Abstract
The effect of various process parameters on the stability of TiO2 nanofluid, which can mostly be defined as zeta potential and particle size, was studied using response surface methodology (RSM) by the design of experiments and was predicted through a trained artificial neural network (ANN). The process parameters studied were weight percentage of surfactant (sodium lauryl sulfate) (0.01–0.2 wt%) and the value of pH (10–12). Central composite design and the RSM were employed to develop a mathematical model as well as to define the optimum condition. A three-layered feed-forward ANN model was designed and used for the prediction of the stability parameters. From the analysis of variance, the significant factors that affected the experimental design responses were also identified. The predicted stability parameters using the RSM and ANNs were compared using figures and tables. It is shown that the trained ANN outperformed the RSM in terms of accuracy and prediction of obtained results.
1 Introduction
Nanofluid as a new class of cooling medium was brought to attention 16 years ago by Choi [1]. Homogeneous dispersion and stability of nanoparticles in a base fluid including water, lubricant, refrigerant, and so forth are matters of concern among scientists and industries [2–5].
Therefore, numerous techniques including zeta potential measurement, agglomerate particle size analyzer, photo capturing, transmission electron microscopy, scanning electron microscopy, and ultraviolet-visible (UV-VIS) spectrophotometry are classified to verify dispersion of a nanosuspension [6].
Among these methods, zeta potential and particle size analysis are the main important measurements numerically and visually, respectively. Design of experiments (DOE) is a tool to optimize the planning of experimental research.
The role of DOE is to estimate the effect of several variables separately, simultaneously, or as combinations [7]. Box and Draper [8] claimed that response surface methodology (RSM) is a statistical method for modeling and analyzing the relationships between several input and response variables.
RSM is an empirical modeling approach that uses polynomials as local approximations to the true input or output relationship. The objective is to optimize the response (output variables) that is influenced by several independent variables (input variables). The advantage of RSM is to minimize the need for repetition of experiments for experiments with multiple factors [9]. Recently, Low et al. [10] applied RSM on the optimization of the mechanical properties of composite materials.
An artificial neural network (ANN) is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. The ANN consists of dense interconnected computing units (artificial neurons) that represent simple mathematical models for complex neurons in biological systems.
The knowledge is acquired during a learning/training process and is stored in the synaptic weights of the internodal connections. The main advantage of ANNs is their ability to represent complex input/output relationships. They are well suited for use in data classification, function approximation, and signal processing, among others. Hassoun [11] introduced most concepts of ANNs, whereas Hertz et al. [12] described the mathematics of ANNs very thoroughly.
Recently, Sha and Edwards [13] investigated the applications of ANNs in material science comprehensively. Hassan et al. [14] predicted the density, porosity, and hardness of an aluminium-copper based composite material using ANNs. Xiao and Zhu [15] studied friction material development using the ANNs and RSM.
Singh et al. [16] applied ANNs to predict the effective thermal conductivity of moist porous materials. A review paper [17] showed that the implementation of ANN technique has greatly promoted research in materials science and technology.
On the basis of the literature review [6], there are numerous inspection measurements that monitor the stability of nanofluids including zeta potential, particle size, UV-VIS spectrophotometry, photo capturing, sedimentation balance, and 3-omega methods. However, the most popular and reliable measurements among researchers are zeta potential and particle size. Therefore, these two parameters are considered as criteria in stability measurements. Using the zeta potential response, the isoelectric point of the suspension can be found to control the pH value far away from that to be stable. Using the particle size measurement, the radius of gyration can be found for fast sedimentation in the suspension or otherwise.
The purpose of this study is to verify the effect of zeta potential and particle size on the stability of nanofluids along with optimization of the aforementioned parameters using RSM and ANN to predict the stability parameters of nanofluids. Statistical and optimization processes using RSM accompanied by ANNs were performed and the results were compared.
The remainder of this paper is organized as follows: In section 2, the TiO2 nanofluid and applied methods including ANN and DOE are described. In section 3, the prediction results obtained from RSM and ANN are compared and discussions are presented in detail. Optimization of the stability parameters to obtain stable TiO2 nanofluid is discussed in section 4. Finally, the conclusions are given in section 5.
