This paper investigates an out-of-autoclave (OoA), embedded-resistive heating method to precisely control the bondline temperature when curing high strength adhesives for joining composite adherends. A challenge with OoA methods is that nonuniform heat loss, e.g., due to substructures that act as local heat sinks, can lead to nonuniform temperatures in the bondline, which in turn, can result in uneven curing, residual stresses, and potentially weak joints. The main contribution of this work is to apply a voltage pattern at the boundary of the embedded heater to control the distribution of the electrical power at the interior bondline, and thereby reduce temperature variations. Additionally, this work devises an empirical model (that can be applied when material parameters and models are not readily available) to predict the desired power generation, and to design the embedded heater and voltage pattern that minimizes the bondline temperature variation. The technique is demonstrated experimentally for bonding a single-lap joint, and the maximum temperature variation in the bond area was reduced by five times from 31.6 °C to 6.0 °C.

## Introduction

This paper presents a method for precisely controlling bondline temperatures to cure adhesive in a bonded joint using an embedded heater. The intended application is an out-of-autoclave (OoA) process for bonding composite adherends where a major challenge is uneven heat loss in the presence of substructures, which act as heat sinks. Uneven heat loss can result in a nonuniform temperature distribution in the bond area, which can lead to uneven curing, increased residual stresses, improper air removal causing defect formation [1], and a weakened joint. The method proposed here solves the nonuniform-temperature problem by designing the boundary heating pattern to compensate for uneven heat loss through heat sinks. This is done by generating more heat in areas where there is increased heat loss to enhance the temperature uniformity at the bondline.

Joining composite materials comes with a set of challenges that are uniquely different from traditional materials such as steel or aluminum. These traditional materials are often joined using fasteners or bolts, which usually necessitates drilling holes. In composite materials, drilling holes creates stress concentrations by cutting fibers [2,3], which can cause delamination near joints [4]. In response to these challenges, a common way of joining composites is through adhesive bonding. Compared to bolted or fastened joints, adhesive bonding has higher joint stiffness, superior fatigue performance, and higher structural efficiency [5,6].

High strength adhesives used to join composites require high curing temperatures (upward of 100 °C) for faster curing and better overall performance [7]. The temperature should be uniform throughout the bonding region for an even cure. A common way to accomplish a uniform temperature is to bond within an autoclave, where the entire part can be heated evenly, and both vacuum and pressure are applied to remove air bubbles from the bondline that might become points of weakness in a cured joint [8]. Large parts require large autoclaves, which leads to high manufacturing costs [9,10]. Even for smaller parts, the autoclave or oven approaches are not possible in the presence of components that are sensitive to elevated temperatures. For these cases, OoA methods for curing adhesive are used. These methods include heat blankets [11,12], UV curing [13], and microwave heating [1416]. External heating methods such as blankets heat the outside surface of the adherends, which can limit part thickness and geometry, because carbon fiber composite materials have poor through-thickness heat conductivity. Similarly, UV can be used with photo-curable adhesives but are not suited to carbon fibers, which are not UV transparent. The use of microwaves is also limited to specialized applications since the presence of graphite fibers in composite materials can lead to arcing and local hot spots [17]. These limitations motivate research on new methods of curing adhesive bonds for composites.

The focus of this paper is on joining using an embedded-resistive heater (ERH) made of carbon fiber to heat high strength adhesives. To accomplish this, a single layer of carbon fiber, either dry or pre-impregnated with resin, is sandwiched between two layers of electrically isolating film adhesive, and placed within the bondline. Voltage is applied to the edges of this single layer of carbon fiber, and the carbon fiber acts as a resistive heater resulting in Joule heating at the bondline. This technique has the advantage of adding heat directly at the bondline, minimizing the area of elevated temperature near the bond, and thereby reducing the overall energy usage. Previous works have used embedded-resistive heating for curing and bonding composite materials. The technique has been demonstrated to cure composite adherends using carbon mats [1820] or carbon nanotube (CNT) buckypaper [21]. Bonded joints have been produced using embedded-resistive heaters such as a metal mesh [22,23], CNT buckypaper [24], and carbon fiber [25,26]. The advantage of using carbon fiber is that the heater could be of the same material as the composite adherends, so the bonding process would not add any extrinsic materials besides the adhesive in the bondline.

