Controlled Resistive Heating of Carbon Fiber Composites

Transcription

1 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 ( ) 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

2 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 ( ) 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

3 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 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 R = ρ l 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, σ e, is a measure of the material s ability to allow electrical current to flow and is defined as the reciprocal of resistivity [3] σ e = 1 ρ. (2.4) Table 2.1: Room Temperature Electrical Conductivities of Common Engineering Materials [3] Metals and Alloys σ e, (Ω m) 1 Nonmetals σ e, (Ω m) 1 Silver 6.3 x 1 7 Graphite 1 5 (average) Copper, commercial purity 5.8 x 1 7 Germanium 2.2 Gold 4.2 x 1 7 Silicon 4.3 x 1 4 Aluminum, commercial purity 3.4 x 1 7 Polyethylene (PE) 1 14 Polystyrene (PS) Diamond 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 2

4 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 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

5 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 2cm length of polymer resin-coated carbon fiber tow. An applied voltage (generated in Simulink and dspace and then amplified by a ±2V, ±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 Th 1 Th 4 Th 6 dspace Controller V + V - i Th amb Data V(t) Power Supply/Amp Personal Computer 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

6 Th 1 Th 2 Th 3 V + Composite Sample 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, R strands, the output voltage, V out, was measured directly during the tests. Knowing the voltage drop across a known 23

7 R 2 R 1 V - - i V + + OP V in R strands + V - R 3 V out - - Figure 2.4: Current sensing circuit used to measure current flow and material resistance. resistance, R 3, allowed the current flowing through R 3 (and also the sample, R strands ) 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. V in V out R 2 /R 1 R 3 V V out + V = = V V = V V i = R + 3 R 2 in ( R + 1 ) 1 R st ( 1 + ) in = R 3 out R 2 ( 1 + ) V V + R R 1 st V = R 3 V in out R 2 ( + 1 ) R 1 R 3 R st V + V 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-/4C) with a range of 4 o C were used to provide the cold-junction reference point for the thermocouples and then output 1V 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

8 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 *PID Controller Algorithm Simulink (Lab PC) Low- Pass RC Filtering IN dspace Controller OUT SC 1 SC 2 SC 3 SC amb Th 1 Th 2 Th 3 Th amb V + V - i *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

9 Channel 1 Voltage V1 T1 Ro Channel 2 Voltage V2 T2 1 1 Ro Channel 3 Voltage V3 T3 Tav g M Tset Data Tset Tav g V V(t) Channel 4 Voltage V4 Tamb Tamb Channel Outputs from D-Space Channel 5 Voltage Temperature Conversion PID Control Channel Inputs to D-Space Current Voltage Current Current Monitoring 2 Tset 3 Tavg error PID Control PID Loop f(t)=(v^2)/r+hast_inf sqrt 1 V 4 Tamb 1 Ro.83 has.4 Kp 1 error.4 Ki 1 s 1 PID Control 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 1V with 3 minutes of heating and 3 minutes of cooling. The measured responses illustrate 26

10 t2sample5 2.mat Th1 Th4 Th Voltage Max Power = W Max Current = A Vin Vout Figure 2.9: Typical temperature and voltage responses measured for an input voltage of 1V. 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, T i, up to a higher final temperature, T f. Cooling is denoted by an exponential decay (when voltage is OFF) from an initially high temperature, T i, to cooler final temperature, T f. Each of these responses are approximated by the following equation T(t) = Ae ct + B (2.5) where, A = T i T f (2.6) B = T i (2.7) and c represents the exponential growth (heating) or decay (cooling) rate. The corresponding heating and cooling time constants, τ = 1 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, V in, represents the voltage 27

11 applied across the strands. In addition, a red, dotted line plots the values of an output voltage, V out, 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 V out 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, ( ( )) Vin R2 R st = R R 3 R st = G R 3, (2.9) V out R 1 V out 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 R st V out = G V 2 out R st = G Vout 2 V out. (2.1) As derived, a change in resistance of the sample, R st, is proportional to a change in the output voltage, V out. Since Figure 2.9 shows a relatively unchanging measurement of V out 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.5v to 1V in.5v increments) and the corresponding temperature response was measured (Figure 2.1). 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

12 11 Plot of all Thavg Responses Run1 Run2 Run3 Run4 Run5 Run6 Run7 Run8 Run9 Run1 Run11 Run12 Run13 Run14 Run15 Run16 Run17 Run18 Run19 Run Figure 2.1: Average temperature response of the sample for various input voltages (Run1=.5V and Run2=1.V). 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 2 R = V 2 R 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 2 R = V 2 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

