink
(Lab PC)
SC1
SC2
Th1
IN
dSpace
Controller
V+
Th2
i
SC3
Th3
SCamb
Thamb
V-
OUT
*Control
Signal
Power
Amplifier
Figure 2.7: Experimental schematic used for feedback temperature control.
In order to visualize the procedure of applying a PID control to this system, the
Simulink Block diagrams used to control temperature are shown Figures 2.8. In these
diagrams, the conversion of measured temperature to control voltage is traced.
Using dSpace to accept thermocouple voltages, the input signals were then converted
to temperature and the average temperature was computed. The error between the measured temperature and the temperature set point was then scaled by the PID control gains,
which generated a control effort power value. Taking into account changes in the ambient
temperature (an external disturbance) and using a constant initial resistance value for the
25
Channel 1 Voltage
T1
V1
Ro
1
T2
Channel 2 Voltage
V2
Channel 3 Voltage
V3
M
T3
Channel 4 Voltage
V4
10
Channel Inputs
to D-Space
Tset
V(t)
V
Tset Data
Tav g
Tav g
Tamb
Tamb
Temperature
Conversion
Channel 5 Voltage
Ro
Channel Outputs
from D-Space
PID Control
Current Voltage
Current
Current Monitoring
2
f(t)=(V^2)/R+hAsT_inf
error
Tset
3
PID Control
sqrt
PID Loop
Tavg
4
1
V
0.083
Tamb
hAs
1
Ro
0.4
Kp
1
1
s
0.04
error
1
PID Control
Ki
0
du/dt
Kd
Figure 2.8: a. Simulink model created to house the temperature control algorithm (top). b. Temperature feedback control loop used to compute the
corrective control voltage (middle). c. PID controller (bottom) located within
the temperature feedback loop (b).
sample, a corrective control voltage was calculated. This signal was then sent out of dSpace,
through the power supply/amplifier, and into the material.
2.3
Open-Loop Heating Results and Discussion
The resistive heating process was initially examined by inputting a single voltage pulse and
recording the temperature response along the strands. The pulse length was selected to
allow the material to reach a final, or steady-state, value. Then the voltage was turned off
and the temperature was observed as the material cooled down. Figure 2.9 shows the temperature responses measured at the three locations along the samples for an input voltage of
10V with 3 minutes of heating and 3 minutes of cooling. The measured responses illustrate
26
t2sample52.mat
Temperature - C
150
Th1
Th4
Th6
100
50
0
0
50
100
150
200
250
300
350
400
15
Vin
Vout
Voltage
10
5
Max Power = 16.458 W
Max Current = 1.6172 A
0
-5
-10
0
50
100
150
200
Time - s
250
300
350
400
Figure 2.9: Typical temperature and voltage responses measured for an input
voltage of 10V.
some important aspects of the heating and cooling properties of this type of composite material. First and foremost, the material experiences an exponential temperature rise (when
voltage is turned ON) from an initial temperature, Ti , up to a higher final temperature, Tf .
Cooling is denoted by an exponential decay (when voltage is OFF) from an initially high
temperature, Ti , to cooler final temperature, Tf . Each of these responses are approximated
by the following equation
T (t) = Ae−ct + B
(2.5)
A = Ti − T f
(2.6)
B = Ti
(2.7)
where,
and c represents the exponential growth (heating) or decay (cooling) rate. The corresponding heating and cooling time constants,
1
τ= ,
c
(2.8)
which refer to how quickly the material responds to a given input, can also be measured
from these plots. Values of the heating constants are then later used to refine the predicted
response model developed for controlled, open-loop heating.
The voltage plot in Figure 2.9 gives additional insight into the power requirements
of the sample. Shown as a blue, dotted line, the input voltage, Vin , represents the voltage
27
applied across the strands. In addition, a red, dotted line plots the values of an output
voltage, Vout , measured from the post-processing circuit during the experiment. Recall that
two additional goals of this test were to measure the current flowing through the material
and, from that measurement, track changes in the electrical resistance of the material.
Returning to Figure 2.9, it is noticed that Vout remains “flat” during the heating process.
Examining the top equation in Figure 2.5 further clarifies how the resistance of the strands
changes for a given change in the output voltage,
G
Vin R2
+1
− R3 ⇒ Rst =
− R3 ,
Rst = R3
Vout R1
Vout
(2.9)
where G is a constant if all other resistances and the input voltage remain fixed. Taking
the partial derivative of Equation 2.9 with respect to the measured output voltage yields
the following
G
∂Rst
G
= − 2 ⇒ ∂Rst = − 2 ∂Vout .
∂Vout
Vout
Vout
(2.10)
As derived, a change in resistance of the sample, Rst , is proportional to a change in the
output voltage, Vout . Since Figure 2.9 shows a relatively unchanging measurement of Vout
over the length of the heating cycle, we can infer that the resistance of the strands also
remains steady. However, a steady resistance is also dependent on the material having
a resistivity that does not vary with temperature. The amount of resistance change per
temperature increase, then, will also be considered during this test. As it will be shown
later, a fixed value of resistance greatly simplifies the both the simulation of the temperature
response as well as the eventual temperature control of the process.
Previously mentioned in the goals for this experiment, it was desired to record the
maximum temperature (an average of the three thermocouple measurements along the
sample) achieved for varying amounts of applied voltage. For each sample, the input voltage
was varied (from 0.5V to 10V in 0.5V increments) and the corresponding temperature
response was measured (Figure 2.10). It was observed that the temperature of the tow
sample was proportional to the square of the applied voltage.
Further, by measuring the current with the sensing circuit shown in Figure 2.4, the
amount of current to obtain these temperatures was also examined. Again, the temperature
was proportional to the square of the current. A second-order polynomial trend-line verifies
this relationship graphically.
The quadratic relationships between temperature and voltage (and current) shown
28
Plot of all Thavg Responses
110
Run1
Run2
Run3
Run4
Run5
Run6
Run7
Run8
Run9
Run10
Run11
Run12
Run13
Run14
Run15
Run16
Run17
Run18
Run19
Run20
100
90
Temperature - C
80
70
60
50
40
30
20
0
50
100
150
200
Time - s
250
300
350
400
Figure 2.10: Average temperature response of the sample for various input
voltages (Run1=0.5V and Run20=10.0V).
in Figure 2.11 are consistent with Equation 2.2, which states that the heat produced by an
electrical signal in a resistive element is proportional to the square of the current. Further,
if Equation 2.1 is substituted into Equation 2.2, the heat generated is equally proportional
to the square of the applied voltage
Ė =
V
R
2
R=
V2
.
R
(2.11)
As it will be further shown in later sections, the temperature of the material is directly
proportional to the amount of heat generated within it. In addition, the heat generated
within the material is proportional to the power supplied to the material. This relation,
then, states that the temperature attained is also linearly proportional to the power supplied
(Figure 2.12)
P = V I = I 2R =
V2
.