2 Materials and applied methods
2.1 Nanofluid preparation and property measurement
The experiments were conducted using 0.1 wt% titania (TiO2) nanoparticles with average diameter of 25 nm and Brunauer Emmett Teller (particle surface area measurements) of 200–220 m2/g dispersed in distilled water. Titanium(IV) nanopowder with 99.7% metal basis from Sigma-Aldrich Company (Saint Louis, MO, USA) was used in this study. An anionic surfactant, sodium lauryl sulfate (referred to as SLS), in chemical grade (Sigma-Aldrich) was used to stabilize the suspension.
In this research, pH was controlled by adding NaOH and HCl. A two-step method was used to prepare the nanosuspension, which is the most practical method on the basis of the literature survey [18–21]. A 3-h ultrasonication bath was applied to disperse nanoparticles homogeneously. Fresh nanofluid was brought to measurement of zeta potential and particle size using a Malvern 3000 HSA particle size analyzer with ±0.1 nm accuracy.
2.2 Design of experiments
The effects of process parameters on the stability of TiO2 nanofluid were studied using the RSM. In this paper, central composite design and RSM were applied to develop the two operating variables. The process parameters selected for this study were surfactant (SDS) concentration and the value of pH.
The Design-Expert software (version 8.0) was used for the statistical DOE and data analysis. Accordingly, the response parameters that were categorized for stability parameters were zeta potential (in millivolts) and particle size (in nanometers). Three repeated runs used as the center of the experiment were carried out to measure the reproducibility at different combinations of process parameters.
The experimental results are shown in Table 1. Due to the difficulty and complexity of the experimental tests, only 9 tests were performed. Additionally, 15 more tests were conducted ordinarily to check the input factors (i.e., pH value and surfactant weight percentage) and responses which were zeta potential and particle size.
Experimental results obtained from 24 independent runs for TiO2 nanofluid.
| Run | Factor 1: surfactant (0.01–0.2%) | Factor 2: pH (10–12) | Response 1: zeta potential (mV) | Response 2: particle size (nm) |
|---|---|---|---|---|
| 1 | 0.01 | 10 | -40 | 295 |
| 2 | 0.01 | 10.5 | -42.1 | 272.8 |
| 3 | 0.01 | 11 | -46.5 | 268.7 |
| 4 | 0.01 | 11.5 | -48 | 431.8 |
| 5 | 0.01 | 12 | -49 | 350 |
| 6 | 0.02 | 10.5 | -42.9 | 243.5 |
| 7 | 0.02 | 11 | -44.5 | 294 |
| 8 | 0.02 | 11.5 | -48 | 302 |
| 9 | 0.02 | 12 | -40.5 | 260.3 |
| 10 | 0.06 | 11 | -48 | 397 |
| 11 | 0.1 | 10 | -48.6 | 424.7 |
| 12 | 0.1 | 10.5 | -47.9 | 427.7 |
| 13 | 0.1 | 11 | -50 | 372.8 |
| 14 | 0.1 | 11.5 | -46.3 | 422.6 |
| 15 | 0.1 | 12 | -45.8 | 383.8 |
| 16 | 0.11 | 10.5 | -48 | 370 |
| 17 | 0.11 | 11 | -44.23 | 481.73 |
| 18 | 0.11 | 11.5 | -42.5 | 450 |
| 19 | 0.15 | 11 | -42.4 | 409.1 |
| 20 | 0.2 | 10 | -51.6 | 232.3 |
| 21 | 0.2 | 10.5 | -37.7 | 380.4 |
| 22 | 0.2 | 11 | -50.2 | 338.6 |
| 23 | 0.2 | 11.5 | -38.7 | 420.6 |
| 24 | 0.2 | 12 | -37.2 | 327 |
2.3 ANN implementation
2.3.1 Introduction
ANNs are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the connections between elements largely determine the network function. ANNs consist of a number of artificial neurons (processing units), which are arranged in a series of layers (input, hidden, and output layers).
The architecture of such ANNs is referred to as a multilayer perceptron (MLP). The MLP networks consist of an input layer representing the input parameters, an output layer representing the output parameters, and one or more hidden layers. Numerous algorithms have been proposed for training ANNs. The most popular and effective learning algorithm is the back-propagation (BP) algorithm [22].