13 y =.6253x x R 2 = y = 2.58x x R 2 = Voltage - V Current - A Figure 2.11: Measured average maximum temperature per input voltage (a) and current (b) applied y = x R 2 = 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, τ h and an average cooling time constant, τ c, 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 (2.8 ± 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 3

14 Time Constants - s Heating Cooling 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 measured indirectly and tracked relative to an initial resistance value (Figure 2.14). Resistance Resistance - W Ro = 1 Ω Resistance - Ω Figure 2.14: Sample resistance as a function of temperature. does decrease with increasing temperature, but only slightly. At 1 o 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 (16 o C-17 o C) have yet to be achieved even in open-loop resistive heating. In order for these higher temperatures to be generated, 31

15 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 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 3V-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 G4-8 carbon fiber coated with Unyte21 polymer resin was conducted for voltages varying from 1 to 18V y =.9898x x R 2 = y = x R 2 = Voltage - V Power - W 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 35 o C were achieved, but at the cost of more than 5W of power. With such a large power requirement, 32

16 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 21, used in this study also helps to diminish the power needed for high temperature cure. This material cures at temperatures in the range of 15 2 o C, whereas a resin such as PETU cures at more than 25 o 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] 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

17 y q c x i ρ, m, C p q g A c z L -V +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, A c ) and material parameters (resistivity, ρ, and specific heat, C p ) is subjected to an applied voltage, V which results in a current, i, flowing through the sample. This model accounts for the heat generated, q g, by Joule-effect heating and the heat given off due to free-convection, q c. 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, B i, 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 = hl c k <.1, (2.13) where, h is the convection coefficient between the solid and the surrounding air, L c is the characteristic length (L c 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 2 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

18 element Ė in Ėout = Ė. (2.14) For this process, the heat generated, q g, comes from the applied electrical signal, the energy given off is due to convection, q c, and the change in energy of the sample results in a temperature increase q g q c = mc p 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, V (t) 2 R ha s (T(t) T ) = mc p dt dt, (2.16) where V (t) is the voltage potential across the length of the sample, R is the effective resistance of the sample, A s is the surface area, T is the ambient (or film) temperature of the surrounding air, m is the mass of the sample, c p 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: mc p dt dt + ha st = V (t)2 R + ha st. (2.17) Written in standard form [36], Equation 2.17 is simplified to where and dt dt + pt(t) = f(t), (2.18) p = ha s mc p (2.19) [ ] ( ) V (t) 2 1 f(t) = R + ha st. (2.2) mc p 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, [ ( C T(t) = e pt p ept + T C )], (2.21) p where again, p = ha s mc p (2.22) 35

19 and [ ] ( ) V (t) 2 1 C = R + ha st. (2.23) mc p Comparing Equation 2.5 to Equations 2.21 and 2.22, and recalling Equation 2.8, the time constant for the temperature response, τ = 1 p = mc p ha s, (2.24) is a function of the material properties of the sample (m and c p ), the sample dimensions (A s ), 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 [ ] V (t) 2 L = V 2 R R L[u(t)] = V 2 R 1 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 (mc p )[st(s) T()] + (ha s )T(s) = V 2 R 1 s + (ha s)t. (2.26) Grouping like terms and re-arranging yields an expression for the input power, V 2 R = s[mc ps + ha s ] T(s) mc p T() ha s T, (2.27) as a function of instantaneous temperature, T, initial temperature, T, 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 Notice that both the ambient temperature, T, and initial temperature, T, 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()), (2.28) 36

20 CONTROLLER T (s) has mc p s+ha s - T (s) + mc p s+ha s V(s) 2 /R T (s) mc p mc p s+ha s - 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 mc p s + ha s. (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 2.1). 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 Initial Temperature Controller 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

21 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, c p, A s, etc.) weakens the effectiveness of an open-loop controller. Nonetheless, predictive open-loop heating was attempted on CFRP samples Predictive Joule Heating Results Using Equation 2.21, the temperature response was simulated for both the heating (V ) cooling regions (V = ). 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 mc p and ha s (shown in Figure 2.18 and Equation 2.27) were approximated for this system and implemented into the predictive response simulation Heating (V is ON) Cooling (V is OFF) Voltage - V During cycles: Voltage is OFF for: s. Voltage is ON for: s Figure 2.2: Simulated temperature response for an input of 6V and a temperature window of 1 o 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

