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A multi-objective Gaussian process approach for optimization and prediction of carbonization process in carbon fiber production under uncertainty

  • Milad Ramezankhani
  • Bryn Crawford
  • Hamid Khayyam
  • Minoo Naebe
  • Rudolf Seethaler
  • Abbas S. MilaniEmail author
Original Research
  • 17 Downloads

Abstract

During composite fiber production, carbon fibers are normally derived from polyacrylonitrile precursor. Carbonization, as a key step of this process, is significantly energy-consuming and costly, owing to its high temperature requirement. A cost-effective approach to optimize energy consumption during the carbonization is implementing predictive modeling techniques. In this article, a Gaussian process approach has been developed to predict the mechanical properties of carbon fibers in the presence of manufacturing uncertainties. The model is also utilized to optimize the fiber mechanical properties under a minimum energy consumption criterion and a range of process constraints. Finally, as the Young’s modulus and ultimate tensile strength of the fibers did not show an evident correlation, a multi-objective optimization approach was introduced to acquire the overall optimum condition of the process parameters. To estimate the trade-off between these material properties, the standard as well as an adaptive weighted sum method were applied. Results were summarized as design chart for potential applications by manufacturing process designers.

Graphical abstract

Keywords

Carbon fiber Gaussian process Carbonization Optimization Adaptive weighted sum method Machine learning 

1 Background and introduction

Carbon fibers, a lightweight, high-stiffness, and high-strength material type, have found wide interest and applications in leading industries such as aerospace, automotive, and energy [1]. Over the past few decades, the quality of carbon fibers has been continually enhanced using newly developed precursors and processes. In particular, much research and effort have been put into improving the tensile properties of fibers, and simultaneously optimizing the costs and energy consumption of carbon fiber production processes [2]. However, the existence of uncertainties in carbon fiber manufacturing can still negatively affect the performance and robustness of the final product and, accordingly, the analysis and decision-making steps for designers. The random behavior of hundreds of thousands of filaments in the production line is one of the primary causes of uncertainty [3, 4], which if not accounted for, it can make the design analysis of carbon fiber production complex, time-consuming, and costly. Yet, the implementation of a powerful and flexible stochastic process model can facilitate the tasks of prediction and optimization of carbon fiber production for designers [5].

Gaussian process (GP) is a useful probabilistic inference method, which effectively deals with complex data structures in the presence of uncertainty [6]. GP provides a robust prediction model and avoids over-fitting [7, 8]. Moreover, GP has been successfully implemented in numerous engineering case studies and has proved to provide more reliable and robust predictions compared with classic regression methods [9, 10, 11]. Consequently, GP may be deemed as an appropriate statistical framework for the prediction and optimization of carbon fiber production processes under uncertain data structures.

1.1 Carbonization process

The conversion of PAN precursor to carbon fibers is essentially comprised of four major stages, namely, polymerization, stabilization, carbonization, and graphitization. Figure 1 illustrates the carbon fiber manufacturing process. The fibers typically go through a two-stage carbonization process (low- and high-temperature steps) after being subjected to an oxidative stabilization process. The goal of these heat treatments, which operate at different controlled temperatures, is to remove all the atoms, except carbon, which eventually results in producing carbon fibers with high modulus and tensile strength [13]. In the carbonization process, the previously stabilized precursor PANs, which are now non-flammable and melt-resistant, are subjected to two different phases of the heating process in an inert atmosphere [14]. The heating process raises the temperature of the fibers up to 800–3000 °C and, consequently, increases its carbon content up to 95% [2].
Fig. 1

Carbon fiber manufacturing process steps (adapted from [12])

In the first stage of the carbonization process, the low-temperature (LT) furnace heats the stabilized PAN fibers up to 950 °C [14]. The LT furnaces usually consume electricity for generating the required heat. The temperature increases gradually by a rate of 5 °C/min in order to lessen the mass transfer, which is primarily due to low-fiber formation capacity. One primary purpose of this stage is to remove tars from the fibers. Temperatures higher than 1000 °C can result in the decomposition of tars, which cause inadequate modifications in the fibers’ properties [2].