R
(2.12)
Heating and cooling time constants for each sample test were also computed from the
average temperature response and were later used to help model the heating process. This
knowledge of the “system’s” response to a known input was also valuable when applying
29
120
120
2
y = 20.508x + 23.985x + 17.329
2
R = 0.9972
100
100
y = 0.6253x + 2.1939x + 18.666
80
Temperature - C
Temperature - C
2
2
R = 0.9968
60
40
20
80
60
40
20
0
0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0
0.5
1
Voltage - V
1.5
2
Current - A
Figure 2.11: Measured average maximum temperature per input voltage (a)
and current (b) applied.
120
Temperature - C
100
80
y = 5.3283x + 23.903
2
R = 0.9908
60
40
20
0
0
5
10
15
20
Power - W
Figure 2.12: Material temperature exhibits a linear relation to electrical power.
feedback control to the heating procedure. From this round of tests, an average heating time
constant, τh and an average cooling time constant, τc , were extracted from the measured
data. These values represent the time for the response to reach 1−e−1 (63.2%) of its steadystate value [35]. The time constants measured demonstrate no significant dependence on
the applied voltage (and thus, the temperature attained) and were roughly the same for
heating (19.2 ± 1.8 seconds) and cooling (20.8 ± 2.6 seconds). A discussion of how these
time constants are modeled is later addressed and shows that these values are primarily
dependent on the material itself.
The resistance of the sample was also mapped versus the temperature reached for
each level of applied voltage. It was desired to observe if this property changed greatly or
if it could be assumed to remain constant during the heating process. A constant heating
process translates into a time-invariant system model for use in feedback control. Otherwise, the control strategy must continually update the resistance value as the temperature
30
30.0
Time Constants - s
25.0
20.0
15.0
10.0
Heating
5.0
Cooling
0.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
Input Voltage - V
Figure 2.13: Measured heating and cooling time constants for open-loop resistive heating tests.
changed. Through the circuit described previously, the resistance of the strands were mea-
16
0.6
14
0.4
12
Ro = 10 Ω
10
0.2
0
8
-0.2
6
-0.4
4
0
20
40
60
80
100
∆ Resistance - Ω
Resistance - W
sured indirectly and tracked relative to an initial resistance value (Figure 2.14). Resistance
-0.6
120
Temperature - C
Figure 2.14: Sample resistance as a function of temperature.
does decrease with increasing temperature, but only slightly. At 100o C, the sample resistance has only decreased by 5.25% to 9.48Ω. The change in resistance might be more of
a factor at higher temperatures, though in the temperature range presented, it remained
stable.
While the temperature of the material has yet to be controlled, resistive heating with
temperature monitoring was established and a method of positioning the thermocouples was
achieved. Further, the steady-state temperature attained was measured as a function of the
voltage, current, and power supplied to the sample. It was also observed that the resistance
of the material did not change significantly during these heating tests. Short of feedback
control, curing temperatures for UnyteSet resin (160o C-170o C) have yet to be achieved
even in open-loop resistive heating. In order for these higher temperatures to be generated,
31
greater amounts of power must be supplied. The resulting experimental setup takes into
account the need for increased power supply and in doing so uncovers a better way to
measure current flow.
2.3.1
High-Temperature Open-loop Resistive Heating
Expanding the results from the low temperature experiments, it was desired to track the
final temperatures attained for larger input voltages. In thought, the quadratic trend-line
shown in Figure 2.11 would be stretched to higher voltages and temperatures. This step
required a change of hardware to safely accommodate higher voltages (and currents). Using
a Xantrex XHR 300V-3.5A DC Power Supply, the output voltage was triggered via remote
signaling. Simulink and dSpace were still used to generate the signal shape and the new
power supply then scaled the signal to desired voltage levels. Another benefit of using
this hardware was that both the output voltage (equivalent to the voltage potential across
the sample) and current from the power supply can be directly monitored and recorded as
scaled voltage signals. Simply, by using this power supply, higher power can be sourced to
the composite sample and the post-processing circuit was eliminated from the picture. A
new round of temperature response testing on samples of G40-800 carbon fiber coated with
Unyte201 polymer resin was conducted for voltages varying from 1 to 18V.
400
400
350
2
y = 0.9898x + 0.4696x + 23.232
2
R = 0.9985
300
y = 6.2951x + 30.64
Temperature - C
Temperature - C
350
250
200
150
300
2
R = 0.9945
250
200
150
100
100
50
50
0
0
0
5
10
15
0
20
10
20
30
40
50
60
Power - W
Voltage - V
Figure 2.15: Power requirements measured during high-temperature, open-loop
resistive heating tests.
Again, the voltage and power were monitored for high temperature resistive heating.
This process only further verified that temperature is dependent on the square of the voltage applied or directly proportional to the applied power. Temperatures near 350o C were
achieved, but at the cost of more than 50W of power. With such a large power requirement,
32
the ability to store and then supply high peak power will be crucial for in-space rigidization
via this method. However, the total electrical energy may offset high power requirements if
the material can be cured in a short amount of time. The novel thermoset resin, UnyteSet
201, used in this study also helps to diminish the power needed for high temperature cure.
This material cures at temperatures in the range of 150 − 200o C, whereas a resin such as
PETU cures at more than 250o C. If the temperature-power relationship shown in Figure
2.15 is referenced, the power to reach the cure temperature can be reduced by more than
37% by using the UnyteSet resin (≈ 22W) instead of the PETU thermoset (≈ 35W).
2.4
Active Temperature Control of Resistive Heating
Open-loop resistive heating, which has been addressed up until now, applied a given voltage
to a resistive material and the resulting temperature increase was measured. However, in
order to achieve a desired temperature through voltage selection, a model for the system
must be well-defined. This type of control, called open-loop control, applies a controller,
independent of the output, to the “plant.” In effect, the appropriate control signal is first
computed based on a desired reference temperature. The control signal, which is then
applied to the actual system, generates the desired temperature response. The possibility
of open-loop control as an effective control strategy is investigated for this system.
Disturbance
Input
Reference
Input
Controller
Control
Input
System
(Controlled
Object)
Controlled
Output
Figure 2.16: Open-loop control scheme [9].
2.4.1
Model Formulation
For open-loop control to be effective, an accurate model of the controlled object must be
known. In resistive heating, a model for a known length of the CFRP material can be
developed as a function of applied voltage.
33
qc
y
x
ρ, m, Cp
i
qg
Ac
-V
z
L
+V
Figure 2.17: Theoretical heating element used in developing a system model.
The resistive element (Figure 2.17) of known dimensions (length, L, and crosssectional area, Ac ) and material parameters (resistivity, ρ, and specific heat, Cp ) is subjected
to an applied voltage, ∆V which results in a current, i, flowing through the sample. This
model accounts for the heat generated, qg , by Joule-effect heating and the heat given off
due to free-convection, qc . Further, it is assumed that heat conduction in the radial direction and thermal radiation from the sample are negligible. In doing so, we apply a lumped
capacitance simplification, which assumes that the “the temperature of the solid is spatially
uniform at any instant during the transient process” [33]. Simply, the lumped capacitance
method negates conduction by assuming that temperature gradients within the solid are
negligible. The Biot number, Bi , provides a measure of the temperature drop within a solid
relative to the temperature difference between the solid and its environment. It can be
calculated to validate this assumption [33]
Bi =
hLc
< 0.1,
k
(2.13)
where, h is the convection coefficient between the solid and the surrounding air, Lc is
the characteristic length (Lc is half of the radius for long cylinders [33]), and k is the
thermal conductivity of the material. For carbon-fiber, axial thermal conductivity can
range anywhere from 5 − 200 W/m·K [2]. However, the convection coefficient, h, depends
on temperature, fluid properties (density, viscosity, thermal conductivity, and specific heat),
surface geometry, and flow conditions and is more difficult to approximate. So, for now,
let us assume that lumped capacitance can be applied to this system. The first law of
thermodynamics, which demands conservation of energy, is then applied to the resistive
34
element
Ėin − Ėout = ∆Ė.