22 6 olsample3 1.mat 65 Open Loop Sample 3 1 (3-24-5) Th1 Th4 Th6 Ambient Temp Measured Response Heating (V is ON) Cooling (V is OFF) 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 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

23 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 K p e K i 1/s + + u + K d s Figure 2.23: Transfer function for a PID controller. system depends on the selection of three control gains: proportional gain (K p ), integral gain (K i ), and derivative gain (K d ). 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 K p, outputs a control effort, u(t) = K p e(t), (2.3) directly proportional to the measured error, e(t), between the controlled variable and the set point [37, 4]. 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 K i, generates a control effort proportional to the the sum of all previous errors [37, 4] u(t) = K i e(t)dt. (2.31) t 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 4 t f

24 the end-all for corrective control. Many PI (proportional-integral) controllers respond quite slowly without the use derivative control, K d. This type of control generates a control action proportional to the time derivative of the error signal, d u(t) = K d (e(t)), (2.32) dt 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: t f d u(t) = K p e(t) + K i e(t)dt + K d (e(t)). (2.33) dt t 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) = K p + K i s + K ds. (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, K p, K i, and K d, were varied and their effects on the system were experimentally observed 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

25 this set point was recorded. A typical response measured during these tests is shown and its defining characteristics are labeled t s Proportional Gain, K p :.5 Integral Gain, K i :.1 Derivative Gain, K d : Avg. Final Temperature, T f : 3.2 C Percent Overshoot, P.O.: 16.83% Settling Time, t s : 22.2 sec. Damping Ratio, zeta: 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 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, t s, 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%, respec- 42

26 Proportional Gain, t s K p :.2 Integral Gain, K i :.1 Derivative Gain, K d : Avg. Final Temperature, T f : C Percent Overshoot, P.O.: 43.16% Settling Time, t s : 81.6 sec. Damping Ratio, zeta: t s Proportional Gain, K p :.5 Integral Gain, K i :.1 Derivative Gain, K d :.1 Avg. Final Temperature, T f : 3. C Percent Overshoot, P.O.: 14.51% Settling Time, t s : 21.3 sec. Damping Ratio, zeta: 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 3 o C, the actual measured temperature reached 34 o 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 t s Proportional Gain, K p :.4 Integral Gain, K :.4 i Derivative Gain, K : d Avg. Final Temperature, T : 3. C f Percent Overshoot, P.O.: 4.15% Settling Time, t s : 19.8 sec. Damping Ratio, zeta: t s Proportional Gain, K p :.4 Integral Gain, K :.4 i Derivative Gain, K : d Avg. Final Temperature, T : 4.8 C f Percent Overshoot, P.O.: 5.91% Settling Time, t s : 31.5 sec. Damping Ratio, zeta: 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.4,.4, and for K p, K i, and K d 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

27 matched a desired temperature of 3 o C. These same control gains, when used to match a step input of 4 o 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. 8 7 PO = 4.31% PO = 3.56% PO = 2.91% PO =.57% Tset=35C Tset=45C Tset=55C Tset=65C Tset=75C 3 PO = % * Kp =.4, Ki =.4, Kd = 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, K p, was varied from.5 to 1. and K i and K d were held at for a desired temperature of 3 o C. Proportional gain, K p, 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 K i was held at zero throughout this study, each temperature response exhibited steady-state error and none matched the temperature set point of 3 o C. With the proportional gain affecting the initial temperature rate increase, the integral gain was now varied from.25 to.1 while holding K p, K d, 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 K i value (red line). The observed effect that K i had on the 44

28 Set Temperature: 3 C Ki = Kd = Kp=.5 Kp=.6 Kp=.7 Kp=.8 Kp= Figure 2.28: Temperature response versus a varied proporional gain, K p. temperature response was reflected in terms of the controlled system s damping. This control gain, when relatively larger (.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 (K p =.5). Lastly, the derivative gain, K d 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 K p and K i 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

29 Nearly identical slopes! Initial Slope - C/s Set Temperature: 3 C Kd =, Kp =.5 Ki = Ki = PID Proportional Gain, Kp Figure 2.29: By increasing K i to a value of.5, 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) Proportional Gain, K p :.4 Derivative Gain, K d : Set Temperature: 4 C Ki=.4 Ki=.35 Ki=.3 Ki= Ki= Figure 2.3: 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 K p and K i on the ability to track a changing reference temperature were examined. For these tests, K d was held at 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