The BP algorithm is the generalization of the Widrow-Hoff learning rule and is used for training MLP networks. The BP is based on nonlinear differentiable transfer functions (normally a sigmoid function).
Input vectors and the corresponding target (output) vectors are used to train a network until it can approximate a function to an arbitrary degree of accuracy. Figure 1 depicts a typical architecture of a three-layered neural network. The fundamental theory and applications of BP-based ANNs are available in the literature [23–26].
Schematic view of a typical MLP network with three layers based on the BP algorithm.
2.3.2 Input and output preparation
Input of the ANNs should be easily and accurately measured and also should be sensitive to the change of the process parameters of the nanofluid. The weight percent of surfactant and the pH of the nanofluid are selected as the inputs of the ANNs in this paper.
Accordingly, the output of the ANNs represents the stability parameters, which include the zeta potential (in mV) and particle size (in nm). It is assumed that a nonlinear mapping exists between the stability and input parameters.
Because of the difficulty of the experiment tests only 24 tests were performed, as shown in Table 1. In order to train the ANNs, 60% of the samples are randomly assigned to the training set and 15% to the validation set.
Validation vectors are used to stop training early if the network performance on the validation vectors fails to improve. Training continues as long as the training error is reduced on the validation samples. After the network generalizes the training set, the training is stopped.
This technique automatically prevents the training from facing the problem of overfitting, which plagues many optimization and learning algorithms. Finally, the last 25% of the samples (unseen samples) provide an independent test of the trained ANN generalization.
Test vectors are used as a further check that the ANN is generalizing well, but do not have any effect on training [27]. It is recommended that the input and output data are normalized prior to training. This is done to avoid possible training saturation [28, 29]. Furthermore, training is often faster when values are normalized.
2.3.3 ANN architecture
The architecture or topological structure of an ANN can be characterized by the arrangement of the layers and neurons, the nodal connectivity, and the nodal transfer functions. Any mapping from an input to output can be simulated by a multilayered feed-forward neural network with BP algorithm.
Hecht Nielsen [30] proved that a three-layered feed-forward neural network with BP algorithm can be trained to approximate any mapping from n dimensions to m dimensions to an arbitrary degree of accuracy. The ANN structure used in this paper was fully connected and had two hidden layers and one output layer, and the number of neurons in the input, output, first hidden layer, and second hidden layer were 2, 1, 20, and 10, respectively. The increase in the number of neurons increases the computation. However, it may allow the ANN to solve more complicated problems.
Similarly, the increase in the number of ANN layers increases the computation time, although this may result in the network solving complex problems more efficiently [27]. Regarding the effects of the number of hidden layers and the learning factor on the expectation accuracy of ANNs, the reader may refer to [31].
In general, a neural network with a single hidden layer is capable of approximating any nonlinear function to an arbitrary degree of accuracy. However, increasing the number of layers can offer improved ANN training/learning for highly complex functions [32]. The ANNs are also sensitive to the number of neurons in their hidden layers.
Having few neurons may lead to underfitting. In contrast, too many neurons can contribute to overfitting, in which all training points are well fitted. However, the fitting curve oscillates wildly among these points. The transfer functions in the feed-forward BP network are supposed as tan-sigmoid, log-sigmoid, and linear transfer function for the first hidden layer, second hidden layer, and output layer, respectively.
This structure is useful for function approximation (or regression) problems [27]. Multiple layers of neurons with nonlinear transfer functions allow the network to learn nonlinear and linear relationships between input and output vectors.
2.3.4 ANN training and simulation
Properly trained BP-based ANNs tend to give reasonable response when presented with unseen inputs after being trained. The generalization property of ANNs makes it possible to train a network on a representative set of input/target pairs and obtain good results without training the network on all possible input/output pairs.
A learning rule is defined as a procedure for modifying the weights and biases of a network (this procedure can also be referred to as a training algorithm). The learning rule is applied to train the network to perform some particular task. Learning rules fall into two broad categories: supervised learning and unsupervised learning [27].
The ANN simulation work was carried out using MATLAB programming software and run on a Pentium V 2.53 GHz with 4 GB RAM. A feed-forward neural network model with a BP algorithm was employed for the network training and simulation.
The Levenberg-Marquardt training function is often the fastest BP algorithm among other training functions and is highly recommended as a first-choice supervised algorithm, although it does require more memory than other algorithms [33].