When substructures are not present, i.e., with uniform heat loss into the surrounding structure, using a carbon fiber-embedded heater for adhesive curing can produce comparable bond strength with joining performed in an autoclave or oven [25,26]. In practice, there can be nonuniform heat loss from the embedded heater into the adherends being joined resulting in a variation of temperature at the bondline. An example case is the adhesive bonding of a carbon fiber repair patch to a surface with substructures, which act as heat sinks as illustrated in Fig. 1. The repair patch is to be bonded on the outer surface of an aircraft with a supporting stringer underneath on the inner surface. Uniform heating of the bond area would produce a drop in temperature in the region near the supporting stringer. The resulting nonuniform temperatures can lead to residual stresses in the bondline and lower structural integrity. Moreover, better control of bondline temperatures can improve void removal as well, which enhances structural integrity, as studied in Ref. [1]. Such temperature nonuniformity in the presence of substructures will occur with any heater, whether using an embedded heater, or other OoA methods. Therefore, there is a need to develop approaches to generate local variations in the heating profiles to achieve temperature uniformity at the bondline.

Fig. 1
Fig. 1
Close modal

This paper proposes a resistance-based joule heating technique where different voltage patterns are applied at the boundary of an embedded heater to produce nonuniform heat-flux (power) within the heater. To demonstrate the proposed technique, bonding of a single-lap joint will be used as the target application, which is a benchmark setup for investigating adhesive bonding, e.g., see Refs. [27] and [28]. A heat sink will be placed under the center of the bond area, e.g., as in Fig. 2, to evaluate the effect of nonuniform heat loss on temperature uniformity, and the ability of the boundary control approach to correct for this. Boundary control of the heat generated by the heater is achieved by applying a voltage to copper tabs added along the edges of the carbon fiber-embedded heater. To illustrate, consider the schematic of a heater with two copper tabs, one on top and the other on the bottom, as shown in Fig. 2. The copper tabs help to distribute the applied AC voltage (V) across the width of the heater so that a uniform current flows across the heater, which in turn, creates a uniform power distribution in the heater. However, a heat sink (e.g., placed as in Fig. 2) will lead to nonuniform heat loss from the bondline leading to nonuniform temperatures.

Fig. 2
Fig. 2
Close modal

This work focusses on designing a heater with multiple copper tabs for generating nonuniform power as shown in Fig. 3. By adding many such tabs and connecting them to voltage independently in a controlled manner, the current flow through the heater can be adjusted to create a variable power distribution within the heater. For example, if switches S1 are closed and the other switches are open, then the majority of the current flows in zone 1 shown in Fig. 3. After the adhesive is cured and the bonding is completed, only the bond area of the heater remains embedded in the bondline. The remaining heater material outside the bond-area along with the copper tabs can be trimmed off. In case of composite-structure repair as in Fig. 1, the surface finish can be enhanced by grinding and sanding.

Fig. 3
Fig. 3
Close modal

This paper is organized as follows: the problem is described in Sec. 2, followed by modeling and heater design in Secs. 3 and 4. Experimental validation is in Sec. 5, and the conclusions are in Sec. 6.

## Problem Description

This paper uses the well-studied single-lap joint, e.g., see Refs. [27] and [28] as an example system to evaluate the proposed boundary control approach. A schematic of this system can be seen in Fig. 4, where the bond area is marked with cross-hatching, and a heat sink is placed under the lap joint so that it is parallel to the y-dimension of the bond area.

Fig. 4
Fig. 4
Close modal

When heating a joint with an embedded heater that generates uniform power, such as in Fig. 2, a significant drop in temperature is seen toward the center of the bond area. The experimentally obtained temperature variation is shown in Fig. 5, where the measured steady-state temperatures in the bond area Tm,u are shown as circles.

Fig. 5
Fig. 5
Close modal

The problem addressed in this paper is to achieve uniform bondline temperature at the target value of Td,ss, which is the steady-state temperature at the end of the standard ramp-and-hold profile Td(t) shown in Fig. 6. The rationale for focusing on the steady-state temperature Td,ss is that the majority of the curing occurs at this temperature. Moreover, maintaining temperature uniformity in the bondline at the highest temperature Td,ss is more difficult because of a higher heat flux out of the bondline to the external heat sink at the higher bondline temperature. The goal is to maintain temperatures throughout the bond area to within $±6 °C$ of the steady-state target temperature Td,ss. This is the manufacturers recommended range for the carbon fiber/epoxy prepreg system used for the adherends in this study. The problem will be solved in two parts, described below, and then verified experimentally.