During the LT carbonization, the fibers’ structure becomes more durable, and therefore, the chance of damage due to the high temperature and heating rate is not significant [3]. Hence, in the high-temperature (HT) carbonization process, the pre-carbonized fibers go through a furnace, which increases the material temperature up to 1600 °C in the presence of N2 gas. Such an atmosphere immensely reduces the amount of oxygen in the furnace. It prevents the PAN fibers from burning during the HT process. Figure 2 illustrates the HT furnace in the carbonization process [15].
Fig. 2

HT furnace in the carbonization process [15]

The temperature in the carbonization process plays a crucial role in the characterization of the carbon fibers’ mechanical properties, and it must be selected according to the resulting fibers’ application [2]. In other words, the tensile properties of the carbon fibers vastly depend on the production temperature. By increasing the process temperature, a steady growth in the modulus value can be observed [16]. A temperature of 1000 °C produces a low modulus outcome, whereas 1500 °C can result in intermediate modulus carbon fibers (type II carbon fiber) [16, 17]. Furthermore, in order for carbon fibers to reach a higher modulus, higher temperatures, 1600–1800 °C and up to 3000 °C, are necessary [17, 18]. The latter, in which the fibers are heated up to 3000 °C, is called graphitization. Trinquecoste et al. [19] also suggested that, although high tensile strength can be attained by a temperature around 1500 °C, more heating treatment is required to achieve a higher modulus.

1.2 State-of-the-art on modeling of carbon fiber production process

Since the carbon fiber production process is extremely expensive and time-consuming, the preference of the industrial sectors is to minimize the number of the process runs during data collection. The limited number of experimental data and severe uncertainty makes the use of mathematical modeling in the carbon fiber production process highly essential for both process design and optimization purposes. Very recently, different mathematical methods, namely, non-linear multivariable regression, Levenberg-Marquardt neural network algorithm, thin plate splines, and convex hull technique, were implemented to optimize the mean value of the resulting carbon fiber modulus [20]. However, the variability of the carbon fiber mechanical properties was not taken in to account. In this paper, a GP modeling approach for the prediction and, consequently, optimization of the carbonization process is implemented. The existence of excessive amount of noise and potential autocorrelation between data points, which can be emanated due to the continuous or multi-stage time-dependent nature of the processes, are the primary reasons for utilizing GP.

2 Gaussian process

Machine learning is a category of artificial intelligence that is capable of developing statistical models to interpolate, extrapolate, and learn a pattern without the interaction of humans. GP is an effective tool in machine learning; in fact, many models that are used in the field of machine learning are specific types of GPs [7]. GP, a powerful probabilistic method, can cope with the uncertainty. This reliable method provides robust and accurate stochastic prediction models in the presence of noise [21] based on Baye’s rule. One significant benefit of GPs is that, unlike regression methods, it is not necessary to provide a predefined explicit relation between inputs and outputs. GPs establish the required relations in an unsupervised framework based on the properties of the prior functions and the given experimental evidence [7]. Moreover, GPs favorably predict a fit error at each prediction point separately, as opposed to the standard regression methods which output a single overall fit error.