The Levenberg-Marquardt algorithm was used to control the learning process in this paper until a reasonable mean square error (MSE) value is achieved. In the learning process, the weights between the connections were adjusted using the enhanced BP algorithm The training samples were extracted from Table 1.
3 Results and discussions
3.1 Statistical results of design of experiment
The experimental results obtained from the formulation of TiO2 nanofluid were analyzed by multiple regression analysis. Therefore, a quadratic model was chosen for this purpose. The analysis of variance (ANOVA) was used to estimate the effects of main variables and their potential interaction on the stability of nanofluid. The RSM was applied for TiO2 nanofluid on a Pentium V 2.53 GHz with 4 GB RAM. The statistical results of zeta potential response for the TiO2 nanofluid using ANOVA are shown in Table 2.
ANOVA results for the zeta potential response for regression model equation and coefficients of model terms.
| Source | SS | DF | Mean square | F value | Prob.>F |
|---|---|---|---|---|---|
| Model | 0.13 | 5 | 0.025 | 60.20 | <0.0002 |
| A | 0.016 | 1 | 0.016 | 39.30 | <0.0015 |
| B | 0.000 | 1 | 0.000 | 0.000 | <1.0000 |
| AB | 0.068 | 1 | 0.068 | 162.82 | <0.0001 |
| A2 | 0.024 | 1 | 0.024 | 56.85 | <0.0007 |
| A2B | 1.962×10-3 | 1 | 1.962×10-3 | 4.67 | <0.0830 |
In Table 2, the sum of squares (SS) is the sum of the squared deviations from the mean due to the effect of this term, DF is the degree of freedom (DF), and mean square is the SS divided by the DF [33]. The important outputs of the model are the F value and associated probability (Prob.>F).
The model’s F value of 60.20 and Prob.>F value of <0.05 (there is only a 0.02% chance that a “model F value” occurred due to noise) imply that the model is significant at the more than 95% confidence interval (CI) for zeta potential response. Values >0.1 indicate that the model terms are not significant.
After eliminating the insignificant parameters, regression analysis gives the following quadratic model equations based on the coded and actual factors using the RSM as shown in Eqs. (1) and (2), respectively. ζ is a notation for zeta potential response.
where A and S are coded and actual factors for weight percent of the surfactant. Accordingly, B and P are coded and actual factors for the value of pH. Consequently, the obtained quadratic model may be considered to represent the actual process of formulation for TiO2 nanofluid. The ANOVA results show that among the parameters, terms A, AB, and A2 have significant effect on the stability of the nanofluid, as shown in Table 2.
In other words, any changes in the value of this parameter will bring significant change to the stability of TiO2 nanofluid. It means that zeta potential response is more dependent on the weight percent of the surfactant compared with the pH value. The statistical results of particle size response for the TiO2 nanofluid using ANOVA are presented in Table 3.
ANOVA results for the particle size response for regression model equation and coefficients of model terms.
| Source | SS | DF | Mean square | F value | Prob.>F |
|---|---|---|---|---|---|
| Model | 76.16 | 3 | 25.39 | 15.46 | <0.0018 |
| A | 2.91 | 1 | 2.91 | 1.77 | <0.2249 |
| B | 12.76 | 1 | 12.76 | 7.77 | <0.0270 |
| B2 | 60.50 | 1 | 60.50 | 36.85 | <0.0005 |
The model’s F value of 15.46 and Prob.>F value of <0.05 (there is only a 0.18% chance that a model F value occurred due to noise) imply that the model is significant at the more than 95% CI for particle size response.
After omitting the unimportant parameters, regression analysis gives the following quadratic model equations based on the coded and actual factors using the RSM as shown in Eqs. (3) and (4), respectively. η is a notation for particle size response.
Therefore, the quadratic model may be considered to stand for the actual process of formulation for TiO2 nanofluid. The ANOVA results show that among those parameters, terms B and B2 have considerable effect on the stability of the TiO2 nanofluid in terms of particle size, as given in Table 3. It means that particle size response is more dependent on the value of pH compared to the weight percent of the surfactant.