Fig. 6
Fig. 6
Close modal
1. (1)

Part 1: Desired power generation—Find the desired power generation profile $Pd(x,y)$ in the bond area that will produce a uniform temperature Td,ss in the bond area. This will be done empirically since using material properties can be proprietary and difficult to obtain, and therefore, an analytical approach is often not possible.

2. (2)
Part 2: Multizone heater design—Design an embedded heater through simulation that will minimize the maximum temperature error in the bondline
$min(|Td,ss−T(t,x,y)|∞)$
(1)

## Part 1: Desired Power Generation

The initial step in designing a heater and control scheme is to find the power generation profile $Pd(x,y)$ that will produce uniform temperatures in the bond area at the steady-state temperature Td,ss. To determine the desired power profile, an experiment was run that is described below, where a known amount of power was applied with a heater of uniform power generation. With this uniform input at the bondline, the temperature was measured at points along the length of the bond center-line. This experimental data can be used to fit a model at the measurement locations, which is then extrapolated to predict temperatures at more points along the bondline. The resulting model is then inverted to find the desired power profile $Pd(x,y)$ within the bond area. This desired profile, which is found along the center-line of the bond area where temperatures are measured, is then extended to the rest of the bond area (along the y-direction). This is visualized in Fig. 2, with a line through the center of the embedded heater and the bonded region.

### Developing the Model.

A model is developed in this section to describe the transient temperatures within the bond area of a single-lap joint as shown in Fig. 4. The temperature variation in a body with heat energy being added to the system at rate $Q˙$ is described with the heat equation
$ρCp∂T*∂t=∇·(k∇T*)+Q˙$
(2)
where ρ is the material density, Cp is the specific heat capacity, k is the conduction coefficient, and $T*$ is the temperature in the body. This can be represented in the Cartesian coordinate system with a lumped heat transfer equation, where there is some mass (at node xk) losing heat through conduction and is being heated with energy input rate $q˙in$ from an embedded heater. It is assumed that the conduction coefficients in the vertical (kz) and horizontal (kx and ky) directions are independent, and that there is no heat flowing in the y-direction. By subtracting the ambient temperature $T∞$ from the measurements $T*(t,x,y)$, the system can be described in terms of the relative temperature $T(t,x,0)=T*(t,x,0)−T∞$ as
$ρCpVxyzdT(t,xk,0)dt=−2kzAxyLzT(t,xk,0)−kxAyzLx(T(t,xk,0)−T(t,xk+1,0))−kxAyzLx(T(t,xk,0)−T(t,xk−1,0))+Axyq˙in$
(3)
where Vxyz is the element volume, A indicates the area of one side of the node, and L is the length between nodes, as illustrated in Fig. 7. Note that the heat is lost through conduction to the left node $xk−1$ at temperature $T(xk−1,0)$ and right node $xk+1$ at temperature $T(xk+1,0)$ with conduction coefficient, kx. Conduction to the top and bottom (along z direction) at temperature $T∞$ is associated with conduction coefficient Kz. The rate of heat energy Peh added from the heater can be found from Ohm's law as the square of the voltage difference 2V between the tabs divided by the effective resistance R of the heater, i.e.,
$Peh=(2V)2R$
(4)
Fig. 7
Fig. 7
Close modal

The power $q˙in$ in units of $J/(s·m2)$, is calculated as power Peh divided by area Axy. This model in Eq. (3) represents the thermal system of adherends heated with an embedded heater, where heat is lost above and below to the environment, and is also transferred horizontally near the bondline.

The model from Eq. (3) can be simplified to general parameters as
$γT˙(t,xk,0)=−βkT(t,xk,0)−α(T(t,xk,0)−T(t,xk+1,0))−α(T(t,xk,0)−T(t,xk−1,0))+εq˙in$
(5)
with parameters γ, β, α, and ε, as summarized in Table 1. The equation can be normalized by setting the parameter ε to one.
Table 1

Summary of parameter scaling for refined model

ParameterPhysical meaningScaling to get refined parameter
γ$ρCpVxyz$$γ¯=γ(Lx,r/Lx)$
α$kxAyzLx$$α¯=α(Lx/Lx,r)$
β$kyAxyLy$$β¯=β(Lx,r/Lx)$
εAxy$ε¯=ε(Lx,r/Lx)$
ParameterPhysical meaningScaling to get refined parameter
γ$ρCpVxyz$$γ¯=γ(Lx,r/Lx)$
α$kxAyzLx$$α¯=α(Lx/Lx,r)$
β$kyAxyLy$$β¯=β(Lx,r/Lx)$
εAxy$ε¯=ε(Lx,r/Lx)$