2.1 Gaussian process basics

GP is defined as a set of random variables of which any subset has a joint Gaussian distribution. In this Bayesian method, it is assumed that a GP prior is applied to the functions [22]. It can be characterized in terms of mean and covariance functions:
$$ m(x)=E\left[f(x)\right], $$
(1)
$$ k\left(x,{x}^{\prime}\right)=E\left[\left(f(x)-m(x)\right)\left(f\left({x}^{\prime}\right)-m\left({x}^{\prime}\right)\right)\right], $$
(2)
where m(x) is the mean function, k(x, x) is the covariance function, and f(x) is a real process. Using mean and covariance functions, GP can be generally expressed as:
$$ f(x)\kern0.5em \sim \kern0.5em N\ \left(m(x),k\left(x,{x}^{\prime}\right)\right). $$
(3)
Based on Bayesian theorem, a posterior is defined in terms of a prior and observed data:
$$ \mathrm{Posterior}=\frac{\mathrm{Likelihood}\times \mathrm{prior}}{\mathrm{marginal}\ \mathrm{likelihood}} $$
(4)
$$ p\left(f,{f}^{\ast }|y\right)=\frac{p\left(f,{f}^{\ast}\right)\times p\left(y|f\right)}{p(y)} $$
(5)
p(f, f) is a joint distribution of the training set outputs, f, and the test set outputs, f, which can be defined as:
$$ \left[\begin{array}{c}f\\ {}{f}^{\ast}\end{array}\right]\sim N\left(0,\left[\begin{array}{cc}K\left(X,X\right)& K\left(X,{X}_{\ast}\right)\\ {}K\left({X}_{\ast },X\right)& K\left({X}_{\ast },{X}_{\ast}\right)\end{array}\right]\right). $$
(6)
K(X, X) is a n × n matrix where n and n are the number of observations and test points, respectively. This covariance matrix contains all pairs of training and test points. In more realistic cases, the observations contain noise. Therefore, Eq. (6) can be modified to:
$$ \left[\begin{array}{c}y\\ {}{f}^{\ast}\end{array}\right]\sim N\left(0,\left[\begin{array}{cc}K\left(X,X\right)+{\sigma}_n^2I& K\left(X,{X}_{\ast}\right)\\ {}K\left({X}_{\ast },X\right)& K\left({X}_{\ast },{X}_{\ast}\right)\end{array}\right]\right) $$
(7)
where y is the observed response value, and \( {\sigma}_n^2 \) is the variance of the Gaussian noise ε.

2.2 Covariance functions

From an unsupervised learning point of view, the similarity is a fundamental concept underlying the prediction of test points [7]. It implies that points with close xs tend to have similar target values, ys. In GP, the covariance function is responsible for specifying this similarity and closeness of the data points. In fact, the covariance functions govern the properties of the GP prediction model.

If the mean function is assumed to be zero for simplicity (i.e., a system with a stationary data nature, when average of true means remains zero over time or under different manufacturing conditions), the whole behavior of the GP prediction model can be defined by the covariance function. For example, the Squared Exponential (SE) covariance function provides smoothness for the process; however, the Matérn covariance function is appropriate for cases where observations are highly random with arbitrary characteristic lengths [23]. Figure 3 displays how the covariance function selection affects the characteristics of the prior in the GP, which then naturally influences the final predictions.
Fig. 3

a Prior functions with SE covariance function. b Prior functions with Matérn covariance function

The SE, as a widely employed covariance function in literature and past case studies [24], is given by:
$$ k\left(x,{x}^{\prime}\right)={\sigma}_f^2\exp \left(-\frac{{\left|x-{x}^{\prime}\right|}^2}{2{l}^2}\right)+{\sigma}_n^2\delta $$
(8)
where l is the characteristic length scale, \( {\sigma}_f^2 \) is the variance of the signal, \( {\sigma}_n^2 \) is the noise variance, and δ is Kronecker delta. l, \( {\sigma}_f^2 \) and \( {\sigma}_n^2 \) are free parameters to be identified via evidence data. Due to the fact that these parameters are related to a non-parametric model, they are called hyperparameters. These hyperparameters characterize the trend and behavior of the covariance function and strongly affect the GP prediction results. In order to achieve an accurate and reliable prediction model, the hyperparameters need to be optimized using Negative Marginal Likelihood (NML) and cross-validation methods [7]. These are two powerful tools that are utilized to select the covariance and mean functions and their corresponding hyperparameters for the GP model.