3.2 Predicted results of the ANN
In this paper, a progressive ANN procedure was applied to characterize the material properties of TiO2 nanofluid for stability parameters. For comparison with the RSM, two cases were considered. The first case concerned the zeta potential response, and the second case was applied for the particle size response.
The obtained output results from the trained ANNs were validated against the results given by the RSM. The 24 experimental data vectors were randomly divided (14 vectors for training, 4 vectors for validation, and 6 vectors for testing).
3.2.1 Case 1: zeta potential response
For case 1, the zeta potential is considered as a response (output) of the ANN. Figure 2 shows the correlation coefficient for training and validation samples. It is a measure of how well the variation in the output is explained by the targets. If correlation coefficient is equal to 1, then, there is perfect correlation between targets and outputs of the training/testing data. From Figure 2A and B, the correlation factor for training and validation samples are 0.96 and 0.98, respectively, which indicates a good fit.
Correlation coefficient plot for the TiO2 nanofluid for the zeta potential response: (A) training samples, (B) validation samples.
Figure 3 illustrates the linear regression for the six test sample data for case 1. As can be seen, the correlation coefficient is 0.93, which demonstrates a good fit between the input and output for unseen data.
Correlation coefficient plot for test samples of TiO2 nanofluid for the zeta potential response (the vertical and horizontal axes are predicted output and corresponding targets, respectively).
As mentioned in subsection 2.3.4, the typical performance function used for training a feed-forward ANN is the MSE for the network errors. This performance function causes the network to have smaller weights and biases, and the MSE forces the network response to be smoother and less likely to overfit [33].
The MSE plots for data corresponding to training, validation, and test samples are shown in Figure 4. The MSE of the network was started at a large value and decreased to a smaller value, as shown in Figure 4. In other words, it shows that the network is learning. The MSE was reduced to 10-12 during the training session.
MSE plot with respect to the number of epochs for training, validation, and test samples for the TiO2 nanofluid for the zeta potential response (case 1).
The training stopped when the validation error increased for eight iterations, which occurred at epoch 3. In this case, the obtained results are reasonable because of the following considerations: The final MSE is too small, the validation and test set error have similar characteristics, and no significant overfitting has occurred by epoch 3 (where the best validation performance occurs).
3.2.2 Case 2: the particle size response
For case 2, the particle size is considered as a response (output) of the ANN. The input vectors were the weight percent of the surfactant and the value of pH. Figure 5A and B shows a linear regression between the network outputs and the corresponding targets for training and validation samples, respectively. From Figure 5, the correlation factor for training and validation samples are 0.93 and 0.91, respectively, which shows an acceptable fit.
Correlation coefficient plot for TiO2 nanofluid for the particle size response: (A) training samples, (B) validation samples.
Similar to case 1, in this case the performance of trained ANN can be validated using unseen test data. The correlation factors between experimental data and test samples are presented in Figure 6. From the figure, the linear regression is 0.93 for test samples, which is almost good mapping.
Correlation coefficient plot for test samples of TiO2 nanofluid for the particle size response (the vertical and horizontal axes are predicted output and corresponding targets, respectively).
The MSE plots for all samples including training, validation, and test samples errors are shown in Figure 7. By observing Figure 7, the MSE between the outputs and inputs of the training set was tried to come to a small value. The MSE was reduced to almost 10-15 during the training session. The training stopped when the validation error increased for six iterations, which occurred at epoch 2.
MSE plot with respect to the number of epochs for training, validation and test samples for TiO2 nanofluid for case 2.
3.3 Comparison of RSM and ANNs
The obtained results from the ANNs and RSM were compared in terms of error percentage, linear regression between the network outputs and the corresponding experimental results, and the MSE. Table 4 shows the comparison of experimental and predicted results for zeta potential (case 1) obtained using RSM and ANNs. Six unseen test samples and corresponding experimental data were compared as shown in Table 4.
Comparison of experimental and predicted results for the zeta potential response obtained from the RSM and ANNs for case 1.
| Run | Surfactant (wt%) | pH | RSM (error %) | ANNs (error %) | Experimental (mV) |
|---|---|---|---|---|---|
| 1 | 0.01 | 11 | -42.43 (-8.75) | -44.34 (-4.64) | -46.5 |
| 2 | 0.01 | 11.5 | -44.77 (-6.72) | -48.30 (0.06) | -48 |
| 3 | 0.1 | 11 | -46.12 (-7.76) | -47.96 (-4.08) | -50 |
| 4 | 0.11 | 10.5 | -45.78 (-4.62) | -47.37 (-1.31) | -48 |
| 5 | 0.11 | 11.5 | -45.02 (5.92) | -45.03 (5.95) | -42.5 |
| 6 | 0.2 | 12 | -37.20 (0) | -39.03 (4.91) | -37.2 |
Values in bold represent lower error percentages.