### Refine the Model to More Points.

The parameters for the model described in Eq. (5) are found with the least squares method using a limited number of thermocouples (TCs). With the estimated parameters, the predicted temperature profile $Tsim,u$ closely matches the shape of the measured temperature profile Tm,u at locations xk as shown in Fig. 5. Moreover, using the estimated parameters, the model is then refined to predict the temperature at other points $x¯k$ along the bond center-line. This is done by taking the nodal Eq. (5) with node spacing Lx and adjusting the parameters as in Table 1 to match a smaller node spacing $Lx¯$ in the x-direction as shown in Fig. 8. The resulting model with nodes at locations $x¯k$ is given by
$γ¯T˙(t,x¯k,0)=−β¯T(t,x¯k,0)−α¯(T(t,x¯k,0)−T(t,x¯k+1,0))−α¯(T(t,x¯k,0)−T(t,x¯k−1,0))+ε¯q˙in$
(6)
Fig. 8
Fig. 8
Close modal

This model was used to predict the temperature Tsim,e at nodes $x¯k$, which are more closely spaced than the locations where the temperature was measured as shown in Fig. 5 (see solid line Tsim,e).

### Finding the Desired Power.

The desired power profile $Pd(x,0)$ at the bond center-line (shown in Fig. 4 as the x-axis), needed to achieve the steady-state target temperature Td,ss, can be found by first reframing the refined model in Eq. (6) into a system of first-order equations
$T˙(t,x,0)=AT(t,x,0)+BP(t,x,0)$
(7)
$T̂(t,x,0)=CT(t,x,0)$
(8)
where A, B, and C come from empirical model parameters, $T(t,x,0)$ is the predicted temperature profile, $T̂(t,x,0)$ is the output temperature profile at the measured locations, and $P(t,x,0)$ is the power added into the bond center-line. The temperature profiles $T̂(t,x)$ and T(t, x) are assumed to be the optimal target trajectory $Td(t)$, which is a uniform temperature throughout the bond area during a ramp and hold profile, as shown in Fig. 6. Since the steady-state temperature Td,ss will be used for heater design, the steady-state desired power profile can be found independent of time t. By taking the derivative of Eq. (8) and combining with Eq. (7), the desired profile can be solved as
$Pd(t,x,0)=(CB)−1(T˙d(t)−CATd(t))$
(9)
where $Td(t)=Td,ss$ and $T˙d(t)=0$, so that the desired profile at steady-state is
$Pd(x,0)=(CB)−1(−CATd,ss)$
(10)

This desired profile, which is calculated for the bond area centerline, is extended through the rest of the bond area by assuming it to be constant in the y-dimension.

## Part 2: Multizone Heater Design

The goal of designing a heater with multiple tabs is to minimize the temperature variation in the bond area. To do this, a model that can predict the power generation of a heater with a specific tab placement is needed. A two-dimensional steady-state voltage distribution for a tabbed heater was developed using a finite difference method (FDM) model [29] to solve the Laplace Equation
$ΔV=∂V2∂2x+∂V2∂2y=0$
(11)

The inputs to this model can be seen in Table 2, where the heater width and length were determined by the geometry of the single-lap joint experimental setup. Simulations showed that a smaller gap size between the tabs led to more control over the heating in the bond area. Nevertheless, shorting between the copper tabs were observed experimentally when the gap size was reduced below 0.635 cm (0.25 in). Therefore, the gap size was selected to be selected to be 0.635 cm in the simulation-based design of the multizone heater.