(2.14)
For this process, the heat generated, qg , comes from the applied electrical signal, the energy
given off is due to convection, qc , and the change in energy of the sample results in a
temperature increase
q̇g − q̇c = mcp
dT
.
dt
(2.15)
Applying Joule’s law, and substituting in the expression for convection heat transfer, we
obtain an expression for the heat balance of this system,
dT
V (t)2
− hAs (T (t) − T∞ ) = mcp
,
R
dt
(2.16)
where V (t) is the voltage potential across the length of the sample, R is the effective
resistance of the sample, As is the surface area, T∞ is the ambient (or film) temperature of
the surrounding air, m is the mass of the sample, cp is the material’s specific heat, and T is
the temperature of the sample at any point along the length. Re-arranging this expression
yields the following first-order, linear, ordinary differential equation:
mcp
V (t)2
dT
+ hAs T =
+ hAs T∞ .
dt
R
(2.17)
Written in standard form [36], Equation 2.17 is simplified to
dT
+ pT (t) = f (t),
dt
(2.18)
where
p=
hAs
mcp
(2.19)
and
V (t)2
f (t) =
+ hAs T∞
R
1
mcp
.
(2.20)
The solution to the above differential equation can be solved via many techniques, variation of parameters, undetermined coefficients, etc. [36]. More importantly, it provides a
relationship between the sample temperature, T , the material parameters, and the input
voltage, V ,
T (t) = e
−pt
C
C pt
e + T∞ −
,
p
p
(2.21)
where again,
p=
hAs
mcp
35
(2.22)
and
V (t)2
C=
+ hAs T∞
R
1
mcp
.
(2.23)
Comparing Equation 2.5 to Equations 2.21 and 2.22, and recalling Equation 2.8, the time
constant for the temperature response,
τ=
mcp
1
=
,
p
hAs
(2.24)
is a function of the material properties of the sample (m and cp ), the sample dimensions
(As ), and the convection coefficient, h.
Having a relationship between temperature and applied voltage allows for an openloop controller to calculate the exact voltage input based on a desired temperature. However, when an exact understanding of the system breaks down and accurate material parameters are not known, open-loop control suffers. For the sake of seeing how closely the
current model can control temperature, we can model the input power required to cause
a desired temperature increase. It is important to note that the applied signal, V (t), is
modeled as a unit step function multiplied by the magnitude of the voltage. The Laplace
transform of this term can be written as
V21
V (t)2
V2
L [u(t)] =
L
=
R
R
R s
(2.25)
where u(t) is a unit step input. For the entire solution, we start with the original differential
equation (Equation 2.17) and apply a Laplace Transformation
(mcp ) [sT (s) − T (0)] + (hAs )T (s) =
V21
+ (hAs )T∞ .
R s
(2.26)
Grouping like terms and re-arranging yields an expression for the input power,
V2
= s [mcp s + hAs ] T (s) − mcp T (0) − hAs T∞ ,
R
(2.27)
as a function of instantaneous temperature, T , initial temperature, T0 , and ambient temperature, T∞ . This expression can then be written in block form and defines the open-loop
controller applied to the system in Figure 2.16. Notice that both the ambient temperature,
T∞ , and initial temperature, T0 , are factors in the performance of this controller (Figure
2.18). If these values are the same and the temperature, T (s), is defined as the increase in
temperature from its initial value,
T (s) = ∆ (T (s) − T (0)) ,
36
(2.28)
CONTROLLER
T (s)
hAs
-
mcps+hAs
T (s)
T0(s)
mcps+hAs
+
mcp
V(s)2/R
-
mcps+hAs
Figure 2.18: Block diagram representation of the open-loop temperature controller.
then the transfer function, G(s), between temperature increase and applied power can be
written
G(s) =
1
.
mcp s + hAs
(2.29)
Equation 2.29 states that an increase in temperature for a step input voltage is first order,
represented by an exponential growth. The general shape of this first-order model conforms
to the measured exponential growth temperature responses (Figure 2.10).
A change in ambient temperature, which can affect the system during resistive heating, can also be thought of as an external disturbance (Figure 2.19).
Ambient
Temperature
Desired
Temperature
Controller
Initial
Temperature
Input
Power
System
(Controlled
Object)
Sample
Temperature
Figure 2.19: Block diagram of temperature control, via an open-loop controller.
If the ambient temperature is fed into the controller, as shown in the block diagram,
then the controller can account for this instantaneous value when it computes the control
signal. The controller, on the other hand, is completely independent from the actual sample temperature (controlled variable) and thus, in general, it cannot account for external
disturbances to the system. An underlying concern with this technique, then, is whether
37
or not the designer has an accurate system model. In this case, our inability to accurately
measure or calculate the convection coefficient and various material properties (ρ, m, cp , As ,
etc.) weakens the effectiveness of an open-loop controller. Nonetheless, predictive open-loop
heating was attempted on CFRP samples.
2.4.2
Predictive Joule Heating Results
Using Equation 2.21, the temperature response was simulated for both the heating (V 6= 0)
cooling regions (V = 0). Matlab was used to generate the predicted response for various
voltage signals. Specifically, the code allowed the user to specify the initial heating and
final cooling time periods as well as define the input voltage value. The user also specifies
a temperature window in which the sample was allowed to heat and cool for a selected
number of cycles at the same voltage level. The code then computed when to turn on and
off the voltage in order to mimic these temperature swings and used these times to generate
the voltage signal pattern automatically. This signal was read directly into Simulink and
an exact replica was sent out of dSpace into the material. In this manner, the predicted
response to a unique voltage signal was directly compared with the measured response
from the same input. By using a combination of estimated material parameters and the
experimentally measured time constants, the terms “mcp ” and “hAs ” (shown in Figure 2.18
and Equation 2.27) were approximated for this system and implemented into the predictive
response simulation.
65
8
60
6
55
4
Voltage - V
Temperature - C
50
45
40
0
35
30
During cycles:
Voltage is OFF for: 4.9357s.
Voltage is ON for:
98.5982s.
-2
25
20
2
Heating (V is ON)
Cooling (V is OFF)
0
100
200
300
400
Time - s
-4
500
600
700
0
100
200
300
400
Time - s
500
600
700
Figure 2.20: Simulated temperature response for an input of 6V and a temperature window of 10o C (left). Corresponding voltage signal generated to cause
the presribed heating (right).