The results obtained using the ANNs outperformed the results obtained by the RSM in four out of six runs, which has lower error percentage with respect to the experimental results, as shown in Table 4. The lower error percentage has been highlighted in bold in Table 4. The MSE for ANNs and RSM are 3.17 and 8.88, respectively. It shows that the ANN offers less MSE compared to the RSM.
The performance of an ANN can be measured to some extent by the errors on the training, validation, and test data. However, it is often useful to investigate the network responses in more detail. One option is to perform a regression analysis between the network responses and the corresponding targets.
Figure 8 represents the linear regression between the predicted and actual experiment results for the zeta potential response as open circles for case 1 using the RSM and ANNs. The best linear fit is indicated by a dashed line.
A parity plot among actual experiments and predicted values for the zeta potential response for the TiO2 nanofluid obtained using the (A) RSM and (B) ANNs (the vertical and horizontal axes are predicted output and corresponding targets, respectively).
The perfect fit (output equal to targets) is indicated by the solid line. m and b are the slope and the y intercept of the best linear regression relating targets to network outputs, respectively. For a perfect fit (outputs exactly equal to targets), the slope would be 1 and the y intercept would be 0.
The correlation coefficient (R value) between the outputs and targets for case 1 are 0.84 and 0.93 for RSM and ANN, respectively, as depicted in Figure 8. The values of m and b for RSM are 0.60 and -16.16, respectively.
Accordingly, the values of m and b for ANN are 0.68 and -14.21, respectively. As can be seen from the compared results in Table 4 and Figure 8, the trained ANNs show superiority to the RSM, having less error percentage, MSE, and high coefficient factor (R value) for case 1.
Table 5 shows the comparison of experimental and predicted results for the particle size response (case 2) obtained from the RSM and ANNs for TiO2 nanofluid. Six unseen sample tests and experimental data were evaluated in the figures and tables. The results obtained using the ANNs and RSM competed with each other in terms of error percentage, as shown in Table 5.
Comparison of experimental and predicted results for the particle size response obtained from the RSM and ANNs for case 2.
| Run | Surfactant (wt%) | pH | RSM (error %) | ANN (error %) | Experimental (nm) |
|---|---|---|---|---|---|
| 1 | 0.02 | 10.5 | 402.35 (65.23) | 264.96 (8.81) | 243.5 |
| 2 | 0.02 | 11 | 474.51 (61.39) | 313.99 (6.79) | 294 |
| 3 | 0.02 | 11.5 | 458.68 (51.88) | 412.82 (36.69) | 302 |
| 4 | 0.2 | 10 | 243.33 (4.74) | 246.67 (6.18) | 232.3 |
| 5 | 0.2 | 11 | 421.95 (24.61) | 462.54 (36.60) | 338.6 |
| 6 | 0.2 | 12 | 319.74 (2.22) | 397.80 (21.65) | 327 |
Values in bold represent lower error percentages.
The lower error percentage has been highlighted in bold in Table 5. In terms of the MSE, the ANN has a lower value of 5620.24 compared with the RSM with the value of 14,914.56. Figure 9 shows the linear regression between the predicted and actual experiment results of particle size response as open circles for case 2.
A parity plot among actual experiments and predicted values of the particle size response for the TiO2 nanofluid obtained using the (A) RSM and (B) ANNs (the vertical and horizontal axes are predicted output and corresponding targets, respectively).
The correlation coefficient (R value) between the outputs and targets for case 2 are 0.41 and 0.93 for the RSM and ANNs, respectively, as shown in Figure 9. The RSM in this case did not fit the output data to the corresponding targets as well as the ANN. The values of m and b for the RSM were 0.85 and 139.97, respectively.