3 Objective: model development

The focus of this research is to incorporate the data collected from carbonization processes in [15] (see also Appendix for raw experimental data) into the training of machine learning algorithms under uncertanity and then use the predicted values to address the process optimization problem. In other words, the objective here is to determine how a robust GP framework can predict and optimize the mechanical properties of carbon fibers as well as the process’s energy consumption under different processing parameters and bounds.

3.1 Carbonization process furnace

The data that is being investigated in this paper was collected from the carbonization processes performed as a part of the carbon fiber production process at the Carbon Nexus Institute at Deakin University in Melbourne, Australia [15]. The multi-zoned carbonization furnace at Carbon Nexus Institute provides a low temperature between 1100 and 1175 °C, and a high temperature between 1400 and 1475 °C. The schematic figure of the furnace is provided in Fig. 2. These temperatures are modified by altering the manageable furnaces’ heat elements. The pure nitrogen gas, which exists in the inert atmosphere of the furnaces, is responsible for removing the gasses evolved in the furnace during the carbonization process. The Process N2 valve (PN) provides enough nitrogen gas to be employed in the furnace. The ratio of nitrogen supplement can vary between 13 and 24 l per min \( \left(\frac{l}{\min}\right) \) [15]. The adjustable speed of the drivers (DR4 and DR5) allows for the application of the required tension to the fibers. The drivers’ speed varies from 25.4 to 28.9 m h \( \left(\frac{m}{h}\right) \). During the carbonization stage, the pre-carbonized fibers pass through the LT and HT furnaces during a specific amount of time and under a pre-determined tension. These process conditions are selected based on the desired final mechanical properties, e.g., as defined by industrial application designers.

3.2 A black box problem

Due to the large amount of uncertainty that exists in carbon fiber production, it is almost impossible to explicitly determine the relation between inputs and outputs of the carbonization process. Therefore, any prediction or optimization investigation would primarily fall into the black box problem category. A black box is a system in which no information and explicit mathematical understanding of the inner function of the system is derivable [25]. The only attainable knowledge is the characteristics of the system evidenced through input and output data. One of the primary benefits of GP is that it can appropriately deal with black box problem scenarios as it does not require supervised modeling of input-output relations; it automatically establishes the relations based on the statistical nature of collected data (i.e., their distances and autocorrelations) in the presence of noise.

In order to formally define different aspects of the current optimization problem, a schematic of inputs and outputs of the carbonization process is illustrated in Fig. 4. The inputs consist of controlled process parameters and some uncertainty or noise. The output includes the mechanical properties of the resulting carbon fibers (the modulus and tensile strength) and the process’ energy consumption. The controlled system parameters are the furnace temperatures of zone 1 and zone 2 and the time the fibers spend in the furnace. Fibers used in the production line in [15] consist of a large number of tows (up to 600), and each tow contains 6000–24,000 filaments. The random behavior of filaments and tows, oven heat transfer non-uniformity, and variation in microstructure of fiber precursors are the main sources of noise in the process response. Fifty random repeats were undertaken for each process condition to capture the variability of the outputs (modulus and tensile strength). Unlike the modulus and tensile strength, the value of energy consumption of each carbonization process does not have any standard deviation. This is expected since for a specific set of process conditions, for instance when the inputs (time and furnace temperatures) are fixed, there is one particular value for energy consumed for the entire system.
Fig. 4

Schematic of inputs and outputs for the carbonization process under study

3.3 Gaussian process model development

The initial step in the proposed optimization of the carbonization process is to develop a GP prediction model for each output (Young’s modulus and tensile strength). The measured tensile properties of fibers in the carbonization process include both the mean and standard deviation (STD). The STDs under each process condition are inconsistent and have been clearly influenced by different heat treatment duration and furnace temperatures. Statistically, a state like this, in which the variability of random variables are not equal, is called heteroscedasticity [26, 27]. However, in the standard GP, the model assumes that standard deviations of actual data are alike (homoscedasticity). Hence, for heteroscedastic data, two separate GP models should be developed. The first model tackles the mean prediction. In the second GP model, the inputs are identical with the first GP model; however, the output response is now the standard deviation. Therefore, five GP models are defined in order to predict the tensile properties of resulting carbon fibers, their variabilities, and the process energy consumption.