Attempts to model and simulate the temperature response of the system proved
38
olsample31.mat
Open Loop Sample 31 (3-24-05)
60
65
55
60
55
50
Temperature - C
Temperature - C
50
45
40
Th1
Th4
Th6
Ambient Temp
35
30
Measured Response
40
35
Heating (V is ON)
Cooling (V is OFF)
30
25
20
45
25
0
100
200
300
Time - s
400
500
20
600
0
100
200
300
400
Time - s
500
600
700
Figure 2.21: Measured temperature response induced by the presribed voltage signal (left). Comparison of the simulated temperature and the average
measured temperature (right).
inaccurate. While the general shapes of the curves matched, the nominal temperature
values differed significantly (Figure 2.21). As a result, temperature control via an open-loop
control does not provide accurate temperature response within the material. The inability
to effectively model the material and correct for external disturbances further negates this
approach.
2.4.3
Feedback Temperature Control
Feedback control, on the other hand, updates the control input based on discrepancies
between a measured controlled variable and the desired reference input. This technique
still requires a controller but does not require the system to be completely known. What
feedback control does require is a way to compare a measured value (say, temperature) with
a desired value throughout the controlled process. It is this step that typically introduces
sensors into the picture. The experimental setup, described in Section 2.2, demonstrates
Disturbance
Input
Reference
Input
+
Error
Controller
Control
Input
-
System
(Controlled
Object)
Controlled
Output
Sensor
Figure 2.22: Typical feedback control structure [9].
39
how temperature was fed back into Simulink/dSpace such that the control algorithm could
compare desired and measured temperatures.
For the application of resistive heating temperature control, a PID controller was
selected. The basis for this control strategy stems from the success of proportional-integralderivative (PID) controllers as being robust, yet simple. The performance of the controlled
Kp
+
e
1/s
Ki
+
u
+
Kd
s
Figure 2.23: Transfer function for a PID controller.
system depends on the selection of three control gains: proportional gain (Kp ), integral
gain (Ki ), and derivative gain (Kd ). The process of determining the “best” values for the
respective gains has been addressed by many and described by VanDoren as “often more of
an art than a science” [37]. Notably, Ziegler and Nichols developed techniques for “tuning”
these control gains in order to match desired performance [38, 39].
An understanding of the three control gains and their functions determines how they
may be adjusted to achieve the desired level of control. Proportional control, with gain Kp ,
outputs a control effort,
u(t) = Kp e(t),
(2.30)
directly proportional to the measured error, e(t), between the controlled variable and the
set point [37, 40]. With greater error, the larger the control effort becomes to diminish this
difference in temperature. However, proportional controllers tend to settle on the wrong
corrective error, leaving an offset between the process variable and the set point. Integral
control, adjusted through the gain Ki , generates a control effort proportional to the the
sum of all previous errors [37, 40]
u(t) = Ki
Ztf
e(t)dt.
(2.31)
t0
External disturbance cancellation is another benefit of using integral control. The addition
of the integral controller provides assurance against steady-state errors, but is not always
40
the end-all for corrective control. Many PI (proportional-integral) controllers respond quite
slowly without the use derivative control, Kd . This type of control generates a control
action proportional to the time derivative of the error signal,
u(t) = Kd
d
(e(t)) ,
dt
(2.32)
and is used to generate a large corrective effort immediately after a load change in order to
eliminate the error as soon as possible [37]. A full PID controller, then, combines the three
types of control and requires selection for all three gains. The corresponding control signal
for this controller is then written as:
u(t) = Kp e(t) + Ki
Ztf
e(t)dt + Kd
d
(e(t)) .
dt
(2.33)
t0
If the ratio of control effort, u(t), to the measured error, e(t), is defined as the control
sensitivity, G(t), then the Laplace Transformation of this equation can be taken to achieve
the transfer function for the controller [41]
G(s) = Kp +
Ki
+ Kd s.
s
(2.34)
Shown in Figure 2.23, the block diagram form of this transfer function demonstrates
that the summation of these control gains provides the overall control signal. Using PID,
temperature control was applied to the resistive heating process. Specifically, the three
gains, Kp , Ki , and Kd , were varied and their effects on the system were experimentally
observed.
2.4.4
Feedback Control Experimental Results
Using Simulink to construct and provide the control algorithm processing, and dSpace
to record temperature measurements and output the control effort, feedback control was
applied to the resistive heating process. The basic feedback structure shown in Figure
2.22 assumes that the dynamics of the plant include the material itself as well as the
thermocouples and that the material temperature is consistent along the length of the
sample. In effect, feedback sensor dynamics from the thermocouples were neglected and an
average measured temperature was used as the controlled variable.
The first round of temperature control experiments involved applying a “step” input
as the desired temperature signal. The ability for the measured temperature to then match
41
this set point was recorded. A typical response measured during these tests is shown and
its defining characteristics are labeled.
36
34
32
Temperature - C
30
28
ts
26
Proportional Gain, Kp: 0.5
Integral Gain, Ki: 0.1
Derivative Gain, Kd: 0
24
22
Avg. Final Temperature, Tf : 30.02 C
Percent Overshoot, P.O.: 16.83%
Settling Time, t s : 22.20 sec.
20
Damping Ratio, zeta: 0.493
18
0
50
100
150
Time - s
Figure 2.24: Representative temperature response taken during the implementation and tuning of a PID controller.
Several aspects of this figure were used to evaluate the effects of the selected control
gains, which are shown within the plotting area in Figure 2.24. First, the average final
temperature, or steady-state temperature, was recorded as a measure of accuracy. Also,
the percent overshoot, a measure of how much the measured temperature (blue line) “overshoots” or goes beyond the desired temperature signal (red line), was recorded. The ability
for the control system to reach a desired temperature without severely overshooting it is
important when prescribing a desired temperature-versus-time curing profile. Lastly, the
settling time, ts , of the measured response was calculated as the time that it took the
measured (actual) temperature to reach and maintain a temperature within ±5% (green
boundary lines) of the desired temperature. In evaluating each of the controller gain settings, these pieces of information were reported and helped to establish the final level of
control. The results of this section are organized so as to reflect the gain selection process
that was performed. The effects of each gain are evaluated for step, ramp, and tracking
desired temeperatures.
The plots in Figure 2.25 demonstrate that the average temperature can be driven
to match the desired temperature. However, temperature overshoot (43% and 15%, respec42
36
38
36
34
34
32
30
Temperature - C
Temperature - C
32
30
28
t s K p: 0.2
Proportional Gain,
Integral Gain, Ki: 0.1
Derivative Gain, K d: 0
26
24
20
Proportional Gain, K p: 0.5
Integral Gain, K i: 0.1
Derivative Gain, K d: 0.1
ts
26
24
Avg. Final Temperature, Tf : 30.00 C
Avg. Final Temperature, Tf : 29.97 C
Percent Overshoot, P.O.: 43.16%
Settling Time, ts : 81.60 sec.
Damping Ratio, zeta: 0.258
22
28
18
22
Percent Overshoot, P.O.: 14.51%
Settling Time, ts : 21.30 sec.
20
Damping Ratio, zeta: 0.524
18
0
50
100
150
0
50
Time - s
100
150
Time - s
Figure 2.25: The first few attempts at selecting gains resulted in marginal
control.
tively) proves to be quite significant. The first plot shows that for a desired temperature of
30o C, the actual measured temperature reached 34o C at its maximum point. The measured
temperature in the first plot also experiences significant oscillation and as a result takes
almost 82 seconds to steady out.