Accordingly, the values of m and b for the ANNs were 1.89 and -198.49, respectively. From the compared results in Table 5 and Figure 9, the ANN shows superiority to the RSM, having less MSE and high coefficient factor (R value) for the second case.
In addition, the value of linear regression (R) for the zeta potential response is higher than the value for the particle size response. This means that the RSM and ANNs have fitted the results for the zeta potential response better than the results for the particle size response.
4 Optimization of stability parameters
The goal of this section is to bring about the optimum TiO2 nanofluid preparation in terms of the pH value and the SDS weight concentration. This task is to achieve the maximum and minimum values for the zeta potential and particle size responses, respectively.
The desirability is a multiple response method with numerical optimization. This response finds a point that maximizes the desirability function. The goal of optimization is to find a good set of conditions that will meet all the goals, not to get to a desirability value of 1.0 [34]. The optimum desirability results are tabulated in Table 6. In the table, seven optimized prediction points based on the mentioned limits (maximum and minimum) are presented.
Optimum values and desirability for the seven predicted points.
| No. | Surfactant (wt%) | pH | Zeta potential | Particle size | Desirability |
|---|---|---|---|---|---|
| 1 | 0.20 | 10 | -52.1207 | 243.332 | 0.980 |
| 2 | 0.19 | 10 | -51.9478 | 245.677 | 0.976 |
| 3 | 0.14 | 10 | -48.7702 | 253.702 | 0.862 |
| 4 | 0.12 | 10 | -47.3168 | 256.465 | 0.801 |
| 5 | 0.05 | 12 | -60.9281 | 354.135 | 0.754 |
| 6 | 0.06 | 10 | -42.5219 | 267.438 | 0.569 |
| 7 | 0.20 | 12 | -37.2101 | 319.749 | 0.022 |
Values in bold represent best optimum values.
On the basis of the literature review by Vandsburger [35], an absolute value of 60 and above for the zeta potential response would bring an excellent stability with little settling likely. Therefore, this limiting condition allowed us to choose the fifth point (highlighted in bold in Table 6) with the zeta potential value of -60.9281 as the best optimum value with desirability of 0.754.
Hence, this point leads us to the particle size value of 354.135 nm, which is not the minimum value among the values in the obtained optimum range. In order to further clarify, Figure 10 depicts the zeta potential response vs. the particle size response for seven optimized points.
Zeta potential response in terms of the particle size response in optimum condition.
The corresponding quantities for the pH value and surfactant weight concentration are 12 and 0.05, accordingly, which are in line with the preparation of a stable titania nanofluid. It is worth mentioning that the rest of the predicted points also have good stability. Figure 11 demonstrates the desirability plot for the obtained model in 2D and 3D views.
Desirability plot for the obtained mathematical model in (A) 2D view and (B) 3D view.
5 Conclusions
In this paper, application of RSM to determine the major factors and interactions that affect the stability characteristics of the nanofluid and ANNs in terms of parameter recognition were presented. The stability parameters of TiO2 nanofluid were zeta potential and particle size. The process parameters were the weight percent of the surfactant and the value of pH. The current research was conducted to obtain a better understanding of the factors that have the greatest effect on the stability of TiO2 nanofluid, thereby proposing a mathematical formulation using the RSM. The RSM showed that the weight percent of the surfactant had a significant influence on the zeta potential response for TiO2 nanofluid. In contrast, on the basis of the RSM results, the value of pH had a considerable effect on the particle size response. The optimum values of the stability parameters were 12 and 0.05 for the pH value and the surfactant weight concentration, respectively. Three-layered feed-forward ANNs (two hidden layers and an output layer) were designed and used for the prediction of the stability parameters. The results obtained from the RSM and ANNs were compared using figures and tables. The value of the linear regression (R) for the zeta potential response is higher than the value for the particle size response. This means that the RSM and ANNs have fitted the results for the zeta potential response better than the results for the particle size response. In general, the trained ANNs surpassed the RSM in terms of coefficient factor (R value), error percentage corresponding to the actual experiment results, and MSE.
The authors would like to acknowledge the Ministry of Higher Education of Malaysia and the University of Malaya, Kuala Lumpur, Malaysia, for the financial support under grant numbers UM.C/HIR/MOHE/ENG/21(D000021-16001) and UM.TNC2/IPPP/UPGP/628/6/ ER013 /2011A.
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