NML and predicted residual error sum of square (PRESS) [28] (cross-validation method) were utilized to ensure the quality of GP models during training. The lower the values of the NML and PRESS, the more reliable and accurate the model. Multiple sets of covariance and mean functions were investigated in order to find the optimal combination with the smallest NML and PRESS values. Next, the two-dimensional prediction graphs of the top candidates were sketched. This is particularly beneficial for engineering judgment, since the statistical analyses (NML and PRESS) do not take the physical behavior of the process variables into account. Figure 5 indicates the two-dimensional prediction graphs of the modulus mean using SE covariance function and linear mean function (top candidate model). The black lines in the graphs are the mean value of the prediction, black stars are the observed points, and gray area is the 95% confidence interval of the predictions.
Fig. 5

Mean modulus prediction as a function of time using SE covariance function and linear mean function at three different furnace temperature conditions: a at T1= 1100 °C and T2 = 1400 °C, b at T1 = 1150 °C and T2 = 1450 °C, and c at T1 = 1175 °C and T2 = 1175 °C. It is worth mentioning that in the areas where more observed data exists, the confidence intervals become narrower, representing a higher level of prediction reliability around those points

Figure 6 represents the graph of the experimental data versus the model prediction. The closeness of the points to the 45° line demonstrates how well the prediction model performed. The overall performances of the selected GP models are assessed using statistical measures of the residual sum of square (RSS) and R-squared. The RSS determines how close the predictions and observations are [29]. A smaller value for RSS and a higher R-squared indicate a closer fit between the model and data points [30]. Table 1 summarizes the RSS and R-squared values of the GP models, clearly indicating a notable performance of each model’s predictability.
Fig. 6

Experimental data against prediction values for the mean modulus GP model. The closeness of points to the 45° line shows how well the GP model performed in predicting the process variables

Table 1

Statistical measures of the carbonization process’s Gaussian process (GP) prediction models

GP prediction model

Covariance function

Mean function

RSS (GPa)

R-squared

Mean modulus

Squared exponential

Linear

0.161

0.99

Modulus STD

Squared exponential

Linear

0.258

0.99

Mean tensile strength

Squared exponential

Linear

0.00005

1.00

Tensile strength STD

Squared exponential

Linear

0.0000038

1.00

Energy consumption

Squared exponential

Constant

0.028

0.98

4 Multi-objective optimization of carbonization process

Every process optimization problem falls into one of the two general categories, single- or multi-objective. In a single-objective optimization, the goal is to find one solution that usually minimizes or maximizes the objective in the presence of some constraints or process bounds. In a multi-objective optimization, however, there are more than one objective to be satisfied and they typically conflict one another, which results in obtaining more than one optimal (Pareto) solution. Each solution would favor one objective more or less than the other(s), while an overall (compromised) optimal solution would be desired to fit the target application. A common practical strategy in solving multi-objective problems is to create different Pareto solutions (trade-off curve) using multi-criteria decision-making techniques [31], and, accordingly, the decision-maker can select the final optimal solution based on their overall preference, additional practical constraints, and bounds, etc.