45
34
32
40
t
28
ts
26
Proportional Gain, K p: 0.4
Integral Gain, K : 0.04
i
Derivative Gain, K : 0
Temperature - C
Temperature - C
30
d
Avg. Final Temperature, T : 30.00 C
f
Percent Overshoot, P.O.: 4.15%
Settling Time, t s : 19.80 sec.
24
Damping Ratio, zeta: 0.712
22
s
35
Proportional Gain, Kp: 0.4
Integral Gain, K : 0.04
i
Derivative Gain, K : 0
30
Avg. Final Temperature, T : 40.08 C
d
f
Percent Overshoot, P.O.: 5.91%
Settling Time, t s : 31.50 sec.
Damping Ratio, zeta: 0.669
25
20
20
18
0
50
100
150
Time - s
0
50
100
150
Time - s
Figure 2.26: Through a trial-and-error process, control gains that decreased the
overshoot (left) were chosen. Some temperature nonlinearities appear when the
same gains are used for a higher desired temperature (right).
The process of selecting the gains that drive the measured temperature to the desired
temperature set point with minimal overshoot explored many gain values. Figure 2.26
demonstrates that with control gains of 0.4, 0.04, and 0 for Kp , Ki , and Kd respectively,
causes the actual temperature to reach and maintain the desired temperature. Overshoot
for this setting was diminished greatly and the average steady-state temperature accurately
43
matched a desired temperature of 30o C. These same control gains, when used to match a
step input of 40o C, were not as effective. Temperature nonlinearities in the plant resulted in
greater overshoot and longer settling time even though the final temperature matched the
desired temperature. In order to further verify this nonlinearity, the set temperature was
increased even further and the temperature responses were measured for constant control
gains. The controlled temperatures show higher amounts of percent overshoot of increased
desired temperature values.
80
PO = 4.31%
70
PO = 3.56%
Temperature - C
60
Tset=35C
Tset=45C
Tset=55C
Tset=65C
Tset=75C
PO = 2.91%
50
PO = 0.57%
40
PO = 0%
30
* Kp = 0.40, Ki = 0.04, Kd = 0.04
20
0
20
40
60
Time - s
80
100
120
Figure 2.27: Temperature dependent nonlinearities within the “plant” result in
different responses.
A second study investigated the individual effects of each control gain on the resulting
temperature response. For these tests, all but one variable were left constant. First, the
proportional gain, Kp , was varied from 0.5 to 1.0 and Ki and Kd were held at 0 for a desired
temperature of 30o C. Proportional gain, Kp , is a control term that scales the output control
effort proportionally to the measured temperature error. As a result, this control gain sets
the initial slope of the temperature response. For higher desired temperatures and thus
larger initial errors, the proportional gain results in a steeper temperature increase (heating
rate). Since Ki was held at zero throughout this study, each temperature response exhibited
steady-state error and none matched the temperature set point of 30o C.
With the proportional gain affecting the initial temperature rate increase, the integral gain was now varied from 0.025 to 0.100 while holding Kp , Kd , and the set temperature
constant. As discussed before, integral gain is used to minimize the steady-state error of
a controlled variable. Figure 2.29 demonstrates this concept for the temperature response
controlled with a non-zero Ki value (red line). The observed effect that Ki had on the
44
29
28
Temperature - C
27
26
25
24
Kp=0.5
Kp=0.6
Kp=0.7
Kp=0.8
Kp=1.0
Set Temperature: 30 C
Ki = Kd = 0
23
22
21
0
50
100
150
Time - s
Figure 2.28: Temperature response versus a varied proporional gain, Kp .
temperature response was reflected in terms of the controlled system’s damping. This control gain, when relatively larger (≈ 0.1), caused the measured temperature to overshoot
the desired temperature and then oscillate before steadying out. As the integral gain was
reduced to smaller values, the controlled temperature experienced less overshoot, exhibiting more of a “damped” response. So, in order to minimize overshoot, which is a goal for
matching the prescribed temperature response, the integral gain was selected to be relative
small compared to the proportional gain (Kp = 0.5).
Lastly, the derivative gain, Kd was varied for a constant temperature step input at
fixed values of proportional and integral gains. The derivative gain, according to literature
[37], is designed to speed up the response (or reduce its settling time). However, the
measured temperature response was not noticeably affected by the derivative gain at low
levels. The selection of the Kp and Ki values from previous analysis have produced an
accurate, highly damped temperature response. Adding derivative gain to the PI controller
did not quicken the temperature response for the system. Even more, in attempts to reduce
the power required to control the temperature, eliminating one control gain potentially
lessens the magnitude of the control signal and thus reduces power applied to the material.
For step response, it was discovered that a proportional-integral (PI) controller resulted in a temperature response that matched the desired temperature and minimized
overshoot. However, this type of desired signal is not practical when it comes to prescribing
a temperature versus time cure schedule [42]. In this case, it is more common to ramp (increase linearly) the temperature up to the curing temperature, hold it there, and then allow
45
2.5
32
30
2
Nearly identical slopes!
Temperature - C
Initial Slope - C/s
28
1.5
26
24
Set Temperature: 30 C
Kd = 0, Kp = 0.5
1
Ki = 0
Ki = 0.1
22
0.5
0.4
0.5
0.6
0.7
0.8
0.9
PID Proportional Gain, Kp
1
1.1
20
0
50
100
150
Time - s
Figure 2.29: By increasing Ki to a value of 0.05, the steady-state error disappears, but the initial slope remains (left). Calculated initial slopes as a function
of increasing proportional gain solidifies its effect on the temperature response
(right).
50
45
Temperature - C
40
Proportional Gain, Kp: 0.4
35
Ki=0.040
Derivative Gain, Kd: 0
Ki=0.035
Set Temperature: 40 C
30
Ki=0.030
Ki=0.025
Ki=0.100
25
20
0
10
20
30
40
50
Time - s
60
70
80
90
100
Figure 2.30: Integral gain effects on controlled temperature response.
it to cool. With that said, the ability to track the desired temperature becomes important
for evaluating the effectiveness of this controller. By applying a ramping desired temperature, the effects of Kp and Ki on the ability to track a changing reference temperature
were examined. For these tests, Kd was held at 0 since it provided no additional control
advantage during the step tests.
The plots in Figure 2.32 provide several insights into the tracking ability of a PI
controller. First, it is observed that the proportional gain had little effect on the controller’s
tracking ability. For each gain value, the controller “lagged” the desired temperature signal.
Secondly, varying the integral gain resulted in a more noticeable effect. As the value of
46
45
Temperature - C
40
Set Temperature: 40 C
Proportional Gain: 0.4
Integral Gain: 0.03
35
30
Kd=0
Kd=0.02
Kd=0.04
25
20
0
50
100
150
Time - s
Figure 2.31: The measured temperature response demonstrated a weaker dependence on the value of the derivative gain at low levels of Kd .