Table 2

FDM model design parameters for heater with multiple tabs as in Fig. 3

ParameterValue(s)
Heater width in x (cm)17.78
Heater length in y (cm)12.7
Tab gap width (cm)0.635
Grid size (mm)1.27
Gap locations θ (cm)x1, x2, x3, $…$
Tabs activated (bin)ci
ParameterValue(s)
Heater width in x (cm)17.78
Heater length in y (cm)12.7
Tab gap width (cm)0.635
Grid size (mm)1.27
Gap locations θ (cm)x1, x2, x3, $…$
Tabs activated (bin)ci
A FDM simulation was used to calculate the power generated $P(θj,ci)$ for a set of parameters (spatial locations) $θ=x1,x2,…,xn−1$, where n is the number of tabs as indicated in Fig. 3, and the tab activation configuration c is a binary number where each digit indicates whether a tab is “on” (attached to a voltage), or “off” (floating voltage). For example, one tab activation configuration is $c=[1 0 0 0 0]$, which indicates a five tabbed heater with the first tab on, and the rest off. A multiple tabbed heater power profile Psim is found as a linear combination of simulations of ci configurations, so that
$Psim(ζ,θj)=ζ1P(θj,c1)+ζ2P(θj,c2)+ζ3P(θj,c3)+⋯+ζmP(θj,cm)$
(12)
Here $m=2n$ is the total number of configurations ci, and ζi indicates the amount of time that each configuration ci is activated. The sum of all the activation times ζi is constrained to be a normalized time period of length 1, and no individual time ζi is less than 0 or greater than 1, i.e.,
$∑i=1mζi=1, and 0≤ζi≤1$
(13)
Using the FDM simulation, the optimal power profile Psim and times ζ for given heater parameters θj are found by minimizing the cost function
$JP(ζ,θj)=‖Pd(x,y)−Psim(x,y,ζ,θj)2‖$
(14)
with respect to the timing ζ of tab activation patterns c. This was accomplished by using a discrete proportional–integral controller to adjust the error in each control zone n (area within the region of a tab, as in Fig. 3), and the details can be found in Ref. [30]. This minimization gives the optimal ζ
$ζopt=arg minζ Jp(ζ,θj)$
(15)
An example of an optimized power profile $Psim(x,yavg,ζopt,θj)$ for a heater with five tabs can be seen in Fig. 9, along with the desired profile $Pd(x,0)$. The power profile $Psim(x,yavg,ζopt,θj)$ is averaged over the y-dimension of the bond area between the y-bounds of the bond area y1 and y2
$Psim(x,yavg,ζopt,θj)=1y2−y1∫y1y2Psim(x,y,ζopt,θj)dy$
(16)
Fig. 9
Fig. 9
Close modal
The final step for finding the optimal heater design parameters θ is to minimize the temperature error between the simulation and desired steady-state temperature Td,ss in the bond area. This is expressed in the cost function $JT(θ)$
$JT(θ)=‖Td,ss−T(ζopt,θ,x)‖∞$
(17)
where the simulated temperature along the bond center-line $T(ζopt,θ,x)$ is found using the optimal configuration times ζopt from Eq. 15, and putting this back into the model in Eq. 6. The cost is then minimized with respect to the heater design parameters θ to find the optimal set θopt
$θopt=arg minθ JT(θ)$
(18)
by initializing the design with equal sized tabs and systematically resizing the tabs until the minimum solution for $JT(θopt)$ is found. This method varies the location of each tab gap on a grid by step size Sx to the left and right and stores the results for all tab locations. The result with the lowest error is kept, and then, this process is repeated until convergence. Then, the step size Sx is decreased by half, and the process is repeated until Sx converges at the smallest grid step. The details of this solution search strategy are described in detail in Ref. [30]. The results of these simulations are shown for heaters designed with two to seven tabs in Fig. 10. This plot shows how the maximum temperature error (as in Eq. (1)) decreases as more tabs are added, and with three or more tabs, there is less than three degrees of predicted temperature variation in the bondline.
Fig. 10
Fig. 10
Close modal

## Experimental Validation

Three experiments were run to evaluate the technique described in this paper. The first experiment was run with a uniform power heater to demonstrate the temperature variation in the bond area and to fit a thermal model of the system. The second experiment is a step response run with a heater designed with multiple tabs in order to tune a controller for each tab. The third experiment was run with the same multiple tabbed heater and controlled to compensate for the temperature variation in the bond area.

### Description of Experimental System.

The experimental setup can be seen in Fig. 4, with a more detailed layup shown in Fig. 11. The adherends for the single-lap joint were produced with HEXCEL HexPly 155 (BMS8-168), plain weave prepreg, with a cure temperature of 125 °C. The adherends are 16 layers, with a $[0]16$ stacking sequence, with a cured thickness of 5.81 mm.