4.1 Constraints and bounds

Defining the decision variables, objectives, constraints, and bounds is a fundamental step in process optimization procedures [31]. After developing the prediction model, an optimization algorithm is needed to obtain the Pareto optimal solutions. In this study, the five previously developed GP models are applied in the optimization of the carbonization process. Next to process variable bounds (e.g., as defined below by the limitations of the machines used), identifying the “application-oriented” constraints can be highly dependent on the method selected for the multi-objective optimization. For example, if the weighted sum method [31] is employed to aggregate the mean responses of the process, the standard deviations of modulus and tensile strength may be considered as the application constraints [32, 33]. In the experiments conducted for this study, the process bounds were identified as:
  • Zone 1’s temperature interval is from 1100 to 1175 °C: 1100 < T1(°C) < 1175

  • Zone 2’s temperature interval is from 1400 to 1475 °C: 1400 < T2(°C) < 1475

  • The amount of time that the fibers spend in the furnaces is between 151 and 167 s: 151 < t(s) < 167.

More discussions and example on the “application-oriented” constraints will be given in the next section.

4.2 A multi-objective optimization scenario

High modulus and high tensile strength are the primary mechanical properties of interest for carbon fiber applications. However, several time-consuming and expensive carbon fiber manufacturing tests must be performed to find such optimal properties. In addition, as discussed in Section 1, the experiments are highly influenced by uncertainty factors such as vibrations within thousands of filaments in each tow, as operator skills, etc. An alternative is the use of black box models to predict the optimum manufacturing conditions for producing carbon fibers with high modulus and tensile strength. Herein, the weighted sum method is implemented to find the overall optimal mean values of both properties. In this method, the weights (wj) of normalized decision variables (mean elastic modulus and mean ultimate tensile strength/UTS) are chosen based on the importance of each objective per preference of the decision-maker (e.g., industry designer) [34]. The normalized values is found by dividing the modulus and tensile strength values by their respective maxima. The STD of the mechanical properties and the value of energy consumption are deemed as secondary decision variables in this scenario and were converted to constraints as follows:

Objective function: maximize w1Modulusnorm+ w2UTSnorm,

Application-oriented constraints:
  • The standard deviation of the modulus to be less than 10 GPa: STDmodulus < 10

  • The standard deviation of the tensile strength to be less than 0.9 GPa: STDUTS < 0.9

  • The consumed energy to be less than 12.45 kWh: E (kWh) < 12.45

Relative weights (preference values) w1 and w2 are specific to the designer’s needs.

Process bounds:
$$ 1100<{T}_1\left({}^{\circ}\mathrm{C}\right)<1175 $$
$$ 1400<{T}_2\left({}^{\circ}\mathrm{C}\right)<1475 $$
$$ 151<t(s)<167 $$
Table 2 summarizes the results of the above multi-objective optimization scenario. Figure 7 is the graph of the Pareto optimal solutions under different decision variable weights. A large gap is observable in the mid-region of the decision space between the two optimal points with the modulus weights of 30% and 40%. In decision spaces with non-constant curvature, the weighted sum method is incapable of predicting all the Pareto optimal solutions regardless of how the weights are chosen [35]. The black dashed line indicates an imaginary trend of the Pareto optimal points in the concave area, which can be mathematically identified using the method explained in the following section.
Table 2

Pareto optimal solutions using different weights

Modulus weight

Tensile strength weight

Temperature of zone 1 (°C)

Temperature of zone 2 (°C)

Time (s)

Modulus (GPa)

Tensile strength (GPa)

Modulus STD (GPa)

Tensile strength STD (GPa)

Energy consumption (kWh)

0.1

0.9

1124

1453

156

216.80

3.082

9.26

0.99

12.43

0.2

0.8

1126

1455

157

217.78

3.081

9.99

0.96

12.45

0.3

0.7

1124

1454

157

218.15

3.079

9.96

0.90

12.44

0.4

0.6

1120

1457

158

221.42

3.056

9.97

0.39

12.44

0.5

0.5

1118

1457

158

222.09

3.049

9.79

0.25

12.43

0.6

0.4

1114

1454

158

222.54

3.041

9.89

0.20

12.41

0.7

0.3

1114

1454

158

222.54

3.041

9.89

0.20

12.41

0.8

0.2

1113

1453

158

222.58

3.04

9.95

0.21

12.41

0.9

0.1

1113

1453

158

222.58

3.04

9.95

0.21

12.41

Fig. 7

The convex region with non-constant curvature in the standard-weighted sum method