40
36
38
34
36
32
Temperature - C
Temperature - C
34
Ki = 0.03
Kd = 0
32
30
Tset
Kp=0.4
Kp=0.5
Kp=0.6
Kp=0.8
Kp=1.0
28
26
24
Kp = 0.5
Kd = 0
30
28
Tset
Ki=0.05
Ki=0.09
Ki=0.13
Ki=0.17
Ki=0.20
26
24
22
20
0
50
100
150
Time - s
200
250
300
22
0
50
100
150
Time - s
200
250
300
Figure 2.32: Ramping temperature responses measured for varying proportional
gains (left) and integral gains (right).
Ki increased, the controlled temperature overshot the desired signal at the end of the
ramping section and exhibited more oscillation during the final zero-order temperature
hold. The values of Ki tested, which were larger than the integral gain (0.03) used in the
first plot, demonstrated that increased integral gain also decreased the lag error during
the temperature ramp. Overall, selecting Ki for a tracking signal reveals the trade-off
between first-order accuracy for larger values of integral gain and zero-order accuracy for
smaller values of Ki . This decision is simplified since some error can be tolerated during
the heating (temperature ramping) phase of a cure schedule. More important, though,
is minimizing the overshoot of the measured temperature and providing accurate steadystate temperature holding. By selecting an integral gain to be “small,” but non-zero, these
47
stipulations were met and the control signal, u(t) was kept to a minimum for given values
of Kp and Kd .
Using the selected values to mimic a full-cure schedule, the effectiveness of the controller is shown. The gain values used during this last demonstration were again varied
to see the effect on a desired temperature profile that combines step, ramp, and hold patterns. The error between the desired temperature and the feedback temperature was also
examined. Each combination provided a controlled temperature within a few degrees of the
1.5
50
Error - C
1
45
Average: 0.030C
0
-0.5
40
-1
50
100
150
200
250
300
350
400
450
400
450
35
6
4
30
Tset
Kp=0.5,Ki=0.1,Kd=0.04
Kp=0.5,Ki=0.08,Kd=0.04
Kp=0.5,Ki=0.1,Kd=0
25
Error - %
Temperature - C
Maximum: 1.025C
Minimum: -0.671C
0.5
Maximum: 4.102%
Minimum: -2.331%
Average: 0.091%
2
0
-2
20
0
50
100
150
200
250
Time - s
300
350
-4
50
400
100
150
200
250
Time - s
300
350
Figure 2.33: PI controller used to control temperature through a full, curingschedule-type profile (left). The associated error for the third combination of
controller gains as a function of time (right).
desired temperature throughout the test. Using the third combination, which employs only
proportional and integral gains (PI), it was shown that the controller was accurate to ±1o C
(ignoring the initial temperature step). So, through an extensive experimental parametric
study of the control gains, the final Kp , Ki , and Kd values chosen are 0.4, 0.04, and 0,
respectively. Without derivative control, but stemming from a PID control strategy basis,
the resulting feedback controller is simply a proportional-integral (PI) controller.
Results from the feedback controller experiments were expanded to higher temperatures by using the Xantrex XHR 300V-3.5A DC Power Supply/Amplifier. Specifically,
it was desired to see if temperature dependent non-linearities in the system affected the
PI controller’s performance at more-realistic curing temperatures. Again, both step and
tracking desired temperature signals were used as references. Looking at the temperature response for the desired temperature step input (Figure 2.34), it is evident that the
overshoot increased slightly for higher set points. In addition to the average measured tem48
100
60
Run 1: Tset=45C
Run 2: Tset=65C
Run 3: Tset=85C
90
50
80
40
Energy values for each run:
60
Power - W
Temperature - C
70
50
40
Run 1: E = 0.075452 W-hr
Run 2: E = 0.14892 W-hr
Run 3: E = 0.23058 W-hr
30
20
30
20
10
10
0
0
10
20
30
40
50
Time - s
60
70
80
90
100
0
0
10
20
30
40
50
Time - s
60
70
80
90
100
Figure 2.34: Temperature response measured for higher temperatures (left).
Associated electrical power and energy for each test (right).
peratures (black lines), the maximum and minimum temperatures are plotted. These values
represent the variation in temperature measured by each thermocouple. Also, the electrical
power and energy (the area under the power versus time curve) supplied to the samples
were measured. For a step input of 85o C, the maximum power supplied was over 50W,
though the total energy during the test was less than 0.24W-hr. This plot demonstrates
that rigidization through resistive heating demands large amounts of power, but if cured
quickly, uses only moderate amounts of energy.
Again, the average measured temperature was used as the feedback control signal,
and temperature difference band between the maximum and minimum recorded temperatures is shown. While the average temperature was controlled to precisely match the desired
Temperature - C
200
150
100
Meas. Temp. Band
Desired Temp.
Average Temp.
50
0
0
50
100
Power - W
30
150
200
250
Time - s
300
350
400
Energy Consumed (area): 1.6301 W-hr
20
10
0
0
50
100
150
200
250
Time - s
300
350
400
Figure 2.35: Tracking ability of the PI controller for a high temperature “curingtype” schedule.
49
temperature signal, the gap between maximum and minimum temperature widened as a
function of temperature. Further, the electrical power data collected shows that power
levels decreased by ramping the desired temperature. Intuitively, this makes sense. For a
large step input, the controller must output a large corrective signal to make up for a great
initial error measurement. In terms of the chosen PI controller gains, these values provided
accurate control even at the much higher curing temperatures.
Discussed later in Chapter 3, the controlled resistive heating of carbon-fiber reinforced polymers is used to induce thermoset resin curing and matrix consolidation, which
rigidizes the material. As an example of a heating schedule used to rigidize the samples,
Figures 2.36 and 2.37 look at the temperature profile, the electrical energy required, and the
error in the controlled temperature. This test forecasts the ability of temperature-controlled
resistive heating to match a desired temperature schedule. This sample was heated up 160o C
180
Desired
Actual
160
Energy Consumed: 1.27 W-hr
20
120
15
Power - W
Temperature - C
140
100
80
10
60
5
40
20
0
50
100
150
200
250
Time - s
300
350
400
450
0
0
50
100
150
200
250
Time - s
300
350
400
Figure 2.36: Controlled temperature (left) and resulting electrical power (right)
measured for the cure of CFRP sample.
at a constant heating rate of 60o C/min and held at the curing temperature for 2 minutes.
The heating process required 1.27W-hr of electrical energy over a total time of 6 minutes
and 40 seconds, with a peak power of roughly 20W. During this controlled heating process,
the material’s temperature was kept within ±8o C of the desired temperature at all times.
Even more, during the dwell time at the curing temperature, the controlled temperature
was within ±2o C of the desired 160o C.
50
180
8
Desired
Actual
160
6
4
140
0
Error - C
Temperature - C
2
120
100
-2
-4
80
-6
60
-8
40
20
-10
0
50
100
150
200
250
Time - s
300
350
400
450
-12
0
50
100
150
200
250
Time - s
300
350
400
450
Figure 2.37: Temperature profile (left) and dynamic error (right) measured
during an actual curing schedule.
2.5
Infrared Thermography: A Visual Approach
In establishing the temperature measurement technique used to monitor sample temperature during the resistive heating process, it was noticed that each of the three thermocouples
measured different values (Figures 2.9 and Figures 2.34 and 2.35). Were the thermocouples
poorly placed (Figure 2.3), or does a real temperature gradient exist along the length of the
material? Up until now, the three thermocouple temperatures were averaged at every point
in time during the tests. This technique was chosen merely to simplify the data through the
assumption that the temperature along the length of the carbon-fiber samples was constant.