Fig. 11
Fig. 11
Close modal

The first two experiments (aimed at the heater design) were run without adhesive. In the bondline on either side of the embedded heater, a nonporous teflon sheet was used to electrically isolate the heater from the adherends. The final bonding experiment was done with supported structural adhesive films (Scotch-Weld AF 163-2U) instead of the teflon sheets. While bonding, this adhesive provides electrical isolation between the embedded heater and adherends, as demonstrated in Refs. [25] and [26]. The heaters used were made of dry carbon fiber fabric of plain weave, 24 × 24 thread count, 0.178 mm thick, and 1k tow (item number CF141 at The Composite Store). Along the center-line of the bond area, 15 thermocouples were placed 1.27 cm apart to measure the bondline temperature.

The entire experiment was performed over a 1.27 cm sheet of silicon rubber for insulation from the table top. An aluminum (6061) bar with a cross section of 0.635 cm × 5.08 cm was used as the heat sink under the center of the single-lap joint. At the ends of the aluminum bar, where it leaves the bonding area, bags of ice were placed to cool the bar during the experiments. A single layer of 3.175 mm thick silicon rubber was placed over the top of the adherends for insulation, along a layer of breather fabric. The entire experiment was vacuum bagged, with a vacuum of 40.6 kPa (0.40 atm). A picture of this experimental setup can be seen in Fig. 12.

Fig. 12
Fig. 12
Close modal

To control the voltage to the embedded heater two Arduino Megas were used: one to collect temperature and calculate the control and the other to control the voltage switching. The voltage was applied with a Variac power supply and modulated with Omron G3NE-220TL solid state relays.

### Step Response With Uniform Heater.

An embedded heater with uniform power generation, as shown in Fig. 2, was used to heat a bond area with a heat sink placed underneath the bottom adherend as shown in Fig. 4. Instead of adhesive, nonporous teflon sheets were used to electrically isolate the heater and simulate adhesive. This experiment was done with a step input of 20 Volts at 50% duty cycle until the maximum temperature reached the target temperature of 125 °C, which took approximately 32 min. The temperatures across the bond area at the end of this experiment Tm,u are shown in Fig. 5. There is a significant dip in temperature of around 30 °C under the center of the bonding area, where the heat sink was located. The temperature also is higher on one side of the bond area. This is consistent with other experiments run with a similar vacuum bagging approach and appears to be skewed due to the placement of the through-bag vacuum connector. The percent variation (PV) of temperature through the center-line of the bond area, calculated as
$PV=Tmax−TminTmax×100$
(19)
is 25.2%, where Tmax is the maximum measured temperature, and Tmin is the minimum measured temperature. These results were used to generate the desired power profile shown in Fig. 9, which was calculated by fitting Eq. (6) and solving for Eq. (10).

### Step Response With Tabbed Heater.

A second experiment was run using a tabbed heater designed by the minimization in Eq. (18). The purpose of this second experiment was to tune controller parameters for each tab, so that for a heater with five tabs on each edge, there would be five individually tuned controllers. This experiment was also run using nonporous teflon in the place of adhesive.

A controller was designed by first fitting a step response of the system to each zone corresponding the area within each tab as in Fig. 3. The results of this were then used to fit a first-order system Gn describing the relationship between the input power and the temperature for each zone
$Gn(s)=Tn(s)Pn(s)=Knτns+1$
(20)
The model parameters Kn and τn were found for each zone using a least squares approximation of the temperature response, i.e.,
$∑k|Tn(tk)−[Pn−Tn˙(tk)][Knτn]|2$
(21)

where Tn indicates the measured temperature in each zone n at time instants tk, and Pn is the power input to the system, which is proportional to the square of the voltage applied to the heater.

Using the model in Eq. (20), a proportional and integral controller
$C(s)=Kp+Kis$
(22)
was designed. The proportional gain Kp and integral gain Ki were found that meet the chosen damping ratio ζ = 1 and natural frequency $ωn=10/τn$ for the closed loop system. The full procedure for selection of the proportional and integral gains for a single heater has been described in Ref. [26] and is omitted here for brevity.

Using the results of a step response to a tabbed heater, the model parameters Kn and τn were determined. From these model parameters, control coefficients Kp and Ki were found and are presented in Table 3.