4.3 Adaptive weighted sum method

As it is shown in Fig. 7, the initial result of the standard-weighted sum method gave a rough estimation of the optimal decision space solutions, containing a gap between two separate sets of the Pareto points. As referred above, this gap is in essence due to the existence or a non-convex Pareto in the current process where some optimal points have not been detected. The normal boundary intersection (NBI) and adaptive weighted sum (AWS) are the two methods frequently used to resolve such Pareto problems [36]. In the NBI method, the utopia points are found by means of single-objective optimizations. Using a utopia line between anchor points, the hidden Pareto optimal solutions are found. The AWS method, however, addresses the problem by applying new inequality constraints and providing a new sub-region [35, 37]. In the current bi-objective optimization problem, the two required constraints are applied to mean modulus and tensile strength. In the next step, a sub-optimization is carried out in order to explore the sub-convex area and discover the undetected points. More sub-regions may need to be defined using new constraints in order to find the remaining hidden optimal points. Figure 8 reveals all the detected Pareto optimal solutions and their trend, from which the decision-maker can select the optimum process conditions for the carbonization process. The numbers adjacent to each data point indicate zone-1 and zone-2 temperatures and the heat treatment time, respectively.
Fig. 8

Mean modulus against tensile strength plot of Pareto optimal solutions achieved by utilizing AWS method. Process variables (T1, T2, and t) are shown at each point, along with ΔT (T2 − T1)

5 Discussions

5.1 The effect of stabilization process temperature on the carbon fiber mechanical properties

It was discussed that the furnace temperature during the carbonization process can closely affect the mechanical properties of the resulting carbon fibers. Early research on the carbon fibers has revealed that the fibers’ modulus has a strong correlation with the process temperature [2]. The experimental data of the current carbonization process, however, states an opposite behavior, such that the carbon fibers’ mean modulus has decreased as the furnace temperatures increase. In other words, the highest mean modulus occurs when the zone-1 temperature is 1100 °C (lowest value). This contrary behavior is believed to have originated from the gap between the furnace temperatures of the stabilization and carbonization process. The process of heating the PAN precursor is highly exothermic. Unless the increase in temperature of fibers is slow and gradual, the thermal runaway reaction will take place and heating process will become uncontrollable, resulting in fibers getting burnt and ending up with poor mechanical properties [2].

Schematically, Fig. 9 depicts how the difference between these temperatures can negatively affect the final mechanical properties of the carbon fibers. It shows the carbon fibers’ temperature versus the time the fibers spend in the furnace. T0 is the stabilization process temperature, T1 and T2 are the temperatures of zone 1 and 2 in the carbonization process, respectively. Figure 9a indicates the carbonization process with the lowest possible furnace temperatures (i.e., T1 = 1100 °C and T2 = 1400 °C). Similarly, Fig. 9b depicts the same process procedure with the highest possible temperatures (T1 = 1175 °C and T2 = 1475 °C). It is assumed that the stabilization process temperature remains constant. The larger difference between T0 and T1 in the process shown in Fig. 9b can cause a more sudden change in the carbon fibers’ behavior. This is observable in the temperature profile slopes, which are drawn at the beginning of each process. The sharp heat rate in Fig. 9b may cause an exothermic thermal explosion and a sudden shock to the structure of the carbon fibers, resulting in unsatisfying product properties.
Fig. 9

Temperature—time graph of carbon fibers passing through low- and high-temperature furnaces in carbonization process. T0 is the stabilization process temperature. The dashed dotted lines indicate the slope of the curves at T0. Compared with (a), the graph (b) has a bigger temperature gap between its stabilization and carbonization process. This causes thermal shock which result in poor mechanical properties