After all, current flowing through a uniformly resistive element should produce the same
temperature at each point. In efforts to validate the “evenness” of the sample temperature,
infrared (IR) imaging was performed. By being able to visualize how the material heats, it
was hoped to better understand if and how temperature gradients are introduced into the
samples.
2.5.1
Introduction to Infrared Imaging
Infrared waves, produced as thermal radiation from heated bodies, are electromagnetic
waves whose wavelengths are beyond the visible wavelength spectrum [3]. The emission
of thermal radiation occurs when the oscillations or transitions of the many electrons that
constitute matter release energy. The temperature of the matter, a form of internal energy,
sustains the electron oscillations and is thus related to the emission of thermal radiation
51
[33]. Infrared (IR) cameras detect thermal radiation by columnating the thermal radiation
and focusing it onto a detector of known material.
Figure 2.38: Infrared detection scheme as found in an Inframetrics 760 IR
Camera [10].
Infrared imaging, or thermography, is not new to the study of carbon-fiber and other
composites. Much work as been performed on using IR imaging to detect inhomogeneities
that affect the performance of a composite. Specifically, factors such as constituent concentrations (fiber-resin ratio), orientation and distribution of reinforcement, voids, and matrixreinforcement bonding can be identified through IR. Further, this technique can also locate
foreign material, fiber breakage, and degradation [43]. A form of non-destructive evaluation
or testing (NDE or NDT), thermography can also be used to quickly analyze composites
without further damaging, or even contacting, the material. Jones and Berger performed
thermographic inspection on glass-reinforce composite marine vessel hulls [44]. Favro, et al
used high power photographic flash lamps to generate a heat pulse on the surface of fiberreinforced polymers and ceramics. Their work also included the study of crack propagation
as identified through thermography [45]. Further, Sakagami and Ogura, of Osaka University, investigated the transient temperature distribution result from through-thickness and
surface cracks in steel plates as well as delaminated CFRP composite samples [46]. In their
study, they used Joule effect heating, through the application of an electric current, to induce thermal radiation emission from the samples. In the area of spacecraft, Trètout, et
al researched the feasibility of applying thermographic NDT to the evaluation of satellite
structures during assembly. The authors establish an experimental procedure for acquiring
the thermal scans and present a method for processing the data.
52
In contrast to previous work [43, 44, 45, 46], the work performed in this document
applies thermography in a different manner. Instead of using temperature distributions to
detect inhomogeneities within the CFRP samples, the temperature distributions themselves
will be used to quantifiy the “evenness” of the heating process. Particularly, this study
investigated the temperature distribution within the samples during resistive heating and
included sample length, sample twist, and temperature as possible deterrents from uniform
heating. Thermographic imaging was also used to confirm temperature values measured
using thermocouples.
2.5.2
Thermographic Imaging Results and Discussion
An Inframetrics 760 Model IR Imaging Radiometer was used to capture still images of
the sample’s temperature map during resistive heating. The application of power and
measurement of temperature remain as they were used before, with the only additions to
the setup being the IR camera and a television (for real-time viewing). It should be noted
that the images obtained during this test were used qualitatively to investigate temperature
distribution, and not quantitatively to look at nominal temperature values.
Additional infrared thermography images of the resistive heating process were obtained using a Flir ThermaCAM EX320 infrared camera. This instrument provided increased accuracy in not only visualizing temperature gradients but also in measuring nominal temperature values. With this capability, it was desired to verify the temperatures
measured by the thermocouples as well as detect if the thermocouples were affecting the
temperature of the sample.
The tests consisted of prescribing a temperature-time profile for the control system
to match (using an average measured temperature as the feedback signal). During the
heating process, the camera was focused on to the heated sample and still images were
taken. The matrix of test conditions evaluated through IR imaging included sample length
(and thus resistance), sample twist, and also temperature. If these factors contribute to
non-uniform heating, it was hoped to notice this effect in the IR images. While this instrument can display infrared images in both black and white and color modes, the images
will be presented in color. The black and white mode, while difficult to distinguish small
temperature differences in gray scale, was used to focus the camera on the object. Then,
with the camera in color mode, the temperature gradient was monitored and recorded. Two
53
Hot
Th1
Th4
V+
Th6
V-
i
Television
IR Camera
Cold
Figure 2.39: Experimental setup for thermographic imaging of CFRP samples
during resistive heating.
drawbacks of this device were its rather poor resolution in color mode and its inability to
record real-time videos.
First, the effect that sample length had on the resistive heating process was examined. The measured temperatures were plotted versus the desired temperature and the
electrical power and energy were also recorded. Increasing the length of the sample increases
20
0
50
100
150
Time - s
Sample Length = 5"
40
20
0
0
50
100
Time - s
150
200
40
20
0
200
60
60
Power - W
40
0
50
100
150
Time - s
Sample Length = 6"
4
2
0
200
0
20
40
60
80
100
120
140
Time - s
Electrical Energy Consumed
160
180
200
0.15
60
40
20
0
Length=3"
Length=4"
Length=5"
Length=6"
6
Energy - W-hr
Temperature - C
Temperature - C
60
0
Electric Power Consumed
Sample Length = 4"
Temperature - C
Temperature - C
Sample Length = 3"
0
50
100
Time - s
150
0.1
Linear Trendline: E = 0.020L + 0.019 W-hr
0.05
0
200
2
3
4
5
6
Length - in
7
8
9
Figure 2.40: Measured temperature responses (left) and associated electrical
power supply (right).
its resistance and mass. As a result, more power was required to achieve the same temperature and thus more energy was used during the heating process (Figure 2.40). However,
infrared stills captured for these samples demonstrate even heating. As shown, the samples
have a hot (dark red) core region running the length of the sample and the temperature
54
Sample Length: L = 3"
Sample Length: L = 4"
Sample Length: L = 5"
Sample Length: L = 6"
Figure 2.41: Still images captured for various lengths at a temperature of 60o C.
cools radially outward.
Secondly, the maximum temperature attained was thought to possibly induce and/or
widen a temperature gradient. Four different maximum temperatures were used to obtain
IR still images of the heating process. Again, the samples heated evenly along their lengths,
with the only noticeably temperature gradient in the radial direction.
100
50
0
Maximum Temperature: Tmax = 80 C
150
Temperature - C
Temperature - C
Maximum Temperature: Tmax = 60 C
150
50
Temperature - C
100
150
Time - s
Maximum Temperature: Tmax = 100 C
150
Temperature - C
0
100
50
0
0
50
100
Time - s
150
Focusing in on a
Maximum Temperature: Tmax = 60 C
Maximum Temperature: Tmax = 80 C
Maximum Temperature: Tmax = 100 C
Maximum Temperature: Tmax = 120 C
100
50
0
0
50
100
150
Time - s
Maximum Temperature: Tmax = 120 C
150
100
50
0
0
50
100
Time - s
150
Figure 2.42: Controlled temperatures (left) and temperature gradients (right)
via IR imaging of samples heated to different temperatures.
segment of one of the samples allows for the radial temperature gradient to be seen more
clearly. Heating of the thermocouples during this process was also noticed through the IR
imaging. In addition, it was observed (Figure 2.43) during the “cool down” phase of the
heating tests that the samples reach room temperature much sooner than the thermocouples.