Table 3

Model parameters Kn and τn, and calculated control constants Kp and Ki for each zone on a tabbed heater as shown in Fig. 3

Zone #Kn ($°C/V2$)τn (s)Kp ($V2/°C$)Ki ($V2/°C$)
10.17378.9114.96631.5969
20.20608.494.58080.8182
30.22753.388.21110.6163
40.24749.180.21450.5636
50.22617.385.35780.7278
Zone #Kn ($°C/V2$)τn (s)Kp ($V2/°C$)Ki ($V2/°C$)
10.17378.9114.96631.5969
20.20608.494.58080.8182
30.22753.388.21110.6163
40.24749.180.21450.5636
50.22617.385.35780.7278

### Bonding With Tabbed Heater.

Using an embedded heater with five tabs as shown in Fig. 3, with optimal tab locations $θ (cm)=6.02, 9.65, 14.06, 16.87, 20.32$, and control coefficients from Table 3, a single-lap joint was bonded with a heat sink under the center of the bond area. The tabs of this heater were designed and placed as described in Sec. 4. The goal of this experiment was to demonstrate the ability to improve uniformity of temperature in the bond area when there is a heat sink causing nonuniform heat loss from the bond area.

The results of this experiment can be seen in Fig. 13. In this plot, the temperatures measured with the tabbed heater $Tm,t$ are averaged over the first 15 min after reaching the target steady-state temperature Td,ss (125 °C), i.e.,
$Tm,t(x)=1t2−t1∫t1t2Tm,t(t,x,0)dt$
(23)

where t1 is the time when the target temperature $Td(t)$ first reaches the steady-state temperature Td,ss, and t2 is 15 min after t1, which is the time frame when most of the curing takes place [31]. All of the thermocouple data for the first and third experiments is shown in Table 4. The thermocouples are numbered such that thermocouple 1 is located on the left side of Fig. 4. The PV of temperature in the center-line of the bond area was measured to be 4.7%, and was predicted to be 4.5% in simulation.

Fig. 13
Fig. 13
Close modal
Table 4

Temperature data for each TC along the bond area center-line. Data are shown for the single tab heater experiment (uniform power) Tm,u, and the multiple tab heater experiment $Tm,t$, which is averaged over the first 15 min after reaching the steady-state temperature $Td,ss=125 °C$.

TC# (n)Uniform heating, Tm,u ($°C$)Boundary control, $Tm,t$ ($°C$)
1117.7125.1
2118.1124.9
3117.3125.5
4114.0126.7
5108.7127.2
6102.5126.6
797.5125.5
893.8125.7
994.0125.0
1099.2127.6
11108.7129.4
12116.2129.0
13121.3127.3
14124.5125.2
15125.4123.4
TC# (n)Uniform heating, Tm,u ($°C$)Boundary control, $Tm,t$ ($°C$)
1117.7125.1
2118.1124.9
3117.3125.5
4114.0126.7
5108.7127.2
6102.5126.6
797.5125.5
893.8125.7
994.0125.0
1099.2127.6
11108.7129.4
12116.2129.0
13121.3127.3
14124.5125.2
15125.4123.4

### Discussion of Results.

The experimental results in Table 4 show a more than five times reduction in the maximum temperature variation with the proposed boundary control approach when compared to a uniform power (single tab) embedded heater, from 31.6 °C with the uniform power to 6.0 °C with the proposed boundary control. Moreover, the temperature variation in the bondline while using the multiple tabbed heater of $±3 °C$ matches well with the prediction of the empirical model used to design the heater. Finally, the reduction of the temperature variation to $±3 °C$ with the proposed boundary control approach is well within the goal of reaching better than $±6 °C$ in the bond area. Thus, the results demonstrate the ability of the proposed boundary control method to reduce temperature variations at the bondline in the presence of local heat sinks such as substructures.

## Conclusions

This paper examined the use of boundary control of an embedded resistive heater to improve the temperature uniformity in the presence of local heat sinks such as substructures. The specific application studied was a single-lap joint with a heat sink under a section of the bond area. A model, in combination with experimental data, was developed to predict the desired power profile $Pd(x,y)$ needed to reduce deviations from the steady-state uniform temperature Td,ss within the bond area. The power produced by a tabbed heater was simulated with an FDM model, which was used along with the desired power profile to design a tabbed heater to minimize temperature variation in the bond area. The proposed boundary control approach reduced the temperature variation in the bondline to $±3 °C$, which was well within the goal of reaching better than $±6 °C$ in the bond area. While this paper focused on rectangular heaters and bond areas, this approach is general and could be extended to other geometries and different heat sink patterns.

## Funding Data

• Division of Civil, Mechanical, and Manufacturing Innovation, U.S. National Science Foundation (CMMI 1536306).

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