5.2 The benefit of using a two-zone furnace in the carbonization process

Another key factor of the carbonization process that affects the tensile properties of the carbon fibers is the process heating rate. Theoretically, it is essential to maintain a slow heating rate, which is not cost- and energy-efficient. One solution is to use a furnace consisting two separately controlled zones. This gives the fibers the chance to experience an approximately constant heating rate during the process. To elucidate, the green line in Fig. 10 indicates how the employment of two zones can bring the process closer to a constant heating rate. The blue line indicates a process in which a one zone furnace is used.
Fig. 10

Comparison of two carbonization processes with different furnace designs: the blue line indicates the process with one zone furnace; the green line shows a two-zone furnace carbonization process

5.3 Recommended design chart for multi-objective optimization of carbonization process

A summary of the results of the performed multi-objective optimization scenario is shown in Fig. 11. This particular scenario was selected for demonstration as it is one of the most practical cases for carbon fiber customers with the goal of optimizing the design of their composite structures under reasonable variability tolerances. The hidden mid-region of the Pareto is not shown in Fig. 11 due to the existence of sharp changes in the design space (observable in Fig. 8), which is not a preferable solution for manufacturers in practice. Due to the non-linear trade-off between carbon fiber mean modulus and tensile strength, shown in Fig. 11, the optimized solutions highly depend on the designer’s preference on the criteria weighting factors. The marginal rate of substitution between the two fiber mechanical properties is non-linear (i.e., the designer must consider a weight much higher than 50% for the tensile strength in order to increase it notably). While maximizing the modulus is achievable by relatively lower energies, the energy consumption has a direct positive relation with the maximization of tensile strength. In addition, the modulus and tensile strength STD bars indicate a very disproportionate decision space for designers (e.g., a higher mean value of modulus does not necessarily mean a higher STD from the carbonization process). Figure 11 is deemed to be a useful general trade-off graph for the carbonization process that may be used by process engineers and structural designers for fast review and recommendations.
Fig. 11

Recommended design chart (n-dimensional Pareto) for multi-objective optimization of mechanical properties of fibers, while the carbonization process energy is constrained, and the robustness of fibers performance is taken into account

6 Conclusion

This paper presented the application of GP in the prediction and optimization of carbon fiber production process. It was demonstrated that GP is an effective and robust tool for capturing the behavior of complex manufacturing processes under noisy responses.

The optimum covariance functions and hyper-parameters of the GP model have been specified using NML and PRESS. The overall performance of the selected models was evaluated using statistical techniques such as RSS and R-squared, to measure the closeness of the predicted values to the experimental data. The assessment results of the models clearly confirm that GPs are able to successfully capture the carbonization process response variables with goodness of fit values over 99% under highly fluctuating data, such as modulus and tensile strength STDs.

In the optimization stage of the research, the weighted sum method was adapted and applied under a process design scenario based on different assumed needs of carbon fiber customers. The standard-weighted sum method was employed for the Pareto optimization of the carbon fibers’ modulus and the tensile strength. However, it was concluded that this method does not capture all of the Pareto optimal solutions, regardless of how the weights are assigned. Hence, a new approach, called the AWS method, was adapted; the results of which indicated that the model is capable of capturing all of the hidden Pareto optimal solutions in the decision space of carbonization process.

Notes

Acknowledgments

The support of colleagues and the stimulating discussions at the Composites Research Network and Carbon Nexus Institute are greatly valued.

Funding information

This study was financially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Milad Ramezankhani
    • 1
  • Bryn Crawford
    • 1
  • Hamid Khayyam
    • 2
  • Minoo Naebe
    • 3
  • Rudolf Seethaler
    • 1
  • Abbas S. Milani
    • 1
    Email author
  1. 1.Composites Research Network, School of EngineeringUniversity of British ColumbiaKelownaCanada
  2. 2.School of EngineeringRMIT UniversityMelbourneAustralia
  3. 3.Institute for Frontier Materials, Carbon NexusDeakin UniversityWaurn PondsAustralia

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