As a result, data collected during this phase was skewed by an artificially high thermocouple
55
Figure 2.43: Up-close thermographic image of the CFRP material (horizontal)
and thermocouple (vertical) during a controlled resistive heat.
temperatures. This effect is not significant since during the cooling phase, the material has
already under-gone its “curing” process and is merely being brought back to a cooled state.
The effect of sample twist on the temperature distribution was also studied. The
initial resistance for the twisted samples was recorded prior to each heating test.
Table 2.2: Sample resistance measured as a function of the number of axial
twists.
Trial
1
2
3
4
# of Twists
0
5
10
20
Initial R
12.7
10.9
8.7
8.4
Image #
15
16
17
18
filename
irtwist1
irtwist2
irtwist3
irtwist4
Increasing the number of twists did have a noticeable effect on the resistance of
the sample (Table 2.2), which in turn reduced the amount of power required to heat the
sample. The temperature along the length of the sample, though, was not affected by the
twisting. This find ensured that twisting can be used to help secure the thermocouples in
place without introducing new thermal gradients. Further, as Figure 2.43 demonstrates, the
hottest region of the sample was centered in the cross-section with a slight thermal gradient
noticed in the radial direction.
In general, thermographic imaging through the use of an IR camera has shown
that the temperature is fairly constant along the length of the samples. The ability to
measure the hottest core region of the sample at any point along its length determines
how well the thermocouple measurements agree. Since poor thermocouple placement can
56
Electrical Power Consumed
Number of Twists: 0
Power - W
Number of Twists: 5
0 Twists
5 Twists
10 Twists
20 Twists
8
6
4
2
0
0
20
40
60
80
100
120
Time - s
Sample Resistance
140
160
180
Number of Twists: 20
Resistance - Ohms
15
10
5
0
0
5
10
15
20
25
Twists - #
Figure 2.44: Electrical power and sample resistance (left) and thermographic
imaging (right) of twisted samples. As the number of twists increased, the resistance and electrical power decreased, but temperature distribution remained
“even.”
result in false temperature readings being counted, an average measured value is not a good
choice for the feedback control signal. When an averaged temperature is used, and one
or more thermocouples is not accurately measuring the sample temperature, the average
value is lowered. This results in the control signal increasing its output in order to correct
for a low temperature and in turn, actually raising the true sample temperature above
the desired value. Instead, using the maximum measured temperature (from any of the
three thermocouples) more accurately represents the true core temperature and reduces the
chance of incorporating false temperatures into the control algorithm. The net effect of
using the maximum temperature as the feedback signal is to better represent the actual
temperature within the sample, eliminate the risk of accidentally overshooting the setpoint,
and minimize the control effort required to drive it.
The actual temperature of a coated, carbon fiber tow sample was validated during
resistive heating using an Flir ThermaCAM EX320 IR camera. While this device technically
measures heat flux, temperature values can be obtained for objects of known emissivity in
a defined environment (i.e. ambient temperature). The emissivity of carbon fiber, which
was experimentally measured by Eto, et al [47], ranges between 0.90 and 1.00.
The increased resolution of the images obtained with this IR camera provide clear
evidence that the thermocouples are affecting the sample temperature. While the sample
reaches an nearly constant temperature along its length between the two thermocouples, it
57
359.7°C
SP02*: 171.5
SP01*: 357.5 SP03*: 165.2
20.6°C
Figure 2.45: Temperature gradient obtained on a sample during resistive heating
is drastically lower near the thermocouples. For the resistive heating schedule prescribed in
Figure 2.45, the image was taken at a point when the thermocouples measured a maximum
temperature of 160o C. The IR image verifies that the temperatures measured through thermography are also near this value when in the vicinity of the thermocouples. However, away
from the thermocouples the sample reaches a maximum temperature of roughly 360o C.
The size of the thermocouples (20-gauge thermocouple wire is 0.81mm in diameter)
in relation to the twisted tow size ( 1 − 2mm in diameter). The thermocouples, in efforts to
place them securely and measure the maximum internal temperature of the tow, are lodged
into the twisted fibers. Because of their size, slight misplacements of the thermocouples
result in drastically different temperatures through the thickness of the sample. Further, the
“large” thermocouples disrupt the fiber alignment and tow orientation in the near vicinity.
This effect, which may cause changes in the flow of electrical current and/or introduce more
surface area subject to convective cooling, results in decreased sample temperature at the
location of the thermocouples. It is seen that for a temperature of 160o C measured by the
thermocouples, IR thermography records a maximum temperature closer to 360o C.
Selecting smaller, 36-gauge (0.13mm wire diameter) thermocouples provides a lessintrusive temperature measurement technique. Placing the thermocouples becomes easier
and fiber orientation in the tow is preserved.
A resistive heating schedule that prescribes a maximum temperature of 160o C and
58
500
Desired
36 ga. J-type
20 ga. J-type
450
400
Temperature - C
350
300
250
200
150
100
50
0
0
50
100
150
200
250
Time - s
300
350
400
Figure 2.46: Temperatures measured by small and large thermocouples on the
same sample differed drastically.
feeds back the temperature measured by the original thermocouple accurately tracks the
desired temperature. However, the smaller thermocouple measures a much higher sample
temperature during this process. This smaller profile of this thermocouple is less intrusive
on the sample and measures a maximum temperature of nearly 400o C. Comparing the
two thermocouple readings and the IR image obtained for the same heating profile shows
that the smaller thermocouple more accurately measures the maximum temperature of the
sample. In both cases, the large thermocouple affected the sample temperature locally,
significantly underestimating the true temperature and causing the sample to overheat.
2.6
Conclusions
Through the use of feedback control, via an experimentally-tuned PI controller, temperature
control of CFRP materials during resistive heating has been established. First, uncontrolled,
open-loop heating tests were performed in order to observe the heating behavior of these
materials as well as initiate a method for implementing resistive heating and temperature
measurement. Temperature control, both in open-loop and closed-loop configurations were
then addressed. Using a lumped-capacitance heating model of the composite sample, predictive Joule heating was performed with minimal success. In its place, a feedback control
algorithm was effectively implemented; the controller stemming from PID control theory.
In this structure, a feedback controller compares a measured temperature to a desired set
point and adjusts its corrective control effort to minimize the difference. Through many
59
experimental variations of PID control gains, proportional and integral gains of 0.4 and 0.04
were chosen, respectively. The tests revealed that the derivative control did not noticeably
improve the controlled temperature response and was thus eliminated (Kd = 0). Further,
the selected PI control was shown to accurately mimic a desired temperature profile for step,
ramp, and arbitrary tracking patterns. Lastly, thermographic images from an infrared (IR)
camera were used to visually observe the heating process and confirmed that the samples
exhibit consistent temperatures along their length. The accuracy of measuring temperature
with thermocouples was also observed through thermography, and it was determined that
smaller thermocouples provided more accurate readings.
60